Upload
trinhtu
View
224
Download
0
Embed Size (px)
Citation preview
Research ArticleMassive Exploration of Perturbed Conditions of the BloodCoagulation Cascade through GPU Parallelization
Paolo Cazzaniga123 Marco S Nobile24 Daniela Besozzi235
Matteo Bellini4 and Giancarlo Mauri24
1 Dipartimento di Scienze Umane e Sociali Universita degli Studi di Bergamo 24129 Bergamo Italy2 SYSBIO Centre for Systems Biology 20126 Milano Italy3 Istituto di Analisi dei Sistemi ed Informatica ldquoAntonio Rubertirdquo Consiglio Nazionale delle Ricerche 00185 Roma Italy4Universita degli Studi di Milano-Bicocca Dipartimento di Informatica Sistemistica e Comunicazione 20126 Milano Italy5 Universita degli Studi di Milano Dipartimento di Informatica 20135 Milano Italy
Correspondence should be addressed to Paolo Cazzaniga paolocazzanigaunibgit
Received 21 February 2014 Revised 7 May 2014 Accepted 18 May 2014 Published 16 June 2014
Academic Editor Horacio Perez-Sanchez
Copyright copy 2014 Paolo Cazzaniga et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited
The introduction of general-purpose Graphics Processing Units (GPUs) is boosting scientific applications in BioinformaticsSystems Biology and Computational Biology In these fields the use of high-performance computing solutions is motivated bythe need of performing large numbers of in silico analysis to study the behavior of biological systems in different conditionswhich necessitate a computing power that usually overtakes the capability of standard desktop computers In this work we presentcoagSODA a CUDA-powered computational tool that was purposely developed for the analysis of a large mechanistic modelof the blood coagulation cascade (BCC) defined according to both mass-action kinetics and Hill functions coagSODA allowsthe execution of parallel simulations of the dynamics of the BCC by automatically deriving the system of ordinary differentialequations and then exploiting the numerical integration algorithm LSODA We present the biological results achieved with amassive exploration of perturbed conditions of the BCC carried out with one-dimensional and bi-dimensional parameter sweepanalysis and show that GPU-accelerated parallel simulations of this model can increase the computational performances up to a181times speedup compared to the corresponding sequential simulations
1 Introduction
Mathematical modeling and computational analysis of bio-logical systems nowadays represent an essential methodol-ogy complementary to conventional experimental biologyto achieve an in-depth comprehension of the functioning ofthese complex systems [1 2] Given a model that describesthe physical or logical interactions between the componentsof a biological system different algorithms can be exploitedto make predictions on the way this system behaves in bothphysiological and perturbed conditions For instance start-ing from distinct parameterizations of the model simulationalgorithms can be used to devise the different emergentbehaviors that the system can present the massive explo-ration of high-dimensional parameter spaces allows us to
better understand the system functioning across a wide spec-trum of natural conditions as well as to derive statisticallymeaningful properties Indeed standard investigations ofbiological systems usually rely on computational methodsthat require the execution of a large number of simulationssuch as parameter sweep analysis [3] sensitivity analysis [4]structure and parameter identifiability [5] parameter estima-tion [6ndash8] and reverse engineering of model topologies [9ndash13]
In this context the use of general-purpose GraphicsProcessing Units (GPUs) has recently boosted many applica-tions in scientific computing where CPUs have traditionallybeen the standard workhorses As a matter of fact whenseveral batches of simulations need to be executed the neces-sary computing power can rapidly overtake the capabilities
Hindawi Publishing CorporationBioMed Research InternationalVolume 2014 Article ID 863298 20 pageshttpdxdoiorg1011552014863298
2 BioMed Research International
of standard desktop computers therefore requiring high-performance computing solutions After the introductionof general-purpose GPUs and of Compute Unified DeviceArchitecture (CUDA Nvidiarsquos GPU programming language)the adoption of these graphics engines largely increasedin research fields as Bioinformatics Systems Biology andComputational Biology (see an overview in [14ndash16]) Anywaydespite the remarkable advantages concerning the computa-tional speedup computing with GPUs usually requires thedevelopment and the implementation of ad hoc algorithmssince GPU-based programming substantially differs fromCPU-based computing as a consequence scientific appli-cations of GPUs might undergo the risk of remaining aniche for few specialists To avoid such limitations severalpackages and software tools have recently been released(see eg [16ndash18]) so that also users with no knowledgeof GPUs hardware and programming can access the high-performance computing power of graphics engines
To investigate the dynamics of biological systems eitherdeterministic or stochastic approaches can be exploited [19]which are based on numerical integration (eg Eulerrsquos orRunge-Kutta methods [20]) or on Markov processes (egGillespiersquos algorithm [21]) respectively To date the most effi-cient algorithms to integrate a system of ordinary differentialequations (ODEs) or to perform stochastic simulations ofreaction-basedmodels are LSODA [22] and tau-leaping [23]respectively In [24ndash26] we previously presented cupSODAand cuTauLeaping the GPU-powered implementation ofLSODA and tauleaping respectively cupSODA allows torun parallel deterministic simulations of a given mass-actionbased system of biochemical reactions using the LSODAalgorithm cuTauLeaping represents a novel restructuring ofthe tau-leaping workflow that fits the GPU architecture andavoids any inefficiency drawback for coarse-grain massiveparallel stochastic simulations
In this work we introduce coagSODA an extension ofcupSODA that was specifically designed for the analysis ofa model of the blood coagulation cascade (BCC) Blood is thesubject of an intense scientific research thanks to its key roleinmaking diagnosis of numerous diseases [27] Humans haveevolved a complex hemostatic system that is able to maintainblood in a fluid state and allow the circulation through anintricate network of vessels in particular the presence ofseveral fine-tuned feedback mechanisms in the BCC allowskeeping all blood components within appropriate concen-tration ranges The BCC consists in a complex network ofcellular reactions which under physiological conditions invivo are inhibited by the presence of intact endothelium [28]Anyway in response to any vascular injury the hemostaticsystem is able to stop the blood leakage by rapidly sealing thedefects in the vesselsrsquo wall [29]
In order to investigate the variations in blood coagu-lation components among individuals and to understandthe corresponding response of the system to perturbedconditions we consider here a computational perspectiveto study the BCC we analyze the alterations (prolongationor reduction) of the time required to form the clot (iethe clotting time) by exploiting a reduced version of themathematical model defined in [30] This model describes
the intrinsic extrinsic and common pathways of the BCCand more importantly it accounts for platelets activationas well as the presence of several inhibitors (eg the TissueFactor Pathway Inhibitor antithrombin III and C1-inhibitor)[31]
Numerous mathematical models of blood coagulationwere developed in the last years as they represent a useful toolfor systematic studies of the intricate network of the coagu-lation cascade and allow obtaining a suitable reconstructionof empirical observations (see eg [32ndash34]) The earliestmodels considered only simple steps of the whole BCC suchas the conversion of the clot fibrin by thrombin [35] latelyHockin et al [36] developed a comprehensive ODE-basedmodel of the extrinsic blood coagulation system This modelwas then considered as reference by several research groupsto investigate the thrombotic risk in healthy and ill popula-tions [37 38] or to understand other complex biochemicalprocesses for instance in [39] the roles of protein C proteinS and phospholipid surface actions were considered whilein [40] the influence of trace amounts of key coagulationproteases on thrombin generation was investigated Recentlyother works modeling the blood clotting process in a com-prehensive manner have been published besides the alreadymentioned model developed by Chatterjee et al [30] wemention the model defined by Wajima et al [32] whichsimulates the intrinsic extrinsic and common pathways thevitamin K cycle the therapy with the anticoagulant drugswarfarin and heparin and the laboratory tests PT and aPTTas well as the Taipan snake bite which causes coagulopathy
coagSODA the GPU-accelerated simulator that wepresent and exploit in this work for the analysis of the BCCmodel is a user-friendly and efficient tool that circumventsthe need of manually defining the system of ODEs thatdescribe the blood coagulation network More preciselycoagSODA is able to automatically derive the system of(mass-action and Hill function-based) ODEs and then per-form their numerical integration starting from the given set of96 biochemical reactions which fully describe the molecularinteractions between all the species involved in the BCCin vivo We show that coagSODA allows us to efficientlyexecute a large number of parallel deterministic simulationsof the BCC at a considerable reduced computational costwith respect to CPUs In particular we exploit coagSODAto carry out one-dimensional and bi-dimensional parametersweep analysis of the BCC to the purpose of investigatingthe prolongation and the reduction of the clotting time inresponse to perturbed values of some reaction constants andof the initial concentration of somemolecular species chosenaccording to their meaning within the whole pathway
The paper is structured as follows In Section 2 we fullydescribe the mechanistic model of the BCC used in thiswork and present the simulation method at the basis ofcupSODA tool to introduce the coagSODA simulator InSection 3 we present the results obtained from the parametersweep analysis of the BCC model as well as a comparison ofthe computational performance of coagSODAwith respect toaCPU-based implementation of LSODA Finally in Section 4we conclude the paper with a discussion of the presentedwork and future research directions
BioMed Research International 3
2 Materials and Methods
21 A Mechanistic Model of the Blood Coagulation CascadeBlood is an essential component in human life whoseprimary functions are to feed cells by delivering a multitudeof nutrients such as oxygen and to carry away the cellularwastes such as carbon dioxide Specialized cells and fluidsin blood perform many physiological functions and can beisolated and analyzed through specific laboratory tests givingthe opportunity to settle a personrsquos health condition Allblood components are kept within appropriate concentrationranges by means of fine-tuned regulatory mechanisms ruledby several feedback controls the constancy of blood com-position is maintained thanks to the circulation through anintricate network of vessels In particular humans evolved acomplex hemostatic system that under physiological condi-tions maintains blood in a fluid state however in responseto any vascular injury this system is able to rapidly react andseal the defects in the vesselsrsquo wall in order to stop the bloodleakage [29] Indeed the circulatory system is self-sealingotherwise a continuous blood flow from even the smallestwound would become a threat for the individualrsquos life
To allow blood coagulation in humans there exist 13blood clotting proteins called coagulation factors which areusually designated by Roman numerals I through XIII Asa consequence of a vascular injury platelets become activeand the Tissue Factor (TF also called factor III) is exposedin the subendothelial tissue starting the blood coagulationcascade (BCC) The ultimate goal of the BCC is to convertprothrombin (factor II) into thrombin (factor IIamdashie theactive factor II) the enzyme that catalyzes the formation ofa clot Traditionally the BCC is divided into the extrinsic andintrinsic pathways both leading to the activation of factor X[41] The last part of the cascade downstream of this factoris called the common pathway and leads to the formationof fibrin monomers whose polymers finally constitute thebackbone of the clot
Excluding thrombin all the enzymes involved in bloodclotting are characterized by a low activity which increasesupon binding to a specific protein cofactor (eg factors Vand VIII) or to appropriate phospholipid surfaces (eg theplasmamembranes of active platelets) [41] Even calcium ionshave a central role in coagulation since they are essentialto start and enhance numerous reactions without calciumions the blood coagulation cannot occur [42] In the BCCpathways the activity of the various active proteases is limitedby several inhibitory factors which allow regulating thewhole cascade When the hemostatic system is unregulatedthrombosis (ie the formation of a blood clot obstructing theblood flow in vessels) may occur due to impairment in theinhibitory pathway or because the functioning of the naturalanticoagulant processes is overwhelmed by the strength of thehemostatic stimulus [29]
The BCC model we consider in this work is a slightlyreduced version of the ldquoPlatelet-Plasmardquo deterministic modeldefined in [30] built upon a previous model [36] whichdescribes all parts of blood coagulation the plateletsactivation and aggregation the extrinsic intrinsic andcommon pathways (with the exception of factor XIII)
the action of several inhibitory molecules (Tissue Fac-tor Pathway Inhibitor antithrombin III C1-inhibitor 1205721-antitrypsin and 1205722-antiplasmin) In addition to simulate thecoagulation process in vitro Chatterjee et al also modeledthe action of corn trypsin inhibitor (CTI) which inhibits theactivation of the so-called contact system as well as the actionof the fluorogenic substrate Boc-VPR-MCA widely used inlaboratories for thrombin titration [30] The role of calciumwas not explicitly included in the model but considered asa nonlimiting factor as in living organisms this ion veryrarely drops to such low levels able to alter the kinetics of theformation of the fibrin clot [43]
Since the aim of this work is the investigation of bloodcoagulation in vivo we exclude a small set of reactions givenin [30] that have no effect on the clotting time To be moreprecise we do not consider the reactions occurring in vitro(namely entries 28 and 35 in Table 1 in [30]) as well asthe reactions downstream the fibrinogen conversion that isthe interactions between the fibrin polymers thrombin andantithrombin III (namely entries 55 56 and 57 in Table 1 in[30])
For the sake of completeness we describe in Table 1 theBCC model considered in this work overall consisting in 96reactions among 71 molecular species A graphical sketch ofthemainmolecular interactions among the BCC componentsis given in Figure 1
The model can be partitioned into four functional mod-ules
(1) Thefirstmodule corresponds to the extrinsic pathwaywhich consists in
(i) the formation of a complex between TissueFactor and factor VII (modeled by reactions1199031 119903
4)
(ii) the activation of factor VII by the complexbetween Tissue Factor and factor VIIa (reaction1199035)
(iii) the activation of factor IX by the complexbetween Tissue Factor and factor VIIa (reac-tions 119903
13 11990314 11990315) and by factor VIIa (reactions
11990378 11990379 11990380)
(iv) the activation of factor X by the complexbetween Tissue Factor and factor VIIa(1199038 119903
12) and by factor VIIa (119903
81 11990382 11990383)
(2) The second module corresponds to the intrinsic path-way which consists in
(i) the formation of a complex between factor VIIIaand factor IXa (modeled by reactions 119903
18and
11990319)
(ii) the activation of factor X by the complexbetween factor VIIIa and factor IXa (reactions11990320 11990321 11990322) and by factor IXa (reactions 119903
72 11990373
11990374)
(iii) the activation of factor XII by factor XII itself(reaction 119903
44) by factor XIIa (reactions 119903
45 11990346
11990347) and by kallikrein (reactions 119903
51 11990352 11990353)
4 BioMed Research International
Table 1 Reaction-based model of the blood coagulation cascade (reduced version of the ldquoPlatelet-Plasmardquo model described in [30]) Themodel consists in 96 reactions among 71 molecular species With the exception of reaction 119903
44(see Section 31) the values of all reaction
constants were taken from [30]
119899 Reactants Products Constant (119896)1199031
TF + fVII TF-fVII 320 sdot 106Mminus1 sminus1
1199032
TF-fVII TF + fVII 310 sdot 10minus2 sminus1
1199033
TF + fVIIa TF-fVIIa 230 sdot 107Mminus1 sminus1
1199034
TF-fVIIa TF + fVIIa 310 sdot 10minus5 sminus1
1199035
TF-fVIIa + fVII TF-fVIIa + fVIIa 440 sdot 105Mminus1 sminus1
1199036
fXa + fVII fXa + fVIIa 130 sdot 107Mminus1 sminus1
1199037
fIIa + fVII fIIa + fVIIa 230 sdot 104Mminus1 sminus1
1199038
TF-fVIIa + fX TF-fVIIa-fX 250 sdot 107Mminus1 sminus1
1199039
TF-fVIIa-fX TF-fVIIa + fX 105 sdot 10minus2 sminus1
11990310
TF-fVIIa-fX TF-fVIIa-fXa 600 sminus1
11990311
TF-fVIIa-fXa TF-fVIIa + fXa 1900 sminus1
11990312
TF-fVIIa + fXa TF-fVIIa-fXa 220 sdot 107Mminus1 sminus1
11990313
TF-fVIIa + fIX TF-fVIIa-fIX 100 sdot 107Mminus1 sminus1
11990314
TF-fVIIa-fIX TF-fVIIa + fIX 240 sminus1
11990315
TF-fVIIa-fIX TF-fVIIa + fIXa 180 sminus1
11990316
fII + fXa fIIa + fXa 750 sdot 103Mminus1 sminus1
11990317
fIIa + fVIII fIIa + fVIIIa 200 sdot 107Mminus1 sminus1
11990318
fVIIIa + fIXa fIXa-fVIIIa 100 sdot 107Mminus1 sminus1
11990319
fIXa-fVIIIa fVIIIa + fIXa 100 sdot 10minus4 sminus1
11990320
fIXa-fVIIIa + fX fIXa-fVIIIa-fX 100 sdot 108Mminus1 sminus1
11990321
fIXa-fVIIIa-fX fIXa-fVIIIa + fX 100 sdot 10minus5 sminus1
11990322
fIXa-fVIIIa-fX fIXa-fVIIIa + fXa 820 sminus1
11990323
fVIIIa fVIIIa1-L + fVIIIa2 600 sdot 10minus5 sminus1
11990324
fVIIIa1-L + fVIIIa2 fVIIIa 220 sdot 104 Mminus1 sminus1
11990325
fIXa-fVIIIa-fX fVIIIa1-L + fVIIIa2 + fX + fIXa 100 sdot 10minus3 sminus1
11990326
fIXa-fVIIIa fVIIIa1-L + fVIIIa2 + fIXa 100 sdot 10minus3 sminus1
11990327
fIIa + fV fIIa + fVa 200 sdot 107Mminus1 sminus1
11990328
fXa + fVa fXa-fVa 400 sdot 108Mminus1 sminus1
11990329
fXa-fVa fXa + fVa 02 sminus1
11990330
fXa-fVa + fII fXa-fVa-fII 100 sdot 108Mminus1 sminus1
11990331
fXa-fVa-fII fXa-fVa + fII 10300 sminus1
11990332
fXa-fVa-fII fXa-fVa + fmIIa 6350 sminus1
11990333
fXa-fVa + fmIIa fXa-fVa + fIIa 150 sdot 107Mminus1 sminus1
11990334
fXa + TFPI fXa-TFPI 900 sdot 105Mminus1 sminus1
11990335
fXa-TFPI fXa + TFPI 360 sdot 10minus4 sminus1
11990336
TF-fVIIa-fXa + TFPI TF-fVIIa-fXa-TFPI 320 sdot 108 Mminus1 sminus1
11990337
TF-fVIIa-fXa-TFPI TF-fVIIa-fXa + TFPI 110 sdot 10minus2 sminus1
11990338
TF-fVIIa + fXa-TFPI TF-fVIIa-fXa-TFPI 500 sdot 107Mminus1 sminus1
11990339
fXa + ATIII fXa-ATIII 150 sdot 103Mminus1 sminus1
11990340
fmIIa + ATIII fmIIa-ATIII 710 sdot 103Mminus1 sminus1
11990341
fIXa + ATIII fIXa-ATIII 490 sdot 102Mminus1 sminus1
11990342
fIIa + ATIII fIIa-ATIII 710 sdot 103Mminus1 sminus1
11990343
TF-fVIIa + ATIII TF-fVIIa-ATIII 230 sdot 102Mminus1 sminus1
11990344
fXII fXIIa 500 sdot 10minus4 sminus1
11990345
fXIIa + fXII fXIIa-fXII 100 sdot 108Mminus1 sminus1
11990346
fXIIa-fXII fXIIa + fXII 75000 sminus1
11990347
fXIIa-fXII fXIIa + fXIIa 330 sdot 10minus2 sminus1
11990348
fXIIa + PKal fXIIa-PKal 100 sdot 108Mminus1 sminus1
11990349
fXIIa-PKal fXIIa + PKal 360 sdot 103 sminus1
BioMed Research International 5
Table 1 Continued
119899 Reactants Products Constant (119896)11990350
fXIIa-PKal fXIIa + Kal 4000 sminus1
11990351
fXII + Kal fXII-Kal 100 sdot 108Mminus1 sminus1
11990352
fXII-Kal fXII + Kal 4530 sminus1
11990353
fXII-Kal fXIIa + Kal 570 sminus1
11990354
PKal + Kal Kal + Kal 270 sdot 104Mminus1 sminus1
11990355
Kal Kal119894
110 sdot 10minus2 sminus1
11990356
fXIIa + C1inh fXIIa-C1inh 360 sdot 103Mminus1 sminus1
11990357
fXIIa + ATIII fXIIa-ATIII 2160Mminus1 sminus1
11990358
fXI + fIIa fXI-fIIa 100 sdot 108Mminus1 sminus1
11990359
fXI-fIIa fXI + fIIa 500 sminus1
11990360
fXI-fIIa fXIa + fIIa 130 sdot 10minus4 sminus1
11990361
fXIIa + fXI fXIIa-fXI 100 sdot 108Mminus1 sminus1
11990362
fXIIa-fXI fXIIa + fXI 200 sminus1
11990363
fXIIa-fXI fXIIa + fXIa 570 sdot 10minus4 sminus1
11990364
fXIa + fXI fXIa + fXIa 319 sdot 106Mminus1 sminus1
11990365
fXIa + ATIII fXIa-ATIII 320 sdot 102Mminus1 sminus1
11990366
fXIa + C1inh fXIa-C1inh 180 sdot 103Mminus1 sminus1
11990367
fXIa + A1AT fXIa-A1AT 100 sdot 102Mminus1 sminus1
11990368
fXIa + A2AP fXIa-A2AP 43 sdot 103Mminus1 sminus1
11990369
fXIa + fIX fXIa-fIX 100 sdot 108Mminus1 sminus1
11990370
fXIa-fIX fXIa + fIX 4100 sminus1
11990371
fXIa-fIX fXIa + fIXa 770 sminus1
11990372
fIXa + fX fIXa-fX 100 sdot 108Mminus1 sminus1
11990373
fIXa-fX fIXa + fX 064 sminus1
11990374
fIXa-fX fIXa + fXa 700 sdot 10minus4 sminus1
11990375
fXa + fVIII fXa-fVIII 100 sdot 108Mminus1 sminus1
11990376
fXa-fVIII fXa + fVIII 210 sminus1
11990377
fXa-fVIII fXa + fVIIIa 0023 sminus1
11990378
fVIIa + fIX fVIIa-fIX 100 sdot 108Mminus1 sminus1
11990379
fVIIa-fIX fVIIa + fIX 090 sminus1
11990380
fVIIa-fIX fVIIa + fIXa 360 sdot 10minus5 sminus1
11990381
fVIIa + fX fVIIa-fX 100 sdot 108Mminus1 sminus1
11990382
fVIIa-fX fVIIa + fX 21000 sminus1
11990383
fVIIa-fX fVIIa + fXa 160 sdot 10minus6 sminus1
11990384
Fbg + fIIa Fbg-fIIa 100 sdot 108Mminus1 sminus1
11990385
Fbg-fIIa Fbg + fIIa 63600 sminus1
11990386
Fbg-fIIa Fbn1 + fIIa + FPA 8400 sminus1
11990387
Fbn1 + fIIa Fbn1-fIIa 100 sdot 108Mminus1 sminus1
11990388
Fbn1-fIIa Fbn1 + fIIa 74260 sminus1
11990389
Fbn1-fIIa Fbn2 + fIIa + FPB 740 sminus1
11990390
Fbn1 + Fbn1 (Fbn1)2 100 sdot 106Mminus1 sminus1
11990391
(Fbn1)2 2Fbn1 640 sdot 10minus2 sminus1
11990392
(Fbn1)2 + fIIa (Fbn1)2-fIIa 100 sdot 108Mminus1 sminus1
11990393
(Fbn1)2-fIIa (Fbn1)2 + fIIa 70100 sminus1
11990394
(Fbn1)2-fIIa (Fbn2)2 + fIIa + FPB 4900 sminus1
11990395
Fbn2 + fIIa Fbn2-fIIa 100 sdot 108Mminus1 sminus1
11990396
Fbn2-fIIa Fbn2 + fIIa 100 sdot 103 sminus1
6 BioMed Research International
fXIITF-fVII
TFfVIIa
fVII
TF-fVIIa
Fbg
FPA
Fbn1
FPB
Fbn2
Fbn1
(Fbn1)2
(Fbn2)2 FPB
fXa
fIIa
TF-fVIIa-fXa
fIX
fIXa
fX
fmIIa
fII
PKal
KalifXI
fXIa
fIX
fIXa
fVIII
fVIIIa
fIXa-fVIIIa
fVIIIa1 -L
fVIIIa2
fVIIIa2
fIXa
fV
fVa
fXa-fVa
fXa-TFPI
TFPI
TF-fVIIa-fXa-TFPI
fXa-AT
AT
AT
fXIa-AT
C1inh
fXIIa-C1inh
A1AT
A2AP
fXIa-A2APfXIa-A1AT
TF-fVIIa
fXIIa Kal
fVIIIa1 -L
Figure 1 Graphical representation of the blood coagulation cascade model considered in this work Legend Blue box coagulation factor redbox inhibitor and related complexes Black arrow complex formation green arrow catalytic activation violet arrow activation red arrowinhibition The reaction is reversible if the arrow tail consists in a dot
BioMed Research International 7
(iv) the activation of prekallikrein by factor XIIa(reactions 119903
48 11990349 11990350) and by kallikrein (reac-
tion 11990354)
(v) the activation of factor XI by factor XIIa (reac-tions 119903
61 11990362 11990363) and by factor XIa (reaction
11990364)
(vi) the activation of factor IX by factor XIa (reac-tions 119903
69 11990370 11990371)
(vii) the dissociation of free factor VIIIa (reactions11990323and 11990324)
(viii) the dissociation of factor VIIIa in complex withother factors (reactions 119903
25and 11990326)
(3) The third module corresponds to the common path-way which consists in
(i) the activation of factor II by factor Xa (modeledby reaction 119903
16) and by the complex between
factor Xa and factor Va through the formationof the intermediate meizothrombin (reactions11990330 119903
33)
(ii) the activation of factor VII by factor Xa (reac-tion 1199036) and by factor IIa (reaction 119903
7)
(iii) the activation of factor VIII by factor Xa (reac-tions 119903
75 11990376 11990377) and by factor IIa (reaction 119903
17)
(iv) the formation of a complex between factor Xaand factor Va (reactions 119903
28and 11990329)
(v) the activation of fibrinogen by factor IIa (reac-tions 119903
84 119903
96)
(vi) the activation of factor V by factor IIa (reaction11990327)
(vii) the activation of factor XI by factor IIa (reac-tions 119903
58 11990359 11990360)
(4) The fourth module describes the inhibition of coagu-lation carried out through the main inhibitors of theBCC (antithrombin Tissue Factor Pathway InhibitorC1-inihibitor 1205721-antitrypsin and 1205722-antiplasmin)These are tight binding inhibitors belonging to theserpin superfamily which form irreversible com-plexes This module consists in
(i) the inhibition of factor Xa by Tissue FactorPathway Inhibitor (modeled by reactions 119903
34and
11990335)
(ii) the inhibition of the complex between TissueFactor factor VIIa and factor Xa by TissueFactor Pathway Inhibitor (reactions 119903
36 11990337 11990338)
(iii) the inhibition of factor IIa by antithrombin(reactions 119903
40and 11990342)
(iv) the inhibition of factor Xa by antithrombin(reaction 119903
39)
(v) the inhibition of factor IXa by antithrombin(reaction 119903
41)
(vi) the inhibition of factor XIa by antithrombin(reaction 119903
65) and by C1-inhibitor (reaction 119903
66)
Table 2 Initial concentrations of molecular species in the bloodcoagulation cascade model (values taken from [30])
Species Symbol Concentration (M)1205721-Antitrypsin A1AT 450 sdot 10
minus5
1205722-Antiplasmin A2AP 100 sdot 10minus6
Antithrombin III ATIII 340 sdot 10minus6
C1-inhibitor C1inh 250 sdot 10minus6
Fibrinogen Fbg 900 sdot 10minus6
Factor II fII 140 sdot 10minus6
Factor V fV 200 sdot 10minus8
Factor VII fVII 100 sdot 10minus8
Active factor VII fVIIa 100 sdot 10minus10
Factor VIII fVIII 700 sdot 10minus10
Factor IX fIX 900 sdot 10minus8
Factor X fX 16 sdot 10minus7
Factor XI fXI 310 sdot 10minus8
Factor XII fXII 340 sdot 10minus7
Prekallikrein Pkal 450 sdot 10minus7
Tissue Factor TF 500 sdot 10minus12
Tissue Factor Pathway Inhibitor TFPI 250 sdot 10minus9
(vii) the inhibition of factor XIIa by C1-inhibitor(reaction 119903
56) by antithrombin (reaction 119903
57)
by 1205721-antitrypsin (reaction 11990367) and by 1205722-
antiplasmin (reaction 11990368)
(viii) the inhibition of the complex between TissueFactor and factor VIIa by antithrombin (reac-tion 11990343)
(ix) the inhibition of kallikrein (reaction 11990355)
The values of the initial concentrations of the molecularspecies occurring in the BCC model are given in Table 2According to [30] the concentrations of complexes and activefactors were set to 0 except for the active factor VII which isphysiologically present in the blood circulation even in theabsence of damage in a concentration that is approximatelyequal to 1 of the corresponding inactive factor [42]
The system of ordinary differential equations (ODEs)needed to carry out the simulations and the parametersweep analysis presented in Section 3 was derived fromthe reactions given in Table 1 according to the mass-actionlaw with the exception of 14 reactions belonging to the set119878120576= 11990319 11990321 11990323 11990329 11990331 11990335 11990346 11990349 11990352 11990362 11990370 11990373 11990376 11990382
The mass-action law is the fundamental empirical law thatgoverns biochemical reaction rates which states that in adilute solution the rate of an elementary reaction is propor-tional to the product of the concentration of its reactantsraised to the power of the corresponding stoichiometriccoefficient [44] The reactions in 119878
120576are instead influenced by
a specifically defined variable 120576 depending on aHill functionfit against experimental data which quantify the plateletactivation status which is used to model the physiologicallevels of thrombin concentration as a function of plateletactivation as thoroughly described in [30] More preciselythe value of 120576 influences the formation of some complexes
8 BioMed Research International
0 0 100 200 300 400 500 600 700
Fact
or II
a con
cent
ratio
n (M
)
Time (s)
1eminus7
2eminus7
3eminus7
4eminus7
5eminus7
6eminus7
7eminus7
8eminus7
9eminus7
Figure 2 Dynamics of thrombin (factor IIa) in physiological condition
occurring on the plateletsrsquo surface by modifying the activityof reactions of the form
119860 + 119861
1198961
999445999468
1198962
119860119861 (1)
Namely 120576 intervenes with the dissociation constant of thesereactions so that the corresponding standard ODEs arechanged to yield new equations of the form
119889 [119860119861]
119889119905
= 1198961sdot [119860] sdot [119861] minus
1198962sdot [119860] sdot [119861]
120576
(2)
The value of 120576 in (2) depends on the following HillfunctionH which quantifies the state of platelets activationaccording to the thrombin concentration (here denoted as[fIIa]) that is the factor catalyzing the formation of the fibrinclot
H ([fIIalowast (119905)]) =[fIIalowast (119905)]16123
[fIIalowast (119905)]16123 + (24279 sdot 10minus9)16123
(3)
where [fIIalowast(119905)] = max1199051015840isin[0119905][fIIa(1199051015840)] for a chosen time
interval [0 119905] of simulationThe value [fIIalowast(119905)] represents the maximum transient
thrombin concentration and is needed to simulate the factthat in physiological conditions the thrombin concentrationstarts decreasing after rising to a peak (Figure 2) So doing[fIIalowast(119905)] never decreases once it reaches its maximummagnitude [fIIalowast(119905)] is equivalent to [fIIa(119905)] until the con-centration of factor IIa reaches the peak while thereafter itremains constant at that value which is the maximum in theconsidered time interval [0 119905] FunctionH allows simulatingthe physiological condition whereby platelets remain activealso when the thrombin concentration decreases
For a given concentration of factor IIa the maximumplatelets activation state 120576max is defined as
120576max = 120576max0
+ (1 minus 120576max0
) sdotH ([fIIalowast (119905)]) (4)
where 120576max0
defines the basal activation state of the plateletsat simulation time 119905 = 0 The value 120576max
0
is initially set to 001assuming a basal 1 binding strength of coagulation factorsto the resting plateletsrsquo surface When the full activationof platelets is reached 120576max is equal to 1 and the complexdissociation constants are minimized (see [30] for moredetails)
22 CUDA Architecture and the coagSODA Simulator Intro-duced by Nvidia in 2006 Compute Unified Device Architec-ture (CUDA) is a parallel computing platform and program-ming model that provides programmers with a frameworkto exploit GPUs in general-purpose computational tasks(GPGPU computing) GPGPU computing is a low-cost andenergy-wise alternative to the traditional high-performancecomputing infrastructures (eg clusters of machines) whichgives access to the tera-scale computing on common work-stations of mid-range price However due to the innovativearchitecture and the intrinsic limitations of GPUs a directporting of sequential code on the GPU is most of the timesunfeasible therefore making it challenging to fully exploit itscomputational power and massive parallelism [45]
CUDA combines the single instruction multiple data(SIMD) architecture with multithreading which automat-ically handles the conditional divergence between threadsThe drawback of such flexibility is that any divergence ofthe execution flow between threads causes a serialization ofthe execution which affects the overall performances UnderCUDArsquos naming conventions the programmer implementsthe kernel that is a CC++ function which is loaded fromthe host (the CPU) to the devices (one or more GPUs)and replicated in many copies named threads Threads canbe organized in three-dimensional structures named blockswhich in turn are contained in three-dimensional grids(a schematic description is given in Figure 3 left side)Whenever the host runs a kernel the GPU creates thecorresponding grid and automatically schedules each blockon one free streaming multiprocessor available on the GPUallowing a transparent scaling of performances on different
BioMed Research International 9
Host
Kernel 1
Kernel 2
Device
Grid 1
Grid 2
Block(0 0)
Block(1 0)
Block(2 0)
Block(0 1)
Block(1 1)
Block(2 1)
Block (1 1)
Thread(0 0)
Thread(1 0)
Thread(2 0)
Thread(3 0)
Thread(4 0)
Thread(0 1)
Thread(1 1)
Thread(2 1)
Thread(3 1)
Thread(4 1)
Thread(0 2)
Thread(1 2)
Thread(2 2)
Thread(3 2)
Thread(4 2)
Global memory
Texture memory
Constant memory
Shared memory Shared memory
Block (0 0) Block (1 0)
Localmemory
Localmemory
Localmemory
Localmemory
Thread(0 0)
Thread(1 0)
Registers Registers Registers Registers
Thread(0 0)
Thread(1 0)
Grid
Figure 3 Schematic description of CUDArsquos architecture in terms of threads and memory hierarchy Left Side Threads organization a singlekernel is launched from the host (the CPU) and is executed in multiple threads on the device (the GPU) Threads can be organized in three-dimensional structures named blocks which can be in turn organized in three-dimensional grids The dimensions of blocks and grids areexplicitly defined by the programmer Right Side Memory hierarchy threads can access data from many different memories with differentscopes Registers and local memories are private for each thread Shared memory lets threads belonging to the same block communicate andhas low access latency All threads can access the global memory which suffers high latencies but it is cached since the introduction of theFermi architecture Texture and constant memory can be read from any thread and are equipped with a cache as well in this work we exploitthe constant memory Figures are taken from Nvidiarsquos CUDA programming guide [60]
devices Threads within a block are executed in groups of 32threads named warps
The GPU is equipped with different kinds of memoryIn this work we exploit the global memory (accessible fromall threads) the shared memory (accessible from threadsof the same block) the local memory (registers and arraysaccessible from owner thread) and the constant memory(cached and not modifiable) A schematic representation ofthis memory hierarchy is shown in Figure 3 right side Toachieve the best performances the sharedmemory should beexploited as much as possible however it is very limited (ie49152 bytes for each multiprocessor since the introductionof the Fermi architecture) and it introduces restrictions onthe blocksrsquo size On the other hand the global memory isvery large (thousands ofMBs) but suffers from high latenciesA solution to this problem was implemented on the Fermiarchitecture where the global memory is equipped with aL2 cache This architecture also introduced the possibility tobalance 64KB of fast on-chip memory between the sharedmemory and L1 cache using two possible configurations48KB for the shared memory and 16KB for L1 cache or16 KB for the shared memory and 48 for L1 cacheThe Keplerarchitecture used in this paper allows a third and perfectly
balanced configuration where shared memory and L1 cacheobtain the same amount of memory (32KB) See Figure 4 fora schematization of the memory architecture
The systematic analysis of models of biological systemsoften consists in the execution of large batches of simulationsOne of the standard analyses that can be executed on suchkind of models regards an intensive search within the param-eters space which requires large numbers of independentsimulations In order to reduce the computational burdenwe previously implemented on the CUDA architecture oneof the most efficient numerical integration algorithms forODEs LSODA that is able to automatically recognize stiffand nonstiff systems and to dynamically select between themost appropriate integration procedure (ie Adams methodin the absence of stiffness and the Backward DifferentiationFormulae otherwise) [22] This GPU-powered tool is calledcupSODA [24 25] and exploits CUDArsquos massive parallelismto execute different and independent simulations in eachthread thus reducing the computational time required by astandard CPU counterpart of LSODA
Besides being very efficient for the simulation of manyindependent simulations cupSODA is also user-friendlyLSODA was originally designed to solve ODEs systems
10 BioMed Research International
Thread
Shared memory L1 cache Read-onlydata cache
L2 cache
DRAM
Figure 4 Schematic description of memory hierarchies in Fermi and Kepler architectures GPUs relying on these architectures are equippedwith a two-level data cache and a read-only data cache Shared memory and L1 cache share the same on-chip 64KB memory banks theamount of memory can be reconfigured by the user according to the specific needs of the application Figure taken from Nvidiarsquos KeplerGK110 whitepaper [57]
written in the canonical form but the user is supposed tospecify the system of ODEs by implementing a custom Cfunction that is passed to the algorithmmoreover in order tospeed up the computation when dealing with stiff systems inLSODA the Jacobianmatrix associatedwith theODEs systemmust be implemented as a custom C function as well On thecontrary cupSODA was conceived as a black-box simulatorthat can be easily used without any programming skillscupSODA consists in a tool to automatically convert a genericmechanistic reaction-based model of a biological system intothe corresponding set of ODEs to comply with the mass-action kinetics [46] and to directly encode the obtainedsystem along with the corresponding Jacobian matrix as Carrays
This fully automaticmethodology can be exploited for thesimulation of models whose chemical kinetics is based on themass-action law The BCC model described in Section 21though includes a set 119878
120576of reactions that do not follow
the mass-action kinetics [30] the platelets activity is notexplicitly modeled by biochemical reactions but it is realizedby modulating the rate of the dissociation of the complexesformed on a plateletrsquos surface by means of the variable120576 which is calculated with a special equation during theintegration steps For this reason cupSODA was used asa starting point for the development of a new tool ableto compute at run-time the specific kinetics of this set ofreactions
This new CUDA-powered simulation tool named coag-SODA is specifically tailored for the simulation of the BCC
model developed in [30] In particular coagSODA realizesthe run-time calculation of the 120576 value required to correctlysimulate the activity of platelets which is determined accord-ing to the equations described in Section 21 The plateletsactivation state 120576 is calculated at each time instant by solvingthe following differential equation
119889120576
119889119905
= 119896 (120576max minus 120576) (5)
where the constant 119896 is inversely proportional to the timescale of platelets activation and is set to 0005 This isconsistent with the fact that platelets do not instantly achievetheir maximum attainable activation state (120576max) but theyreach it on a physiologically relevant timescale [30]
Dealing with the 14 reactions in the set 119878120576that are
influenced by the Hill functionH (see Section 21) the valueof [fIIalowast(119905)]must be stored on the GPU because during eachintegration step coagSODA recalculates (4) which exploitsthe value [fIIalowast(119905)] to determine a new 120576 value Since CUDArsquosarchitecture does not offer static variables the informationfor each thread has to be memorized in the global memoryThe accesses to the global memory and the computationalcosts due to these additional calculations slow down theintegration process with respect to the integration of ODEsperformed according to strictly mass-action based kinetics(as in the cupSODA simulator) Nevertheless in Section 33we show that our parallel implementation largely outper-forms a sequential counterpart of the LSODA algorithm
BioMed Research International 11
0 100 200 300 400 500 600 700
Fibr
inog
en co
ncen
trat
ion
(M)
Time (s)
30 of fibrinogen
0
1eminus6
2eminus6
3eminus6
4eminus6
5eminus6
6eminus6
7eminus6
8eminus6
9eminus6
Figure 5 Dynamics of fibrinogen in physiological condition The clotting time is defined as the time necessary to convert the 70 of thefibrinogen into fibrin
As in the case of cupSODA [24 25] coagSODA exploitsthe shared memory to improve performances by storing thecurrent state and time of the simulations in these low-latencymemory banks Despite the improvement of performancesensured by this solution it strongly affects the occupancyof GPUrsquos multiprocessors and therefore it represents thelimiting factor for the number of blocks that can be executedsimultaneously as a matter of fact coagSODA is limited to2 blocks per streaming multiprocessor on GPUs based onthe Fermi architecture with a reduced exploitation of theGPU with respect to the theoretical 8 blocks allowed by thisarchitecture
3 Results and Discussion
In this section we discuss the results of the parameter sweepanalysis (PSA) carried out on the BCC model to the aimof investigating either the prolongation or the reduction ofthe clotting time in response to perturbed values of somereaction constants and of the initial concentration of somemolecular species chosen according to their meaning withinthe whole pathway PSA was performed by generating aset of different initial conditions corresponding to differentparameterizations of the model and then automaticallyexecuting the deterministic simulations with coagSODATheuse ofGPU technology is fundamental in this type of analysisespecially for large biological systems as the BCC is becauseit drastically reduces the computational time
The sweep analysis for single parameters (PSA-1D) wasperformed considering a logarithmic sampling of numericalvalues of each parameter under investigation (reaction con-stant or initial molecular concentration) within a specifiedrange with respect to its physiological reference value Thesweep analysis over pairs of parameters (PSA-2D) was per-formed by simultaneously varying the values of two param-eters within a specified range considering a logarithmicsampling on the resulting lattice The logarithmic samplingallows uniformly spanning different orders of magnitude ofthe parameters value using a reduced and fine-grained set
of samples therefore efficiently analyzing the response of thesystem in a broad range of conditions
To determine the response of the BCC to perturbed con-ditions we chose the clotting time (CT) as output of the PSAThe CT is defined as the time necessary to convert the 70of the fibrinogen into fibrin (see Figure 5) conventionallyassumed to correspond to the time required to form the clot[47] and it is generally used in laboratory tests formonitoringthe therapywith anticoagulant drugs According to themodeldefined in [30] the reference value of CT is around 300seconds in physiological conditions We investigate herethe response of the BCC by evaluating the CT in variousconditions corresponding to different values of the reactionconstants varying over six orders of magnitude with respectto their physiological values (ie three below and three abovethe reference values if not otherwise specified) as well asto different values of the initial molecular concentrationsand varying over twelve orders of magnitude with respect totheir physiological values (ie six below and six above thereference values if not otherwise specified)
The total number of parallel simulations executed tocarry out these analyses was 100 for PSA-1D over reactionconstants 200 for PSA-1D over initial concentrations and1600 for PSA-2D
Finally we present the comparison of the performanceof the CPU and GPU to run an increasing number ofsimulations of the BCC model to prove the efficiency ofcoagSODA
31 PSA-1D of the BCC Model
311 Reaction 11990344 Thefirst PSA was performed to determine
the value of the kinetic constant for the autoactivation offactor XII (reaction 119903
44in Table 1) which corresponds to an
upstream process in the intrinsic pathway This analysis wasmotivated by two considerations Firstly by using the fullparameterization given in [30] the action of the intrinsicpathway turns out to be fundamental for the BCC in vivowhich is in contrast to experimental observations which
12 BioMed Research International
324
326
328
330
332
334
336
338
340
Clot
ting
time (
s)
1eminus
10
1eminus
9
1eminus
8
1eminus
7
1eminus
6
1eminus
5
1eminus
4
1eminus
3
k44 (sminus1)
Figure 6 Clotting time according to PSA-1D over constant 11989644of reaction 119903
44 XII rarr XIIaThe reference value of 119896
44used in [30] is 5sdot10minus4sminus1
the sweep range is [5 sdot 10minus10 5 sdot 10minus4]
280
300
320
340
360
380
400
1e4 1e5 1e6 1e7 1e8 1e9 1e10 1e11
Clot
ting
time (
s)
k27 (Mminus1sminus1)
Figure 7 Clotting time according to PSA-1D over constant 11989627of reaction 119903
27 IIa + V rarr IIa + Va The reference value is 2 sdot 107Mminus1 sminus1 the
sweep range is [2 sdot 104 2 sdot 1010]
indicate that the extrinsic pathway is the main responsibleof clot formation Secondly in [30] all constants values havea reference to experimental measurements except for theconstant of this reaction (which corresponds to entry 29 inTable 1 in [30])
Figure 6 shows the results of this PSA-1D where 11989644was
varied over six orders of magnitude considering the valuegiven in [30] as the upper limit of the sweep interval We canobserve that by decreasing the value of 119896
44with respect to
the value considered in [30] the CT increases with respect toits reference value however for values of this constant lowerthan 100 sdot 10minus6 sminus1 the CT remains unaltered and this canbe intuitively explained by the fact that in this situation thefibrinogen is mainly activated by the extrinsic pathway
Consequently we assigned the value 700 sdot 10minus6 sminus1 to11989644 achieving a CT that is comparable to the experimental
observations of the BCC in vivo This new value was used inall PSA discussed in what follows
312 Reactions 11990327
and 11990358 In the next PSA we investigated
the effect of the perturbation of the kinetics of two pivotal
reactions of the BCC model Reaction 11990327 which describes
the catalytic activation of factor V by factor IIa was chosenbecause it represents the main positive feedback within thecommon pathway reaction 119903
58 which is involved in the
activation of factor XI by binding to factor IIa was chosenbecause it represents the main positive feedback in theintrinsic pathway (Figure 1) Moreover preliminary PSA overall reaction constants of the BCC model given in Table 1evidenced that these two reactions are among the mostrelevant steps of the coagulation network
The PSA-1D over 11989627(Figure 7) shows that the CT is very
sensitive to the perturbation of the rate of this reaction whenthe reference value of its constant (ie 119896
27= 2 sdot 10
7Mminus1 sminus1)is either increased or decreased in particular when 119896
27is
very low a plateau in CT is reached since the strength ofthe positive feedback exerted by factor IIa is largely reduceda condition where the contribution of the amplification ofthe hemostatic stimulus (due by the common pathway) tothe formation of the clot is basically not effective On theother side the PSA-1D over 119896
58(Figure 8) shows that while
a decrease of the constant with respect to its reference value
BioMed Research International 13
310
320
330
340
350
360
370
1e5 1e6 1e7 1e8 1e9 1e10 1e11
Clot
ting
time (
s)
k58 (Mminus1sminus1)
Figure 8 Clotting time according to PSA-1D over constant 11989658of reaction 119903
58 IIa + XI rarr IIa-XI The reference value is 1 sdot 108Mminus1 sminus1 the
sweep range is [1 sdot 105 2 sdot 1011]
220
240
260
280
300
320
340
360
380
400
Clot
ting
time (
s)
Factor VIII (M)
Seve
re h
aem
ophi
lia
Mild
hae
mop
hilia
Prot
hrom
botic
stat
e
1eminus
16
1eminus
14
1eminus
12
1eminus
10
1eminus
8
1eminus
6
1eminus
4
1eminus
2
Figure 9 Clotting time at different initial concentrations of factor VIII The reference value is 700 sdot 10minus10M (dashed black line) the sweeprange is [700 sdot 10minus16 700 sdot 10minus4] The blue area indicates a condition in which factor VIII concentration is less than 1 of its physiologicalconcentration corresponding to a situation of severe haemophilia the red area indicates a condition in which factor VIII concentration isbetween 1 and 30 of its physiological concentration corresponding to a situation of mild haemophilia the green area indicates a conditionin which factor VIII concentration is greater than 130 of its physiological concentration corresponding to a situation of hypercoagulability
(ie 11989658= 1 sdot 10
8Mminus1 sminus1) does not have any substantialeffect an increase of its value leads to a progressive increasein the CT This increase is due to the fact that factor IIa issequestered in the formation of a complex with factor XI andhence it is no longer available as a free component in bloodto participate in other reactions especially those reactionsof the extrinsic pathway which principally lead to the clotformation in vivo This behavior highlights that the intrinsicpathway has a secondary role in blood coagulation in vivocompared with the extrinsic pathway as also evidenced byvarious experimental observations [48]
313 Factors VIII IX and II The next set of PSA-1D wasrealized by varying the initial concentrations of factors VIIIIX and II These factors were selected since both an excess
and a deficiency of their concentrations lead to diseasesrelated to blood clotting
The PSA-1D over factor VIII (Figure 9) shows thatincreasing the initial concentration of this factor resultsin decreasing the CT suggesting the possible presence ofhypercoagulable states in these perturbed conditions As amatter of fact high levels of factor VIII cause an increasedrisk of deep vein thrombosis and pulmonary embolism[49] On the other hand individuals with less than 1of the average concentration of factor VIII show a severehaemophilia A characterized by higher CT (Figure 9) andrequire infusions of plasma containing the deficient factorotherwise frequent spontaneous bleeding would occur [50]When the concentration of this factor is between 5 and 30of the average concentration individuals still risk bleeding incase of trauma [50]
14 BioMed Research International
250
300
350
400
450
500
550
600
650
700
Clot
ting
time (
s)
Factor IX (M)
1eminus
14
1eminus
12
1eminus
10
1eminus
8
1eminus
6
1eminus
4
1eminus
2
1e0
Figure 10 Clotting time at different initial concentrations of factor IXThe reference value is 900sdot10minus8M the sweep range is [900sdot10minus14 900sdot10minus2]
0
100
200
300
400
500
600
700
Clot
ting
time (
s)
Factor II (M)
1eminus
12
1eminus
10
1eminus
8
1eminus
6
1eminus
4
1eminus
2
1e0
1e2
Figure 11 Clotting time at different initial concentrations of thrombinThe reference value is 140sdot10minus6M the sweep range is [140sdot10minus12 140sdot100]
The PSA-1D over factor IX (Figure 10) shows only aslight decrease of the CT as the initial concentration offIX increases this is in contrast to recent studies thatdemonstrated how the excess of factor IX leads to anincreased risk of deep vein thrombosis [51] This result canbe explained by considering that (i) a high concentrationof factor IX is not sufficient to bring about coagulationproblems though when the concentration of other factorsis above the average value (yet not at pathological levels)prothrombotic states can be observed (ii) in this model weconsider average values as initial concentrations of factorshowever individuals are characterized by different (balanced)combinations of procoagulant and anticoagulant factor levelsthat altogether contribute to define a unique coagulationphenotype that reflects the developmental environmentalgenetic nutritional and pharmacological influences of eachindividual [52] On the contrary the lack of factor IX causes
haemophilia B characterized by higher CT with respect tothe reference value (Figure 10)
Furthermore by comparing the PSA-1D of factors VIIIand IX (Figures 9 and 10) it is clear that haemophilia A ismore serious than haemophilia B since the CT achieved inconditions of factor VIII deficiency is higher than the CTobtained in the case of factor IX deficiency
In both PSA-1D over factors VIII and IX we observedafter the initial decrease of the CT an unexpected increaseof the CT as the factor concentration increasesThis counter-intuitive behavior arises at very high concentrations of thesefactors (with respect to the average physiological levels) andto the best of our knowledge it was never observed in vivoNonetheless it would be interesting to verify by means of adhoc laboratory experiments if the model correctly describesthe behavior of the BCC even in these conditions or on thecontrary it is not predictive in these extreme situations
BioMed Research International 15
The PSA-1D over factor II (Figure 11) shows a dramaticdecrease of the CT as the initial concentration of this factorincreases (with respect to the average physiological level)This behavior resembles the effects of hypercoagulability (orthrombophilia) a disease caused by mutation G20210A inthe prothrombin gene [53] that causes an increase of theprothrombin level (factor II) in the blood flow resulting inan excessive formation of the active form of this factor thusheightening venous thrombosis risks [54] Hypercoagulabil-ity is usually treated with warfarin therapy or with otheranticoagulants with a similar effect These drugs decrease thecapacity of coagulation factors to become active preventingthe formation of unwanted thrombi
On the other hand when the initial concentration offactor II is low we achieved the effects of prothrombindeficiency a rare autosomal recessive disease that causes atendency to severe bleeding [29 55] As shown in Figure 11 aconcentration equal to 10of the physiological value of factorII (ie 14 sdot 10minus6 M) leads to clotting effects similar to severehaemophilia A
32 PSA-2D of the BCCModel We present here the results ofthe PSA-2D on the BCCmodel where couples of parameterswere varied to analyze the possible effects arising from thecombined perturbation of their values
321 Reactions 11990327
and 11990358 Figure 12 shows the effect of
the simultaneous variation of constants 11989627
and 11989658
(overthe same sweep ranges considered in the two PSA-1DSection 31) This result remarks that reaction 119903
27 involved
in the common pathway has a stronger influence on theBCC and that there is a synergic interplay between thesetwo reactions In particular when the value of 119896
27is low and
the value of 11989658
is high the CT is higher than the valuesachieved when only a single constant is changed because inthis condition both the intrinsic and the common pathwaysare simultaneously inhibited
322 Factor VIII Factor IX and Tissue Factor In the last twoPSA-2Dwe varied the initial concentration of factor VIII andTissue Factor and the initial concentration of factor IX andTissue Factor respectively
The initial concentrations of factors VIII and IX were var-ied over four orders of magnitude using their physiologicalvalues as upper limit for the sweep ranges the concentrationof Tissue Factor was varied over four orders of magnitudetwo above and two below its reference value (see Table 2)The rationale behind this choice is to observe how the BCCmodel in conditions corresponding to different states ofhaemophilia (obtained by decreasing the concentrations offactors VIII and IX) behaves with different initial concentra-tions of the Tissue Factor which is the upstream factor of theextrinsic pathway that is the most important element of theBCC
The results of these PSA-2D show that with respect tothe condition of haemophilia B in the case of haemophiliaA the amount of Tissue Factor (below its reference value)has a negligible influence on the CT as indicated by the
presence of a plateau in Figure 13 on the contrary concerninghaemophilia B a deficiency of Tissue Factor leads to anincrease of the CT especially when factor IX is present in lowconcentrations (Figure 14)
The different results achieved in the two PSA-2D are dueto the presence in the BCC model of a direct interactionbetweenTissue Factor and factor IX bymeans of theTF-fVIIacomplex (see reactions 119903
13 119903
15in Table 1) Indeed the lack
of Tissue Factor directly affects the concentration of activefactor IX which results in a strong alteration of the CT withrespect to physiological conditions
33 CPU versus GPU Performance Comparison In orderto show the relevant speedup achieved by coagSODA wepresent here the comparison of the computational effortrequired by GPU and CPU for the simulation of the BCCmodel The performances of our GPU simulator were com-pared with those obtained using the LSODA algorithmimplemented in the software COPASI [56] executing onthe CPU the same set of simulations that were run on theGPU COPASI is single-threaded and does not exploit thephysical and logical cores of the CPU therefore it representsa good benchmark as a single-node CPU-bound simulator ofbiological systems
In all simulations executed on both GPU and CPU westored 100 samples uniformly distributed in the time intervalconsidered for each simulation that is [0 700] seconds of thedynamics of all chemical species involved in the BCCmodelThe settings for the LSODA algorithm were the followingrelative error 1 sdot 10minus7 absolute error 1 sdot 10minus13 and maximumnumber of internal steps set to 20000
331 Benchmark Details The GPU used for the simulationsis a Nvidia Tesla K20c equipped with 2496 cores organized in13 streamingmultiprocessors GPU clock 706MHz and 5GBofDDR5RAM In all tests coagSODAwas compiled and exe-cuted with version 55 of CUDA libraries Even though thisGPU has compute capability 35 and is based on the GK110Kepler architecture currently coagSODA does not exploitany of the new features (eg Dynamic Parallelism Hyper-QRemote DMA) with the exception of the new ISA encodingwhich allows threads to exploit an increased number ofregisters (255 instead of 63) reducing register spilling intoglobal memory and increasing performances [57] Moreoveras described in Section 22 the Kepler architecture offers thepossibility to reconfigure the 64KB on-chip cache balancingbetween L1 cache and shared memory Since coagSODAexploits the shared memory to reduce the latencies duringthe access to the concentration values of the BCC model weopted for the following configuration 48KB for the sharedmemory 16 KB for the L1 cache
The performance of the GPU was compared against apersonal computer equippedwith a dual-core CPU Intel Corei5 frequency 23 GHz 4GB of DDR3 RAM running theoperating systemMacOSXLion 1075The software used as acoagSODA-equivalent single-threaded CPU implementationis COPASI version 48 (build 35)
16 BioMed Research International
280
300
320
340
360
380
400
420
440
1e41e6
1e91e11 1e5
1e7
1e9
1e11
280 320 360 400 440
Clot
ting
time (
s)
k 58(M
minus1 sminus1 )
k27 (Mminus1sminus1)
Figure 12 Clotting time according to PSA-2D over reaction constants 11989627and 119896
58 The sweep ranges are [2 sdot 104 2 sdot 1010] and [1 sdot 105 2 sdot 1011]
respectively
100
200
300
400
Tissue factor (M) Factor VIII
(M)
Clot
ting
time (
s)
100
150
200
250
300
350
400
450
1eminus131eminus11
1eminus91eminus14
1eminus12
1eminus10
Figure 13 Clotting time at different initial concentrations of Tissue Factor and factor VIII The sweep ranges are [500 sdot 10minus14 500 sdot 10minus10]and [700 sdot 10minus14 500 sdot 10minus10] respectively
A direct comparison of the performances between thesetwo different architectures and implementations is not aneasy task since CPUs are optimized for single-thread exe-cution and exploit a large number of optimizations (eghigher clock frequency instruction caching pipelining outof order execution and branch prediction) whereas GPUsare optimized for graphics processing and parallel exe-cution of identical code relying only on multilevel datacaching For this reason in this paper we propose anempirical comparison of the performances based on therunning times of identical batches of simulations of the BCCmodel
332 Benchmark Results In Figure 15 we show the compar-ison of the running times required to run several batchesof simulations executed to carry out the PSA-1D over thereaction constant 119896
27 The choice of comparing the perfor-
mances of the GPU and CPU implementations by executingbatches of simulations that are related to a PSA instead ofrunning 119899 independent but identical simulations (ie allcharacterized by the same parameterization of the model)is due to the fact that these results represent a more realisticscenario in the computational analysis of biological systemswhereby it is useful to investigate large search spaces ofparameters corresponding to different perturbed conditionsof themodel [26]Moreover for the evaluation of the running
time the execution of a batch of 119899 identical deterministicsimulations would be futile The figure clearly shows thatcoagSODA always performs better than the CPU counter-part In particular while the CPU performance increaseslinearly with the number of simulations the running timesare strongly reduced on the GPU in this case the overallrunning time roughly corresponds to the time required forthe execution of the slowest simulations This is due to thefact that different parameterizations may require differenttime steps for LSODA to converge and the execution ofa block terminates as long as all the threads it containsterminate In turn the execution of a kernel terminates whenall blocks terminate for this reason a single simulationwhose dynamics requires more steps than the others mayaffect the overall running time
In Table 3 we report the running times of all batchesof simulations along with the speedup achieved on theGPU In particular these results highlight that the advantageof exploiting the GPUs for the simulation of the BCCmodel becomes more evident as the number of simula-tions increases with a 181times speedup when a PSA with10000 different parameterizations is executed Therefore theGPU accelerated analysis of the BCC model with coag-SODA represents a novel relevant computational means toinvestigate the behavior of this complex biological systemunder nonphysiological conditions and could be exploited
BioMed Research International 17
0
200
400
600
Tissue factor (M) Factor IX (M
)
Clot
ting
time (
s)
0
100
200
300
400
500
600
700
1eminus12
1eminus10
1eminus8
1eminus131eminus11
1eminus9
Figure 14 Clotting time at different initial concentrations of Tissue Factor and factor IXThe sweep ranges are [500 sdot 10minus14 500 sdot 10minus10] and[900 sdot 10
minus12 900 sdot 10
minus8] respectively
5 10 50 100 500 1000 5000 10000
Runn
ing
time (
s)
Number of simulations
CPUGPU
1e2
1e3
1e4
1e5
1e6
1e7
Figure 15 Comparison of the computational time required to execute an increasing number of simulations of the BCCmodel on CPU (IntelCore i5 23 GHz) and GPU (Nvidia Tesla K20c GPU clock 706MHz) The computational time is expressed in seconds The time valuesrelated to 1000 5000 and 10000 simulations on the CPU were estimated by regression (see also Table 3)
Table 3 CPU versus GPU performance comparison
Number ofsimulations CPU time (sec) GPU time (sec) Speedup
5 3808 1278 298times10 10604 2329 456times50 51704 8058 642times100 116528 10977 1061times500 536058 17394 3081times1000 1073582lowast 19982 5373times5000 5363507lowast 30960 1732times10000 10724975lowast 58951 1819timeslowastValues estimated by regression
to efficiently determine the response of the BCC to differenttherapeutical drugs
4 Conclusions
Thanks to their high-performance computing capabilitiesand the very low costs GPUs nowadays represent a suitable
technology for the development and the application of par-allel computational methods for in silico analysis of complexbiological systems However the implementation of efficientcomputational tools able to fully exploit the large potentialityof GPUs is still challenging since good programming skillsare required to implement GPU-based algorithmic methodsand to handle specific features as an optimal usage ofmemoryor the communication bandwidth between GPU and CPUMoreover algorithms cannot be directly ported on the GPUbecause of the limited programming capabilities allowed byGPUkernels as amatter of fact they need to be redesigned tomeet the requirements of the underlying SIMD architecture
In this work we presented coagSODA a GPU-poweredsimulator specifically developed to carry out fast parallelsimulations of the BCC model coagSODA was designed tooffer a black-box solution usable by any final user in aneasy way It relies on cupSODA [24] a numerical integratorfor ODEs that we previously implemented for the GPUarchitecture based on the LSODA algorithm and capableof automatically translating a reaction-based model into aset of coupled ODEs In addition to mass-action kinet-ics coagSODA implements specific functions to compute
18 BioMed Research International
the kinetics of Hill function based reactions such as thoseinvolved in the platelets activity of the BCC model coag-SODA exploits the massive parallel capabilities of modernGPUs and our results demonstrated that it can achieve arelevant reduction of the computational time required toexecutemany concurrent and independent simulations of theBCC dynamics The mutual independence of the simulationsallows fully exploiting the underlying SIMD architecturemoreover coagSODA benefits from an additional speedupthanks to our choice of storing the state of the systeminto the low-latency shared memory (a solution that wasalready implemented in cupSODA) Since the BCC modelis large (96 reactions 71 chemical species) a large amountof shared memory was assigned to each thread stronglyreducing the theoretical occupancy of the GPU that is theratio of active warps with respect to the maximum numberof warps supported by each streaming multiprocessor of theGPU However the results of the analysis presented in thiswork performed on the BCC model show that coagSODAachieves a relevant boost with respect to a reference CPUimplementation For instance in the case of 10000 simula-tions we achieved a noticeable 181times speedup Interestinglythe performances of coagSODA are better than COPASI evenfor small batches of simulations These results indicate thatfor biological models consisting in many reactions and manyspecies our GPU implementation of LSODA becomes moreprofitable than the CPU counterpart as the number of con-current simulations increases making it suitable especiallywhen performing demanding computational analysis suchas for example parameter sweep parameter estimation orsensitivity analysis
As a test case in this paper we presented several parame-ter sweep analyses over reaction constants and initial concen-trations of factors involved in the BCC model Other com-putational analyses on mathematical models of this pathwaywere previously presented For instance in [58] a parameterestimation of the reaction constants of different modelsinvolving the activation of factor X factor V prothrombinand the inactivation of factors was performed The aim ofthis analysis was to discriminate between different modelsto unravel the mechanisms on the basis of the BCC Similaranalyses were executed on a model describing thrombingeneration in plasma since the reaction mechanism thereaction constants and initial concentrations were unknown[59] In [34] a sensitivity analysis of a model consisting of44 species over 34 chemical species was presented reactionconstants were varied between 10 and 1000 of theirreference value to the aim of identifying the most influentialfactors of the BCC
All these analyses were performed bymeans of sequentialsimulations of the models under investigation thereforein general only small batches of simulations could be run(for instance in [34] only 836 simulations were executed toexecute the sensitivity analysis of the model) On the otherhand in this paper we presented the results of different PSAefficiently executed bymeans of coagSODA overall providinguseful information regarding unknownparameters and inter-esting insights into the functioning of the BCC A thoroughsensitivity analysis of the whole BCC model consisting in
around 5 sdot 105 simulations is currently in progress on GPUsby our group exploiting the great computational efficiency ofcoagSODA
The results of our computational analyses should benow validated by means of specifically designed laboratoryexperiments In particular we identified a plausible valuefor the constant of the reaction describing the autoactivationof factor XII (since no reference values can be found inliterature) moreover the results of the PSA over factor IXsuggest that a deficiency of this factor is not enough to causesevere bleeding disorders as haemophilia B but an alterationof other factors seems to be necessary for the occurrence ofsuch condition (eg the lack of Tissue Factor as suggestedby the PSA-2D over Tissue Factor and factor IX) Finallythe PSA over factors VIII and IX showed in a situationcharacterized by very high concentrations of these factorsa counterintuitive behavior in which the clotting time isincreased with respect to the value obtained in physiologicalconditions Indeed it would be interesting to design ad hoclaboratory experiments to verify if the BCCmodel is actuallypredictive in such extreme situations
The coagSODA software and the SBML version of theBCC model used in this work are available from the authorsupon request
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This work is partially supported by SYSBIO Centre forSystems Biology
References
[1] B B Aldridge JM BurkeDA Lauffenburger andP K SorgerldquoPhysicochemicalmodelling of cell signalling pathwaysrdquoNatureCell Biology vol 8 no 11 pp 1195ndash1203 2006
[2] I-C Chou and E O Voit ldquoRecent developments in parameterestimation and structure identification of biochemical andgenomic systemsrdquoMathematical Biosciences vol 219 no 2 pp57ndash83 2009
[3] D Besozzi P Cazzaniga D Pescini G Mauri S Colomboand EMartegani ldquoThe role of feedback control mechanisms onthe establishment of oscillatory regimes in the RascAMPPKApathway in S cerevisiaerdquo EURASIP Journal on Bioinformaticsand Systems Biology vol 2012 no 10 2012
[4] A Saltelli M Ratto T Andres et al Global Sensitivity AnalysisThe Primer Wiley-Interscience 2008
[5] A Raue C Kreutz T Maiwald et al ldquoStructural and practicalidentifiability analysis of partially observed dynamical modelsby exploiting the profile likelihoodrdquo Bioinformatics vol 25 no15 pp 1923ndash1929 2009
[6] A Drager M Kronfeld M J Ziller et al ldquoModeling metabolicnetworks in C glutamicum a comparison of rate laws incombination with various parameter optimization strategiesrdquoBMC Systems Biology vol 3 article 5 2009
BioMed Research International 19
[7] D Besozzi P Cazzaniga G Mauri D Pescini and L Van-neschi ldquoA comparison of genetic algorithms and particle swarmoptimization for parameter estimation in stochastic biochem-ical systemsrdquo in Proceedings of the 7th European ConferenceEvolutionary Computation Machine Learning and Data Miningin Bioinformatics (EvoBIO rsquo09) C Pizzuti M Ritchie and MGiacobini Eds vol 5483 of Lecture Notes in Computer Sciencepp 116ndash127 Springer 2009
[8] M S Nobile D Besozzi P Cazzaniga GMauri andD PescinildquocupSODA aGPU-basedmulti-swarmPSOmethod for param-eter estimation in stochastic biological systems exploitingdiscrete-time target seriesrdquo in Proceedings of the 10th EuropeanConference Evolutionary Computation Machine Learning andData Mining in Bioinformatics (EvoBIO rsquo12) M Giacobini LVanneschi and W Bush Eds vol 7246 of Lecture Notes inComputer Science pp 74ndash85 Springer
[9] M S Nobile D Besozzi P Cazzaniga D Pescini and GMaurildquoReverse engineering of kinetic reaction networks by meansof Cartesian Genetic Programming and Particle Swarm Opti-mizationrdquo in Proceedings of the IEEE Congress on EvolutionaryComputation (CEC rsquo13) pp 1594ndash1601 June 2013
[10] M S Nobile ldquoEvolutionary inference of biochemical inter-action networks accelerated on Graphics Processing Unitsrdquoin Proceedings of the 11th International Conference on HighPerformance Computingamp Simulation 2013 (HPCS rsquo13) pp 668ndash670 Helsinki Finland 2013
[11] K Kentzoglanakis and M Poole ldquoA swarm intelligence frame-work for reconstructing gene networks searching for bio-logically plausible architecturesrdquo IEEEACM Transactions onComputational Biology and Bioinformatics vol 9 no 2 pp 358ndash371 2012
[12] J R Koza W Mydlowec G Lanza Y Jessen and M A KeaneldquoAutomatic computational discovery of chemical reaction net-works using genetic programmingrdquo inComputationalDiscoveryof Scientific Knowledge S Dzeroski and L Todorovski Edsvol 4660 of Lecture Notes in Computer Science pp 205ndash227Springer 2007
[13] S Ando E Sakamoto and H Iba ldquoEvolutionary modeling andinference of gene networkrdquo Information Sciences vol 145 no3-4 pp 237ndash259 2002
[14] L Dematte and D Prandi ldquoGPU computing for systemsbiologyrdquo Briefings in Bioinformatics vol 11 no 3 Article IDbbq006 pp 323ndash333 2010
[15] J L Payne N A Sinnott-Armstrong and J HMoore ldquoExploit-ing graphics processing units for computational biology andbioinformaticsrdquo Interdisciplinary Sciences Computational LifeSciences vol 2 no 3 pp 213ndash220 2010
[16] M J Harvey and G De Fabritiis ldquoA survey of computationalmolecular science using graphics processing unitsrdquoWiley Inter-disciplinary Reviews Computational Molecular Science vol 2no 5 pp 734ndash742 2012
[17] Y Zhou J Liepe X Sheng M P H Stumpf and C BarnesldquoGPU accelerated biochemical network simulationrdquo Bioinfor-matics vol 27 no 6 Article ID btr015 pp 874ndash876 2011
[18] M Vigelius A Lane and B Meyer ldquoAccelerating reaction-diffusion simulations with general-purpose graphics processingunitsrdquo Bioinformatics vol 27 no 2 Article ID btq622 pp 288ndash290 2011
[19] D J Wilkinson ldquoStochastic modelling for quantitative descrip-tion of heterogeneous biological systemsrdquo Nature ReviewsGenetics vol 10 no 2 pp 122ndash133 2009
[20] J C Butcher ldquoTrees and numerical methods for ordinarydifferential equationsrdquo Numerical Algorithms vol 53 no 2-3pp 153ndash170 2010
[21] D T Gillespie ldquoStochastic simulation of chemical kineticsrdquoAnnual Review of Physical Chemistry vol 58 pp 35ndash55 2007
[22] L Petzold ldquoAutomatic selection of methods for solving stiffand nonstiff systems of ordinary differential equationsrdquo SIAMJournal of Scientific and Statistical Computing vol 4 no 1 pp136ndash148 1983
[23] Y Cao D T Gillespie and L R Petzold ldquoEfficient step sizeselection for the tau-leaping simulation methodrdquo The Journalof Chemical Physics vol 124 no 4 2006
[24] M S Nobile D Besozzi P Cazzaniga GMauri andD PescinildquocupSODA a CUDA-powered simulator of mass-action kinet-icsrdquo in Proceedings of the 12th International Conference onParallel Computing Technologies (PaCT rsquo13) V Malyshkin Edvol 7979 of Lecture Notes in Computer Science pp 344ndash357
[25] M S Nobile P Cazzaniga D Besozzi and G Mauri ldquoGPU-accelerated simulations of mass-action kinetics models withcupSODArdquoThe Journal of Supercomputing 2014
[26] M S Nobile P Cazzaniga D Besozzi D Pescini and GMauri ldquocuTauLeaping a GPU-powered tau-leaping stochasticsimulator for massive parallel analyses of biological systemsrdquoPLoS ONE vol 9 no 3 2014
[27] K Rogers Blood Physiology and CirculationTheHuman BodyBritannica Educational Publishing 2010
[28] A D Michelson Platelets Elsevier 2nd edition 2007[29] R W Colman J Hirsh V J Marder A W Clowes and J
N George Hemostasis and Thrombosis Basic Principles andClinical Practice Lippincott Williams ampWilkins 2006
[30] M S Chatterjee W S Denney H Jing and S L DiamondldquoSystems biology of coagulation initiation Kinetics of thrombingeneration in resting and activated human bloodrdquo PLoS Com-putational Biology vol 6 no 9 Article ID e1000950 2010
[31] M Bellini D Besozzi P Cazzaniga GMauri andM S NobileldquoSimulation and analysis of the blood coagulation cascadeaccelerated on GPUrdquo in Proceedings of the 22nd EuromicroInternational Conference on Parallel Distributed and Network-Based Processing (PDP) pp 590ndash593 IEEE Computer SocietyConference Publishing Services 2014
[32] T Wajima G K Isbister and S B Duffull ldquoA comprehensivemodel for the humoral coagulation network in humansrdquo Clini-cal Pharmacology and Therapeutics vol 86 no 3 pp 290ndash2982009
[33] K Lo W S Denney and S L Diamond ldquoStochastic modelingof blood coagulation initiationrdquo Pathophysiology of Haemostasisand Thrombosis vol 34 no 2-3 pp 80ndash90 2006
[34] C M Danforth T Orfeo K G Mann K E Brummel-Ziedins and S J Everse ldquoThe impact of uncertainty in a bloodcoagulation modelrdquo Mathematical Medicine and Biology vol26 no 4 Article ID dqp011 pp 323ndash336 2009
[35] S D Lewis P P Shields and J A Shafer ldquoCharacterizationof the kinetic pathway for liberation of fibrinopeptides duringassembly of fibrinrdquoThe Journal of Biological Chemistry vol 260no 18 pp 10192ndash10199 1985
[36] M F Hockin K C Jones S J Everse and K G Mann ldquoAmodel for the stoichiometric regulation of blood coagulationrdquoThe Journal of Biological Chemistry vol 277 no 21 pp 18322ndash18333 2002
[37] K Brummel-Ziedins C YVossen F R Rosendaal KUmezakiand K G Mann ldquoThe plasma hemostatic proteome thrombin
20 BioMed Research International
generation in healthy individualsrdquo Journal of Thrombosis andHaemostasis vol 3 no 7 pp 1472ndash1481 2005
[38] K E Brummel-Ziedins C Y Vossen S Butenas K G Mannand F R Rosendaal ldquoThrombin generation profiles in deepvenous thrombosisrdquo Journal of Thrombosis and Haemostasisvol 3 no 11 pp 2497ndash2505 2005
[39] S D Bungay P A Gentry and R D Gentry ldquoA mathematicalmodel of lipid-mediated thrombin generationrdquo MathematicalMedicine and Biology vol 20 no 1 pp 105ndash129 2003
[40] S Butenas T Orfeo M T Gissel K E Brummel and K GMann ldquoThe significance of circulating factor IXa in bloodrdquoThe Journal of Biological Chemistry vol 279 no 22 pp 22875ndash22882 2004
[41] D Voet and J G Voet Biochemistry Wiley 2011[42] A C Guyton and J E Hall Textbook of Medical Physiology
Saunders 2010[43] M F M James and A M Roche ldquoDose-response relationship
between plasma ionized calcium concentration and thrombe-lastographyrdquo Journal of Cardiothoracic and Vascular Anesthesiavol 18 no 5 pp 581ndash586 2004
[44] D L Nelson and M M Cox Lehninger Principles of Biochem-istry W H Freeman Company 2004
[45] R M Farber ldquoTopical perspective on massive threading andparallelismrdquo Journal of Molecular Graphics and Modelling vol30 pp 82ndash89 2011
[46] OWolkenhauerMUllahWKolch andK-HCho ldquoModelingand simulation of intracellular dynamics choosing an appropri-ate frameworkrdquo IEEETransactions onNanobioscience vol 3 no3 pp 200ndash207 2004
[47] B H Estridge A P Reynolds and N J Walters Basic MedicalLaboratory Techniques Cengage Learning 4th edition 2000
[48] T Renne A H Schmaier K F Nickel M Blomback and CMaas ldquoIn vivo roles of factor XIIrdquo Blood vol 120 no 22 pp4296ndash4303 2012
[49] P V Jenkins O Rawley O P Smith and J S ODonnellldquoElevated factor VIII levels and risk of venous thrombosisrdquoBritish Journal of Haematology vol 157 no 6 pp 653ndash663 2012
[50] C E Walsh and K M Batt ldquoHemophilia clinical gene therapybrief reviewrdquo Translational Research vol 161 no 4 pp 307ndash3122013
[51] A Van Hylckama Vlieg I K Van Der Linden R M Bertinaand F R Rosendaal ldquoHigh levels of factor IX increase the risk ofvenous thrombosisrdquo Blood vol 95 no 12 pp 3678ndash3682 2000
[52] K Brummel-Ziedins ldquoModels for thrombin generation and riskof diseaserdquo Journal ofThrombosis and Haemostasis vol 11 no 1pp 212ndash223 2013
[53] U Seligsohn and A Lubetsky ldquoGenetic susceptibility to venousthrombosisrdquoTheNew England Journal of Medicine vol 344 no16 pp 1222ndash1231 2001
[54] I Martinelli P Bucciarelli and P M Mannucci ldquoThromboticrisk factors basic pathophysiologyrdquo Critical Care Medicine vol38 no 2 pp S3ndashS9 2010
[55] K Brummel-Ziedins A Undas T Orfeo et al ldquoThrombingeneration in acute coronary syndrome and stable coronaryartery disease dependence on plasma factor compositionrdquoJournal ofThrombosis andHaemostasis vol 6 no 1 pp 104ndash1102008
[56] S Hoops R Gauges C Lee et al ldquoCOPASImdasha complexpathway simulatorrdquo Bioinformatics vol 22 no 24 pp 3067ndash3074 2006
[57] Nvidia NVIDIArsquos Next Generation CUDA Compute Architec-ture Kepler GK110 2012
[58] W Stortelder W Hemker and H Hemker ldquoMathematicalmodelling in blood coagulation simulation and parameterestimationrdquoModelling Analysis and Simulation vol 20 pp 1ndash111997
[59] S M C Orfao G Jank and K Mottaghy ldquoA new mathematicalapproach to modelling thrombin generationrdquo InternationalJournal of Artificial Organs vol 29 no 7 pp 703ndash708 2006
[60] Nvidia Nvidia CUDA C Programming Guide v50 2012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Anatomy Research International
PeptidesInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
International Journal of
Volume 2014
Zoology
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Molecular Biology International
GenomicsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
BioinformaticsAdvances in
Marine BiologyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Signal TransductionJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
BioMed Research International
Evolutionary BiologyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Biochemistry Research International
ArchaeaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Genetics Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in
Virolog y
Hindawi Publishing Corporationhttpwwwhindawicom
Nucleic AcidsJournal of
Volume 2014
Stem CellsInternational
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Enzyme Research
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Microbiology
2 BioMed Research International
of standard desktop computers therefore requiring high-performance computing solutions After the introductionof general-purpose GPUs and of Compute Unified DeviceArchitecture (CUDA Nvidiarsquos GPU programming language)the adoption of these graphics engines largely increasedin research fields as Bioinformatics Systems Biology andComputational Biology (see an overview in [14ndash16]) Anywaydespite the remarkable advantages concerning the computa-tional speedup computing with GPUs usually requires thedevelopment and the implementation of ad hoc algorithmssince GPU-based programming substantially differs fromCPU-based computing as a consequence scientific appli-cations of GPUs might undergo the risk of remaining aniche for few specialists To avoid such limitations severalpackages and software tools have recently been released(see eg [16ndash18]) so that also users with no knowledgeof GPUs hardware and programming can access the high-performance computing power of graphics engines
To investigate the dynamics of biological systems eitherdeterministic or stochastic approaches can be exploited [19]which are based on numerical integration (eg Eulerrsquos orRunge-Kutta methods [20]) or on Markov processes (egGillespiersquos algorithm [21]) respectively To date the most effi-cient algorithms to integrate a system of ordinary differentialequations (ODEs) or to perform stochastic simulations ofreaction-basedmodels are LSODA [22] and tau-leaping [23]respectively In [24ndash26] we previously presented cupSODAand cuTauLeaping the GPU-powered implementation ofLSODA and tauleaping respectively cupSODA allows torun parallel deterministic simulations of a given mass-actionbased system of biochemical reactions using the LSODAalgorithm cuTauLeaping represents a novel restructuring ofthe tau-leaping workflow that fits the GPU architecture andavoids any inefficiency drawback for coarse-grain massiveparallel stochastic simulations
In this work we introduce coagSODA an extension ofcupSODA that was specifically designed for the analysis ofa model of the blood coagulation cascade (BCC) Blood is thesubject of an intense scientific research thanks to its key roleinmaking diagnosis of numerous diseases [27] Humans haveevolved a complex hemostatic system that is able to maintainblood in a fluid state and allow the circulation through anintricate network of vessels in particular the presence ofseveral fine-tuned feedback mechanisms in the BCC allowskeeping all blood components within appropriate concen-tration ranges The BCC consists in a complex network ofcellular reactions which under physiological conditions invivo are inhibited by the presence of intact endothelium [28]Anyway in response to any vascular injury the hemostaticsystem is able to stop the blood leakage by rapidly sealing thedefects in the vesselsrsquo wall [29]
In order to investigate the variations in blood coagu-lation components among individuals and to understandthe corresponding response of the system to perturbedconditions we consider here a computational perspectiveto study the BCC we analyze the alterations (prolongationor reduction) of the time required to form the clot (iethe clotting time) by exploiting a reduced version of themathematical model defined in [30] This model describes
the intrinsic extrinsic and common pathways of the BCCand more importantly it accounts for platelets activationas well as the presence of several inhibitors (eg the TissueFactor Pathway Inhibitor antithrombin III and C1-inhibitor)[31]
Numerous mathematical models of blood coagulationwere developed in the last years as they represent a useful toolfor systematic studies of the intricate network of the coagu-lation cascade and allow obtaining a suitable reconstructionof empirical observations (see eg [32ndash34]) The earliestmodels considered only simple steps of the whole BCC suchas the conversion of the clot fibrin by thrombin [35] latelyHockin et al [36] developed a comprehensive ODE-basedmodel of the extrinsic blood coagulation system This modelwas then considered as reference by several research groupsto investigate the thrombotic risk in healthy and ill popula-tions [37 38] or to understand other complex biochemicalprocesses for instance in [39] the roles of protein C proteinS and phospholipid surface actions were considered whilein [40] the influence of trace amounts of key coagulationproteases on thrombin generation was investigated Recentlyother works modeling the blood clotting process in a com-prehensive manner have been published besides the alreadymentioned model developed by Chatterjee et al [30] wemention the model defined by Wajima et al [32] whichsimulates the intrinsic extrinsic and common pathways thevitamin K cycle the therapy with the anticoagulant drugswarfarin and heparin and the laboratory tests PT and aPTTas well as the Taipan snake bite which causes coagulopathy
coagSODA the GPU-accelerated simulator that wepresent and exploit in this work for the analysis of the BCCmodel is a user-friendly and efficient tool that circumventsthe need of manually defining the system of ODEs thatdescribe the blood coagulation network More preciselycoagSODA is able to automatically derive the system of(mass-action and Hill function-based) ODEs and then per-form their numerical integration starting from the given set of96 biochemical reactions which fully describe the molecularinteractions between all the species involved in the BCCin vivo We show that coagSODA allows us to efficientlyexecute a large number of parallel deterministic simulationsof the BCC at a considerable reduced computational costwith respect to CPUs In particular we exploit coagSODAto carry out one-dimensional and bi-dimensional parametersweep analysis of the BCC to the purpose of investigatingthe prolongation and the reduction of the clotting time inresponse to perturbed values of some reaction constants andof the initial concentration of somemolecular species chosenaccording to their meaning within the whole pathway
The paper is structured as follows In Section 2 we fullydescribe the mechanistic model of the BCC used in thiswork and present the simulation method at the basis ofcupSODA tool to introduce the coagSODA simulator InSection 3 we present the results obtained from the parametersweep analysis of the BCC model as well as a comparison ofthe computational performance of coagSODAwith respect toaCPU-based implementation of LSODA Finally in Section 4we conclude the paper with a discussion of the presentedwork and future research directions
BioMed Research International 3
2 Materials and Methods
21 A Mechanistic Model of the Blood Coagulation CascadeBlood is an essential component in human life whoseprimary functions are to feed cells by delivering a multitudeof nutrients such as oxygen and to carry away the cellularwastes such as carbon dioxide Specialized cells and fluidsin blood perform many physiological functions and can beisolated and analyzed through specific laboratory tests givingthe opportunity to settle a personrsquos health condition Allblood components are kept within appropriate concentrationranges by means of fine-tuned regulatory mechanisms ruledby several feedback controls the constancy of blood com-position is maintained thanks to the circulation through anintricate network of vessels In particular humans evolved acomplex hemostatic system that under physiological condi-tions maintains blood in a fluid state however in responseto any vascular injury this system is able to rapidly react andseal the defects in the vesselsrsquo wall in order to stop the bloodleakage [29] Indeed the circulatory system is self-sealingotherwise a continuous blood flow from even the smallestwound would become a threat for the individualrsquos life
To allow blood coagulation in humans there exist 13blood clotting proteins called coagulation factors which areusually designated by Roman numerals I through XIII Asa consequence of a vascular injury platelets become activeand the Tissue Factor (TF also called factor III) is exposedin the subendothelial tissue starting the blood coagulationcascade (BCC) The ultimate goal of the BCC is to convertprothrombin (factor II) into thrombin (factor IIamdashie theactive factor II) the enzyme that catalyzes the formation ofa clot Traditionally the BCC is divided into the extrinsic andintrinsic pathways both leading to the activation of factor X[41] The last part of the cascade downstream of this factoris called the common pathway and leads to the formationof fibrin monomers whose polymers finally constitute thebackbone of the clot
Excluding thrombin all the enzymes involved in bloodclotting are characterized by a low activity which increasesupon binding to a specific protein cofactor (eg factors Vand VIII) or to appropriate phospholipid surfaces (eg theplasmamembranes of active platelets) [41] Even calcium ionshave a central role in coagulation since they are essentialto start and enhance numerous reactions without calciumions the blood coagulation cannot occur [42] In the BCCpathways the activity of the various active proteases is limitedby several inhibitory factors which allow regulating thewhole cascade When the hemostatic system is unregulatedthrombosis (ie the formation of a blood clot obstructing theblood flow in vessels) may occur due to impairment in theinhibitory pathway or because the functioning of the naturalanticoagulant processes is overwhelmed by the strength of thehemostatic stimulus [29]
The BCC model we consider in this work is a slightlyreduced version of the ldquoPlatelet-Plasmardquo deterministic modeldefined in [30] built upon a previous model [36] whichdescribes all parts of blood coagulation the plateletsactivation and aggregation the extrinsic intrinsic andcommon pathways (with the exception of factor XIII)
the action of several inhibitory molecules (Tissue Fac-tor Pathway Inhibitor antithrombin III C1-inhibitor 1205721-antitrypsin and 1205722-antiplasmin) In addition to simulate thecoagulation process in vitro Chatterjee et al also modeledthe action of corn trypsin inhibitor (CTI) which inhibits theactivation of the so-called contact system as well as the actionof the fluorogenic substrate Boc-VPR-MCA widely used inlaboratories for thrombin titration [30] The role of calciumwas not explicitly included in the model but considered asa nonlimiting factor as in living organisms this ion veryrarely drops to such low levels able to alter the kinetics of theformation of the fibrin clot [43]
Since the aim of this work is the investigation of bloodcoagulation in vivo we exclude a small set of reactions givenin [30] that have no effect on the clotting time To be moreprecise we do not consider the reactions occurring in vitro(namely entries 28 and 35 in Table 1 in [30]) as well asthe reactions downstream the fibrinogen conversion that isthe interactions between the fibrin polymers thrombin andantithrombin III (namely entries 55 56 and 57 in Table 1 in[30])
For the sake of completeness we describe in Table 1 theBCC model considered in this work overall consisting in 96reactions among 71 molecular species A graphical sketch ofthemainmolecular interactions among the BCC componentsis given in Figure 1
The model can be partitioned into four functional mod-ules
(1) Thefirstmodule corresponds to the extrinsic pathwaywhich consists in
(i) the formation of a complex between TissueFactor and factor VII (modeled by reactions1199031 119903
4)
(ii) the activation of factor VII by the complexbetween Tissue Factor and factor VIIa (reaction1199035)
(iii) the activation of factor IX by the complexbetween Tissue Factor and factor VIIa (reac-tions 119903
13 11990314 11990315) and by factor VIIa (reactions
11990378 11990379 11990380)
(iv) the activation of factor X by the complexbetween Tissue Factor and factor VIIa(1199038 119903
12) and by factor VIIa (119903
81 11990382 11990383)
(2) The second module corresponds to the intrinsic path-way which consists in
(i) the formation of a complex between factor VIIIaand factor IXa (modeled by reactions 119903
18and
11990319)
(ii) the activation of factor X by the complexbetween factor VIIIa and factor IXa (reactions11990320 11990321 11990322) and by factor IXa (reactions 119903
72 11990373
11990374)
(iii) the activation of factor XII by factor XII itself(reaction 119903
44) by factor XIIa (reactions 119903
45 11990346
11990347) and by kallikrein (reactions 119903
51 11990352 11990353)
4 BioMed Research International
Table 1 Reaction-based model of the blood coagulation cascade (reduced version of the ldquoPlatelet-Plasmardquo model described in [30]) Themodel consists in 96 reactions among 71 molecular species With the exception of reaction 119903
44(see Section 31) the values of all reaction
constants were taken from [30]
119899 Reactants Products Constant (119896)1199031
TF + fVII TF-fVII 320 sdot 106Mminus1 sminus1
1199032
TF-fVII TF + fVII 310 sdot 10minus2 sminus1
1199033
TF + fVIIa TF-fVIIa 230 sdot 107Mminus1 sminus1
1199034
TF-fVIIa TF + fVIIa 310 sdot 10minus5 sminus1
1199035
TF-fVIIa + fVII TF-fVIIa + fVIIa 440 sdot 105Mminus1 sminus1
1199036
fXa + fVII fXa + fVIIa 130 sdot 107Mminus1 sminus1
1199037
fIIa + fVII fIIa + fVIIa 230 sdot 104Mminus1 sminus1
1199038
TF-fVIIa + fX TF-fVIIa-fX 250 sdot 107Mminus1 sminus1
1199039
TF-fVIIa-fX TF-fVIIa + fX 105 sdot 10minus2 sminus1
11990310
TF-fVIIa-fX TF-fVIIa-fXa 600 sminus1
11990311
TF-fVIIa-fXa TF-fVIIa + fXa 1900 sminus1
11990312
TF-fVIIa + fXa TF-fVIIa-fXa 220 sdot 107Mminus1 sminus1
11990313
TF-fVIIa + fIX TF-fVIIa-fIX 100 sdot 107Mminus1 sminus1
11990314
TF-fVIIa-fIX TF-fVIIa + fIX 240 sminus1
11990315
TF-fVIIa-fIX TF-fVIIa + fIXa 180 sminus1
11990316
fII + fXa fIIa + fXa 750 sdot 103Mminus1 sminus1
11990317
fIIa + fVIII fIIa + fVIIIa 200 sdot 107Mminus1 sminus1
11990318
fVIIIa + fIXa fIXa-fVIIIa 100 sdot 107Mminus1 sminus1
11990319
fIXa-fVIIIa fVIIIa + fIXa 100 sdot 10minus4 sminus1
11990320
fIXa-fVIIIa + fX fIXa-fVIIIa-fX 100 sdot 108Mminus1 sminus1
11990321
fIXa-fVIIIa-fX fIXa-fVIIIa + fX 100 sdot 10minus5 sminus1
11990322
fIXa-fVIIIa-fX fIXa-fVIIIa + fXa 820 sminus1
11990323
fVIIIa fVIIIa1-L + fVIIIa2 600 sdot 10minus5 sminus1
11990324
fVIIIa1-L + fVIIIa2 fVIIIa 220 sdot 104 Mminus1 sminus1
11990325
fIXa-fVIIIa-fX fVIIIa1-L + fVIIIa2 + fX + fIXa 100 sdot 10minus3 sminus1
11990326
fIXa-fVIIIa fVIIIa1-L + fVIIIa2 + fIXa 100 sdot 10minus3 sminus1
11990327
fIIa + fV fIIa + fVa 200 sdot 107Mminus1 sminus1
11990328
fXa + fVa fXa-fVa 400 sdot 108Mminus1 sminus1
11990329
fXa-fVa fXa + fVa 02 sminus1
11990330
fXa-fVa + fII fXa-fVa-fII 100 sdot 108Mminus1 sminus1
11990331
fXa-fVa-fII fXa-fVa + fII 10300 sminus1
11990332
fXa-fVa-fII fXa-fVa + fmIIa 6350 sminus1
11990333
fXa-fVa + fmIIa fXa-fVa + fIIa 150 sdot 107Mminus1 sminus1
11990334
fXa + TFPI fXa-TFPI 900 sdot 105Mminus1 sminus1
11990335
fXa-TFPI fXa + TFPI 360 sdot 10minus4 sminus1
11990336
TF-fVIIa-fXa + TFPI TF-fVIIa-fXa-TFPI 320 sdot 108 Mminus1 sminus1
11990337
TF-fVIIa-fXa-TFPI TF-fVIIa-fXa + TFPI 110 sdot 10minus2 sminus1
11990338
TF-fVIIa + fXa-TFPI TF-fVIIa-fXa-TFPI 500 sdot 107Mminus1 sminus1
11990339
fXa + ATIII fXa-ATIII 150 sdot 103Mminus1 sminus1
11990340
fmIIa + ATIII fmIIa-ATIII 710 sdot 103Mminus1 sminus1
11990341
fIXa + ATIII fIXa-ATIII 490 sdot 102Mminus1 sminus1
11990342
fIIa + ATIII fIIa-ATIII 710 sdot 103Mminus1 sminus1
11990343
TF-fVIIa + ATIII TF-fVIIa-ATIII 230 sdot 102Mminus1 sminus1
11990344
fXII fXIIa 500 sdot 10minus4 sminus1
11990345
fXIIa + fXII fXIIa-fXII 100 sdot 108Mminus1 sminus1
11990346
fXIIa-fXII fXIIa + fXII 75000 sminus1
11990347
fXIIa-fXII fXIIa + fXIIa 330 sdot 10minus2 sminus1
11990348
fXIIa + PKal fXIIa-PKal 100 sdot 108Mminus1 sminus1
11990349
fXIIa-PKal fXIIa + PKal 360 sdot 103 sminus1
BioMed Research International 5
Table 1 Continued
119899 Reactants Products Constant (119896)11990350
fXIIa-PKal fXIIa + Kal 4000 sminus1
11990351
fXII + Kal fXII-Kal 100 sdot 108Mminus1 sminus1
11990352
fXII-Kal fXII + Kal 4530 sminus1
11990353
fXII-Kal fXIIa + Kal 570 sminus1
11990354
PKal + Kal Kal + Kal 270 sdot 104Mminus1 sminus1
11990355
Kal Kal119894
110 sdot 10minus2 sminus1
11990356
fXIIa + C1inh fXIIa-C1inh 360 sdot 103Mminus1 sminus1
11990357
fXIIa + ATIII fXIIa-ATIII 2160Mminus1 sminus1
11990358
fXI + fIIa fXI-fIIa 100 sdot 108Mminus1 sminus1
11990359
fXI-fIIa fXI + fIIa 500 sminus1
11990360
fXI-fIIa fXIa + fIIa 130 sdot 10minus4 sminus1
11990361
fXIIa + fXI fXIIa-fXI 100 sdot 108Mminus1 sminus1
11990362
fXIIa-fXI fXIIa + fXI 200 sminus1
11990363
fXIIa-fXI fXIIa + fXIa 570 sdot 10minus4 sminus1
11990364
fXIa + fXI fXIa + fXIa 319 sdot 106Mminus1 sminus1
11990365
fXIa + ATIII fXIa-ATIII 320 sdot 102Mminus1 sminus1
11990366
fXIa + C1inh fXIa-C1inh 180 sdot 103Mminus1 sminus1
11990367
fXIa + A1AT fXIa-A1AT 100 sdot 102Mminus1 sminus1
11990368
fXIa + A2AP fXIa-A2AP 43 sdot 103Mminus1 sminus1
11990369
fXIa + fIX fXIa-fIX 100 sdot 108Mminus1 sminus1
11990370
fXIa-fIX fXIa + fIX 4100 sminus1
11990371
fXIa-fIX fXIa + fIXa 770 sminus1
11990372
fIXa + fX fIXa-fX 100 sdot 108Mminus1 sminus1
11990373
fIXa-fX fIXa + fX 064 sminus1
11990374
fIXa-fX fIXa + fXa 700 sdot 10minus4 sminus1
11990375
fXa + fVIII fXa-fVIII 100 sdot 108Mminus1 sminus1
11990376
fXa-fVIII fXa + fVIII 210 sminus1
11990377
fXa-fVIII fXa + fVIIIa 0023 sminus1
11990378
fVIIa + fIX fVIIa-fIX 100 sdot 108Mminus1 sminus1
11990379
fVIIa-fIX fVIIa + fIX 090 sminus1
11990380
fVIIa-fIX fVIIa + fIXa 360 sdot 10minus5 sminus1
11990381
fVIIa + fX fVIIa-fX 100 sdot 108Mminus1 sminus1
11990382
fVIIa-fX fVIIa + fX 21000 sminus1
11990383
fVIIa-fX fVIIa + fXa 160 sdot 10minus6 sminus1
11990384
Fbg + fIIa Fbg-fIIa 100 sdot 108Mminus1 sminus1
11990385
Fbg-fIIa Fbg + fIIa 63600 sminus1
11990386
Fbg-fIIa Fbn1 + fIIa + FPA 8400 sminus1
11990387
Fbn1 + fIIa Fbn1-fIIa 100 sdot 108Mminus1 sminus1
11990388
Fbn1-fIIa Fbn1 + fIIa 74260 sminus1
11990389
Fbn1-fIIa Fbn2 + fIIa + FPB 740 sminus1
11990390
Fbn1 + Fbn1 (Fbn1)2 100 sdot 106Mminus1 sminus1
11990391
(Fbn1)2 2Fbn1 640 sdot 10minus2 sminus1
11990392
(Fbn1)2 + fIIa (Fbn1)2-fIIa 100 sdot 108Mminus1 sminus1
11990393
(Fbn1)2-fIIa (Fbn1)2 + fIIa 70100 sminus1
11990394
(Fbn1)2-fIIa (Fbn2)2 + fIIa + FPB 4900 sminus1
11990395
Fbn2 + fIIa Fbn2-fIIa 100 sdot 108Mminus1 sminus1
11990396
Fbn2-fIIa Fbn2 + fIIa 100 sdot 103 sminus1
6 BioMed Research International
fXIITF-fVII
TFfVIIa
fVII
TF-fVIIa
Fbg
FPA
Fbn1
FPB
Fbn2
Fbn1
(Fbn1)2
(Fbn2)2 FPB
fXa
fIIa
TF-fVIIa-fXa
fIX
fIXa
fX
fmIIa
fII
PKal
KalifXI
fXIa
fIX
fIXa
fVIII
fVIIIa
fIXa-fVIIIa
fVIIIa1 -L
fVIIIa2
fVIIIa2
fIXa
fV
fVa
fXa-fVa
fXa-TFPI
TFPI
TF-fVIIa-fXa-TFPI
fXa-AT
AT
AT
fXIa-AT
C1inh
fXIIa-C1inh
A1AT
A2AP
fXIa-A2APfXIa-A1AT
TF-fVIIa
fXIIa Kal
fVIIIa1 -L
Figure 1 Graphical representation of the blood coagulation cascade model considered in this work Legend Blue box coagulation factor redbox inhibitor and related complexes Black arrow complex formation green arrow catalytic activation violet arrow activation red arrowinhibition The reaction is reversible if the arrow tail consists in a dot
BioMed Research International 7
(iv) the activation of prekallikrein by factor XIIa(reactions 119903
48 11990349 11990350) and by kallikrein (reac-
tion 11990354)
(v) the activation of factor XI by factor XIIa (reac-tions 119903
61 11990362 11990363) and by factor XIa (reaction
11990364)
(vi) the activation of factor IX by factor XIa (reac-tions 119903
69 11990370 11990371)
(vii) the dissociation of free factor VIIIa (reactions11990323and 11990324)
(viii) the dissociation of factor VIIIa in complex withother factors (reactions 119903
25and 11990326)
(3) The third module corresponds to the common path-way which consists in
(i) the activation of factor II by factor Xa (modeledby reaction 119903
16) and by the complex between
factor Xa and factor Va through the formationof the intermediate meizothrombin (reactions11990330 119903
33)
(ii) the activation of factor VII by factor Xa (reac-tion 1199036) and by factor IIa (reaction 119903
7)
(iii) the activation of factor VIII by factor Xa (reac-tions 119903
75 11990376 11990377) and by factor IIa (reaction 119903
17)
(iv) the formation of a complex between factor Xaand factor Va (reactions 119903
28and 11990329)
(v) the activation of fibrinogen by factor IIa (reac-tions 119903
84 119903
96)
(vi) the activation of factor V by factor IIa (reaction11990327)
(vii) the activation of factor XI by factor IIa (reac-tions 119903
58 11990359 11990360)
(4) The fourth module describes the inhibition of coagu-lation carried out through the main inhibitors of theBCC (antithrombin Tissue Factor Pathway InhibitorC1-inihibitor 1205721-antitrypsin and 1205722-antiplasmin)These are tight binding inhibitors belonging to theserpin superfamily which form irreversible com-plexes This module consists in
(i) the inhibition of factor Xa by Tissue FactorPathway Inhibitor (modeled by reactions 119903
34and
11990335)
(ii) the inhibition of the complex between TissueFactor factor VIIa and factor Xa by TissueFactor Pathway Inhibitor (reactions 119903
36 11990337 11990338)
(iii) the inhibition of factor IIa by antithrombin(reactions 119903
40and 11990342)
(iv) the inhibition of factor Xa by antithrombin(reaction 119903
39)
(v) the inhibition of factor IXa by antithrombin(reaction 119903
41)
(vi) the inhibition of factor XIa by antithrombin(reaction 119903
65) and by C1-inhibitor (reaction 119903
66)
Table 2 Initial concentrations of molecular species in the bloodcoagulation cascade model (values taken from [30])
Species Symbol Concentration (M)1205721-Antitrypsin A1AT 450 sdot 10
minus5
1205722-Antiplasmin A2AP 100 sdot 10minus6
Antithrombin III ATIII 340 sdot 10minus6
C1-inhibitor C1inh 250 sdot 10minus6
Fibrinogen Fbg 900 sdot 10minus6
Factor II fII 140 sdot 10minus6
Factor V fV 200 sdot 10minus8
Factor VII fVII 100 sdot 10minus8
Active factor VII fVIIa 100 sdot 10minus10
Factor VIII fVIII 700 sdot 10minus10
Factor IX fIX 900 sdot 10minus8
Factor X fX 16 sdot 10minus7
Factor XI fXI 310 sdot 10minus8
Factor XII fXII 340 sdot 10minus7
Prekallikrein Pkal 450 sdot 10minus7
Tissue Factor TF 500 sdot 10minus12
Tissue Factor Pathway Inhibitor TFPI 250 sdot 10minus9
(vii) the inhibition of factor XIIa by C1-inhibitor(reaction 119903
56) by antithrombin (reaction 119903
57)
by 1205721-antitrypsin (reaction 11990367) and by 1205722-
antiplasmin (reaction 11990368)
(viii) the inhibition of the complex between TissueFactor and factor VIIa by antithrombin (reac-tion 11990343)
(ix) the inhibition of kallikrein (reaction 11990355)
The values of the initial concentrations of the molecularspecies occurring in the BCC model are given in Table 2According to [30] the concentrations of complexes and activefactors were set to 0 except for the active factor VII which isphysiologically present in the blood circulation even in theabsence of damage in a concentration that is approximatelyequal to 1 of the corresponding inactive factor [42]
The system of ordinary differential equations (ODEs)needed to carry out the simulations and the parametersweep analysis presented in Section 3 was derived fromthe reactions given in Table 1 according to the mass-actionlaw with the exception of 14 reactions belonging to the set119878120576= 11990319 11990321 11990323 11990329 11990331 11990335 11990346 11990349 11990352 11990362 11990370 11990373 11990376 11990382
The mass-action law is the fundamental empirical law thatgoverns biochemical reaction rates which states that in adilute solution the rate of an elementary reaction is propor-tional to the product of the concentration of its reactantsraised to the power of the corresponding stoichiometriccoefficient [44] The reactions in 119878
120576are instead influenced by
a specifically defined variable 120576 depending on aHill functionfit against experimental data which quantify the plateletactivation status which is used to model the physiologicallevels of thrombin concentration as a function of plateletactivation as thoroughly described in [30] More preciselythe value of 120576 influences the formation of some complexes
8 BioMed Research International
0 0 100 200 300 400 500 600 700
Fact
or II
a con
cent
ratio
n (M
)
Time (s)
1eminus7
2eminus7
3eminus7
4eminus7
5eminus7
6eminus7
7eminus7
8eminus7
9eminus7
Figure 2 Dynamics of thrombin (factor IIa) in physiological condition
occurring on the plateletsrsquo surface by modifying the activityof reactions of the form
119860 + 119861
1198961
999445999468
1198962
119860119861 (1)
Namely 120576 intervenes with the dissociation constant of thesereactions so that the corresponding standard ODEs arechanged to yield new equations of the form
119889 [119860119861]
119889119905
= 1198961sdot [119860] sdot [119861] minus
1198962sdot [119860] sdot [119861]
120576
(2)
The value of 120576 in (2) depends on the following HillfunctionH which quantifies the state of platelets activationaccording to the thrombin concentration (here denoted as[fIIa]) that is the factor catalyzing the formation of the fibrinclot
H ([fIIalowast (119905)]) =[fIIalowast (119905)]16123
[fIIalowast (119905)]16123 + (24279 sdot 10minus9)16123
(3)
where [fIIalowast(119905)] = max1199051015840isin[0119905][fIIa(1199051015840)] for a chosen time
interval [0 119905] of simulationThe value [fIIalowast(119905)] represents the maximum transient
thrombin concentration and is needed to simulate the factthat in physiological conditions the thrombin concentrationstarts decreasing after rising to a peak (Figure 2) So doing[fIIalowast(119905)] never decreases once it reaches its maximummagnitude [fIIalowast(119905)] is equivalent to [fIIa(119905)] until the con-centration of factor IIa reaches the peak while thereafter itremains constant at that value which is the maximum in theconsidered time interval [0 119905] FunctionH allows simulatingthe physiological condition whereby platelets remain activealso when the thrombin concentration decreases
For a given concentration of factor IIa the maximumplatelets activation state 120576max is defined as
120576max = 120576max0
+ (1 minus 120576max0
) sdotH ([fIIalowast (119905)]) (4)
where 120576max0
defines the basal activation state of the plateletsat simulation time 119905 = 0 The value 120576max
0
is initially set to 001assuming a basal 1 binding strength of coagulation factorsto the resting plateletsrsquo surface When the full activationof platelets is reached 120576max is equal to 1 and the complexdissociation constants are minimized (see [30] for moredetails)
22 CUDA Architecture and the coagSODA Simulator Intro-duced by Nvidia in 2006 Compute Unified Device Architec-ture (CUDA) is a parallel computing platform and program-ming model that provides programmers with a frameworkto exploit GPUs in general-purpose computational tasks(GPGPU computing) GPGPU computing is a low-cost andenergy-wise alternative to the traditional high-performancecomputing infrastructures (eg clusters of machines) whichgives access to the tera-scale computing on common work-stations of mid-range price However due to the innovativearchitecture and the intrinsic limitations of GPUs a directporting of sequential code on the GPU is most of the timesunfeasible therefore making it challenging to fully exploit itscomputational power and massive parallelism [45]
CUDA combines the single instruction multiple data(SIMD) architecture with multithreading which automat-ically handles the conditional divergence between threadsThe drawback of such flexibility is that any divergence ofthe execution flow between threads causes a serialization ofthe execution which affects the overall performances UnderCUDArsquos naming conventions the programmer implementsthe kernel that is a CC++ function which is loaded fromthe host (the CPU) to the devices (one or more GPUs)and replicated in many copies named threads Threads canbe organized in three-dimensional structures named blockswhich in turn are contained in three-dimensional grids(a schematic description is given in Figure 3 left side)Whenever the host runs a kernel the GPU creates thecorresponding grid and automatically schedules each blockon one free streaming multiprocessor available on the GPUallowing a transparent scaling of performances on different
BioMed Research International 9
Host
Kernel 1
Kernel 2
Device
Grid 1
Grid 2
Block(0 0)
Block(1 0)
Block(2 0)
Block(0 1)
Block(1 1)
Block(2 1)
Block (1 1)
Thread(0 0)
Thread(1 0)
Thread(2 0)
Thread(3 0)
Thread(4 0)
Thread(0 1)
Thread(1 1)
Thread(2 1)
Thread(3 1)
Thread(4 1)
Thread(0 2)
Thread(1 2)
Thread(2 2)
Thread(3 2)
Thread(4 2)
Global memory
Texture memory
Constant memory
Shared memory Shared memory
Block (0 0) Block (1 0)
Localmemory
Localmemory
Localmemory
Localmemory
Thread(0 0)
Thread(1 0)
Registers Registers Registers Registers
Thread(0 0)
Thread(1 0)
Grid
Figure 3 Schematic description of CUDArsquos architecture in terms of threads and memory hierarchy Left Side Threads organization a singlekernel is launched from the host (the CPU) and is executed in multiple threads on the device (the GPU) Threads can be organized in three-dimensional structures named blocks which can be in turn organized in three-dimensional grids The dimensions of blocks and grids areexplicitly defined by the programmer Right Side Memory hierarchy threads can access data from many different memories with differentscopes Registers and local memories are private for each thread Shared memory lets threads belonging to the same block communicate andhas low access latency All threads can access the global memory which suffers high latencies but it is cached since the introduction of theFermi architecture Texture and constant memory can be read from any thread and are equipped with a cache as well in this work we exploitthe constant memory Figures are taken from Nvidiarsquos CUDA programming guide [60]
devices Threads within a block are executed in groups of 32threads named warps
The GPU is equipped with different kinds of memoryIn this work we exploit the global memory (accessible fromall threads) the shared memory (accessible from threadsof the same block) the local memory (registers and arraysaccessible from owner thread) and the constant memory(cached and not modifiable) A schematic representation ofthis memory hierarchy is shown in Figure 3 right side Toachieve the best performances the sharedmemory should beexploited as much as possible however it is very limited (ie49152 bytes for each multiprocessor since the introductionof the Fermi architecture) and it introduces restrictions onthe blocksrsquo size On the other hand the global memory isvery large (thousands ofMBs) but suffers from high latenciesA solution to this problem was implemented on the Fermiarchitecture where the global memory is equipped with aL2 cache This architecture also introduced the possibility tobalance 64KB of fast on-chip memory between the sharedmemory and L1 cache using two possible configurations48KB for the shared memory and 16KB for L1 cache or16 KB for the shared memory and 48 for L1 cacheThe Keplerarchitecture used in this paper allows a third and perfectly
balanced configuration where shared memory and L1 cacheobtain the same amount of memory (32KB) See Figure 4 fora schematization of the memory architecture
The systematic analysis of models of biological systemsoften consists in the execution of large batches of simulationsOne of the standard analyses that can be executed on suchkind of models regards an intensive search within the param-eters space which requires large numbers of independentsimulations In order to reduce the computational burdenwe previously implemented on the CUDA architecture oneof the most efficient numerical integration algorithms forODEs LSODA that is able to automatically recognize stiffand nonstiff systems and to dynamically select between themost appropriate integration procedure (ie Adams methodin the absence of stiffness and the Backward DifferentiationFormulae otherwise) [22] This GPU-powered tool is calledcupSODA [24 25] and exploits CUDArsquos massive parallelismto execute different and independent simulations in eachthread thus reducing the computational time required by astandard CPU counterpart of LSODA
Besides being very efficient for the simulation of manyindependent simulations cupSODA is also user-friendlyLSODA was originally designed to solve ODEs systems
10 BioMed Research International
Thread
Shared memory L1 cache Read-onlydata cache
L2 cache
DRAM
Figure 4 Schematic description of memory hierarchies in Fermi and Kepler architectures GPUs relying on these architectures are equippedwith a two-level data cache and a read-only data cache Shared memory and L1 cache share the same on-chip 64KB memory banks theamount of memory can be reconfigured by the user according to the specific needs of the application Figure taken from Nvidiarsquos KeplerGK110 whitepaper [57]
written in the canonical form but the user is supposed tospecify the system of ODEs by implementing a custom Cfunction that is passed to the algorithmmoreover in order tospeed up the computation when dealing with stiff systems inLSODA the Jacobianmatrix associatedwith theODEs systemmust be implemented as a custom C function as well On thecontrary cupSODA was conceived as a black-box simulatorthat can be easily used without any programming skillscupSODA consists in a tool to automatically convert a genericmechanistic reaction-based model of a biological system intothe corresponding set of ODEs to comply with the mass-action kinetics [46] and to directly encode the obtainedsystem along with the corresponding Jacobian matrix as Carrays
This fully automaticmethodology can be exploited for thesimulation of models whose chemical kinetics is based on themass-action law The BCC model described in Section 21though includes a set 119878
120576of reactions that do not follow
the mass-action kinetics [30] the platelets activity is notexplicitly modeled by biochemical reactions but it is realizedby modulating the rate of the dissociation of the complexesformed on a plateletrsquos surface by means of the variable120576 which is calculated with a special equation during theintegration steps For this reason cupSODA was used asa starting point for the development of a new tool ableto compute at run-time the specific kinetics of this set ofreactions
This new CUDA-powered simulation tool named coag-SODA is specifically tailored for the simulation of the BCC
model developed in [30] In particular coagSODA realizesthe run-time calculation of the 120576 value required to correctlysimulate the activity of platelets which is determined accord-ing to the equations described in Section 21 The plateletsactivation state 120576 is calculated at each time instant by solvingthe following differential equation
119889120576
119889119905
= 119896 (120576max minus 120576) (5)
where the constant 119896 is inversely proportional to the timescale of platelets activation and is set to 0005 This isconsistent with the fact that platelets do not instantly achievetheir maximum attainable activation state (120576max) but theyreach it on a physiologically relevant timescale [30]
Dealing with the 14 reactions in the set 119878120576that are
influenced by the Hill functionH (see Section 21) the valueof [fIIalowast(119905)]must be stored on the GPU because during eachintegration step coagSODA recalculates (4) which exploitsthe value [fIIalowast(119905)] to determine a new 120576 value Since CUDArsquosarchitecture does not offer static variables the informationfor each thread has to be memorized in the global memoryThe accesses to the global memory and the computationalcosts due to these additional calculations slow down theintegration process with respect to the integration of ODEsperformed according to strictly mass-action based kinetics(as in the cupSODA simulator) Nevertheless in Section 33we show that our parallel implementation largely outper-forms a sequential counterpart of the LSODA algorithm
BioMed Research International 11
0 100 200 300 400 500 600 700
Fibr
inog
en co
ncen
trat
ion
(M)
Time (s)
30 of fibrinogen
0
1eminus6
2eminus6
3eminus6
4eminus6
5eminus6
6eminus6
7eminus6
8eminus6
9eminus6
Figure 5 Dynamics of fibrinogen in physiological condition The clotting time is defined as the time necessary to convert the 70 of thefibrinogen into fibrin
As in the case of cupSODA [24 25] coagSODA exploitsthe shared memory to improve performances by storing thecurrent state and time of the simulations in these low-latencymemory banks Despite the improvement of performancesensured by this solution it strongly affects the occupancyof GPUrsquos multiprocessors and therefore it represents thelimiting factor for the number of blocks that can be executedsimultaneously as a matter of fact coagSODA is limited to2 blocks per streaming multiprocessor on GPUs based onthe Fermi architecture with a reduced exploitation of theGPU with respect to the theoretical 8 blocks allowed by thisarchitecture
3 Results and Discussion
In this section we discuss the results of the parameter sweepanalysis (PSA) carried out on the BCC model to the aimof investigating either the prolongation or the reduction ofthe clotting time in response to perturbed values of somereaction constants and of the initial concentration of somemolecular species chosen according to their meaning withinthe whole pathway PSA was performed by generating aset of different initial conditions corresponding to differentparameterizations of the model and then automaticallyexecuting the deterministic simulations with coagSODATheuse ofGPU technology is fundamental in this type of analysisespecially for large biological systems as the BCC is becauseit drastically reduces the computational time
The sweep analysis for single parameters (PSA-1D) wasperformed considering a logarithmic sampling of numericalvalues of each parameter under investigation (reaction con-stant or initial molecular concentration) within a specifiedrange with respect to its physiological reference value Thesweep analysis over pairs of parameters (PSA-2D) was per-formed by simultaneously varying the values of two param-eters within a specified range considering a logarithmicsampling on the resulting lattice The logarithmic samplingallows uniformly spanning different orders of magnitude ofthe parameters value using a reduced and fine-grained set
of samples therefore efficiently analyzing the response of thesystem in a broad range of conditions
To determine the response of the BCC to perturbed con-ditions we chose the clotting time (CT) as output of the PSAThe CT is defined as the time necessary to convert the 70of the fibrinogen into fibrin (see Figure 5) conventionallyassumed to correspond to the time required to form the clot[47] and it is generally used in laboratory tests formonitoringthe therapywith anticoagulant drugs According to themodeldefined in [30] the reference value of CT is around 300seconds in physiological conditions We investigate herethe response of the BCC by evaluating the CT in variousconditions corresponding to different values of the reactionconstants varying over six orders of magnitude with respectto their physiological values (ie three below and three abovethe reference values if not otherwise specified) as well asto different values of the initial molecular concentrationsand varying over twelve orders of magnitude with respect totheir physiological values (ie six below and six above thereference values if not otherwise specified)
The total number of parallel simulations executed tocarry out these analyses was 100 for PSA-1D over reactionconstants 200 for PSA-1D over initial concentrations and1600 for PSA-2D
Finally we present the comparison of the performanceof the CPU and GPU to run an increasing number ofsimulations of the BCC model to prove the efficiency ofcoagSODA
31 PSA-1D of the BCC Model
311 Reaction 11990344 Thefirst PSA was performed to determine
the value of the kinetic constant for the autoactivation offactor XII (reaction 119903
44in Table 1) which corresponds to an
upstream process in the intrinsic pathway This analysis wasmotivated by two considerations Firstly by using the fullparameterization given in [30] the action of the intrinsicpathway turns out to be fundamental for the BCC in vivowhich is in contrast to experimental observations which
12 BioMed Research International
324
326
328
330
332
334
336
338
340
Clot
ting
time (
s)
1eminus
10
1eminus
9
1eminus
8
1eminus
7
1eminus
6
1eminus
5
1eminus
4
1eminus
3
k44 (sminus1)
Figure 6 Clotting time according to PSA-1D over constant 11989644of reaction 119903
44 XII rarr XIIaThe reference value of 119896
44used in [30] is 5sdot10minus4sminus1
the sweep range is [5 sdot 10minus10 5 sdot 10minus4]
280
300
320
340
360
380
400
1e4 1e5 1e6 1e7 1e8 1e9 1e10 1e11
Clot
ting
time (
s)
k27 (Mminus1sminus1)
Figure 7 Clotting time according to PSA-1D over constant 11989627of reaction 119903
27 IIa + V rarr IIa + Va The reference value is 2 sdot 107Mminus1 sminus1 the
sweep range is [2 sdot 104 2 sdot 1010]
indicate that the extrinsic pathway is the main responsibleof clot formation Secondly in [30] all constants values havea reference to experimental measurements except for theconstant of this reaction (which corresponds to entry 29 inTable 1 in [30])
Figure 6 shows the results of this PSA-1D where 11989644was
varied over six orders of magnitude considering the valuegiven in [30] as the upper limit of the sweep interval We canobserve that by decreasing the value of 119896
44with respect to
the value considered in [30] the CT increases with respect toits reference value however for values of this constant lowerthan 100 sdot 10minus6 sminus1 the CT remains unaltered and this canbe intuitively explained by the fact that in this situation thefibrinogen is mainly activated by the extrinsic pathway
Consequently we assigned the value 700 sdot 10minus6 sminus1 to11989644 achieving a CT that is comparable to the experimental
observations of the BCC in vivo This new value was used inall PSA discussed in what follows
312 Reactions 11990327
and 11990358 In the next PSA we investigated
the effect of the perturbation of the kinetics of two pivotal
reactions of the BCC model Reaction 11990327 which describes
the catalytic activation of factor V by factor IIa was chosenbecause it represents the main positive feedback within thecommon pathway reaction 119903
58 which is involved in the
activation of factor XI by binding to factor IIa was chosenbecause it represents the main positive feedback in theintrinsic pathway (Figure 1) Moreover preliminary PSA overall reaction constants of the BCC model given in Table 1evidenced that these two reactions are among the mostrelevant steps of the coagulation network
The PSA-1D over 11989627(Figure 7) shows that the CT is very
sensitive to the perturbation of the rate of this reaction whenthe reference value of its constant (ie 119896
27= 2 sdot 10
7Mminus1 sminus1)is either increased or decreased in particular when 119896
27is
very low a plateau in CT is reached since the strength ofthe positive feedback exerted by factor IIa is largely reduceda condition where the contribution of the amplification ofthe hemostatic stimulus (due by the common pathway) tothe formation of the clot is basically not effective On theother side the PSA-1D over 119896
58(Figure 8) shows that while
a decrease of the constant with respect to its reference value
BioMed Research International 13
310
320
330
340
350
360
370
1e5 1e6 1e7 1e8 1e9 1e10 1e11
Clot
ting
time (
s)
k58 (Mminus1sminus1)
Figure 8 Clotting time according to PSA-1D over constant 11989658of reaction 119903
58 IIa + XI rarr IIa-XI The reference value is 1 sdot 108Mminus1 sminus1 the
sweep range is [1 sdot 105 2 sdot 1011]
220
240
260
280
300
320
340
360
380
400
Clot
ting
time (
s)
Factor VIII (M)
Seve
re h
aem
ophi
lia
Mild
hae
mop
hilia
Prot
hrom
botic
stat
e
1eminus
16
1eminus
14
1eminus
12
1eminus
10
1eminus
8
1eminus
6
1eminus
4
1eminus
2
Figure 9 Clotting time at different initial concentrations of factor VIII The reference value is 700 sdot 10minus10M (dashed black line) the sweeprange is [700 sdot 10minus16 700 sdot 10minus4] The blue area indicates a condition in which factor VIII concentration is less than 1 of its physiologicalconcentration corresponding to a situation of severe haemophilia the red area indicates a condition in which factor VIII concentration isbetween 1 and 30 of its physiological concentration corresponding to a situation of mild haemophilia the green area indicates a conditionin which factor VIII concentration is greater than 130 of its physiological concentration corresponding to a situation of hypercoagulability
(ie 11989658= 1 sdot 10
8Mminus1 sminus1) does not have any substantialeffect an increase of its value leads to a progressive increasein the CT This increase is due to the fact that factor IIa issequestered in the formation of a complex with factor XI andhence it is no longer available as a free component in bloodto participate in other reactions especially those reactionsof the extrinsic pathway which principally lead to the clotformation in vivo This behavior highlights that the intrinsicpathway has a secondary role in blood coagulation in vivocompared with the extrinsic pathway as also evidenced byvarious experimental observations [48]
313 Factors VIII IX and II The next set of PSA-1D wasrealized by varying the initial concentrations of factors VIIIIX and II These factors were selected since both an excess
and a deficiency of their concentrations lead to diseasesrelated to blood clotting
The PSA-1D over factor VIII (Figure 9) shows thatincreasing the initial concentration of this factor resultsin decreasing the CT suggesting the possible presence ofhypercoagulable states in these perturbed conditions As amatter of fact high levels of factor VIII cause an increasedrisk of deep vein thrombosis and pulmonary embolism[49] On the other hand individuals with less than 1of the average concentration of factor VIII show a severehaemophilia A characterized by higher CT (Figure 9) andrequire infusions of plasma containing the deficient factorotherwise frequent spontaneous bleeding would occur [50]When the concentration of this factor is between 5 and 30of the average concentration individuals still risk bleeding incase of trauma [50]
14 BioMed Research International
250
300
350
400
450
500
550
600
650
700
Clot
ting
time (
s)
Factor IX (M)
1eminus
14
1eminus
12
1eminus
10
1eminus
8
1eminus
6
1eminus
4
1eminus
2
1e0
Figure 10 Clotting time at different initial concentrations of factor IXThe reference value is 900sdot10minus8M the sweep range is [900sdot10minus14 900sdot10minus2]
0
100
200
300
400
500
600
700
Clot
ting
time (
s)
Factor II (M)
1eminus
12
1eminus
10
1eminus
8
1eminus
6
1eminus
4
1eminus
2
1e0
1e2
Figure 11 Clotting time at different initial concentrations of thrombinThe reference value is 140sdot10minus6M the sweep range is [140sdot10minus12 140sdot100]
The PSA-1D over factor IX (Figure 10) shows only aslight decrease of the CT as the initial concentration offIX increases this is in contrast to recent studies thatdemonstrated how the excess of factor IX leads to anincreased risk of deep vein thrombosis [51] This result canbe explained by considering that (i) a high concentrationof factor IX is not sufficient to bring about coagulationproblems though when the concentration of other factorsis above the average value (yet not at pathological levels)prothrombotic states can be observed (ii) in this model weconsider average values as initial concentrations of factorshowever individuals are characterized by different (balanced)combinations of procoagulant and anticoagulant factor levelsthat altogether contribute to define a unique coagulationphenotype that reflects the developmental environmentalgenetic nutritional and pharmacological influences of eachindividual [52] On the contrary the lack of factor IX causes
haemophilia B characterized by higher CT with respect tothe reference value (Figure 10)
Furthermore by comparing the PSA-1D of factors VIIIand IX (Figures 9 and 10) it is clear that haemophilia A ismore serious than haemophilia B since the CT achieved inconditions of factor VIII deficiency is higher than the CTobtained in the case of factor IX deficiency
In both PSA-1D over factors VIII and IX we observedafter the initial decrease of the CT an unexpected increaseof the CT as the factor concentration increasesThis counter-intuitive behavior arises at very high concentrations of thesefactors (with respect to the average physiological levels) andto the best of our knowledge it was never observed in vivoNonetheless it would be interesting to verify by means of adhoc laboratory experiments if the model correctly describesthe behavior of the BCC even in these conditions or on thecontrary it is not predictive in these extreme situations
BioMed Research International 15
The PSA-1D over factor II (Figure 11) shows a dramaticdecrease of the CT as the initial concentration of this factorincreases (with respect to the average physiological level)This behavior resembles the effects of hypercoagulability (orthrombophilia) a disease caused by mutation G20210A inthe prothrombin gene [53] that causes an increase of theprothrombin level (factor II) in the blood flow resulting inan excessive formation of the active form of this factor thusheightening venous thrombosis risks [54] Hypercoagulabil-ity is usually treated with warfarin therapy or with otheranticoagulants with a similar effect These drugs decrease thecapacity of coagulation factors to become active preventingthe formation of unwanted thrombi
On the other hand when the initial concentration offactor II is low we achieved the effects of prothrombindeficiency a rare autosomal recessive disease that causes atendency to severe bleeding [29 55] As shown in Figure 11 aconcentration equal to 10of the physiological value of factorII (ie 14 sdot 10minus6 M) leads to clotting effects similar to severehaemophilia A
32 PSA-2D of the BCCModel We present here the results ofthe PSA-2D on the BCCmodel where couples of parameterswere varied to analyze the possible effects arising from thecombined perturbation of their values
321 Reactions 11990327
and 11990358 Figure 12 shows the effect of
the simultaneous variation of constants 11989627
and 11989658
(overthe same sweep ranges considered in the two PSA-1DSection 31) This result remarks that reaction 119903
27 involved
in the common pathway has a stronger influence on theBCC and that there is a synergic interplay between thesetwo reactions In particular when the value of 119896
27is low and
the value of 11989658
is high the CT is higher than the valuesachieved when only a single constant is changed because inthis condition both the intrinsic and the common pathwaysare simultaneously inhibited
322 Factor VIII Factor IX and Tissue Factor In the last twoPSA-2Dwe varied the initial concentration of factor VIII andTissue Factor and the initial concentration of factor IX andTissue Factor respectively
The initial concentrations of factors VIII and IX were var-ied over four orders of magnitude using their physiologicalvalues as upper limit for the sweep ranges the concentrationof Tissue Factor was varied over four orders of magnitudetwo above and two below its reference value (see Table 2)The rationale behind this choice is to observe how the BCCmodel in conditions corresponding to different states ofhaemophilia (obtained by decreasing the concentrations offactors VIII and IX) behaves with different initial concentra-tions of the Tissue Factor which is the upstream factor of theextrinsic pathway that is the most important element of theBCC
The results of these PSA-2D show that with respect tothe condition of haemophilia B in the case of haemophiliaA the amount of Tissue Factor (below its reference value)has a negligible influence on the CT as indicated by the
presence of a plateau in Figure 13 on the contrary concerninghaemophilia B a deficiency of Tissue Factor leads to anincrease of the CT especially when factor IX is present in lowconcentrations (Figure 14)
The different results achieved in the two PSA-2D are dueto the presence in the BCC model of a direct interactionbetweenTissue Factor and factor IX bymeans of theTF-fVIIacomplex (see reactions 119903
13 119903
15in Table 1) Indeed the lack
of Tissue Factor directly affects the concentration of activefactor IX which results in a strong alteration of the CT withrespect to physiological conditions
33 CPU versus GPU Performance Comparison In orderto show the relevant speedup achieved by coagSODA wepresent here the comparison of the computational effortrequired by GPU and CPU for the simulation of the BCCmodel The performances of our GPU simulator were com-pared with those obtained using the LSODA algorithmimplemented in the software COPASI [56] executing onthe CPU the same set of simulations that were run on theGPU COPASI is single-threaded and does not exploit thephysical and logical cores of the CPU therefore it representsa good benchmark as a single-node CPU-bound simulator ofbiological systems
In all simulations executed on both GPU and CPU westored 100 samples uniformly distributed in the time intervalconsidered for each simulation that is [0 700] seconds of thedynamics of all chemical species involved in the BCCmodelThe settings for the LSODA algorithm were the followingrelative error 1 sdot 10minus7 absolute error 1 sdot 10minus13 and maximumnumber of internal steps set to 20000
331 Benchmark Details The GPU used for the simulationsis a Nvidia Tesla K20c equipped with 2496 cores organized in13 streamingmultiprocessors GPU clock 706MHz and 5GBofDDR5RAM In all tests coagSODAwas compiled and exe-cuted with version 55 of CUDA libraries Even though thisGPU has compute capability 35 and is based on the GK110Kepler architecture currently coagSODA does not exploitany of the new features (eg Dynamic Parallelism Hyper-QRemote DMA) with the exception of the new ISA encodingwhich allows threads to exploit an increased number ofregisters (255 instead of 63) reducing register spilling intoglobal memory and increasing performances [57] Moreoveras described in Section 22 the Kepler architecture offers thepossibility to reconfigure the 64KB on-chip cache balancingbetween L1 cache and shared memory Since coagSODAexploits the shared memory to reduce the latencies duringthe access to the concentration values of the BCC model weopted for the following configuration 48KB for the sharedmemory 16 KB for the L1 cache
The performance of the GPU was compared against apersonal computer equippedwith a dual-core CPU Intel Corei5 frequency 23 GHz 4GB of DDR3 RAM running theoperating systemMacOSXLion 1075The software used as acoagSODA-equivalent single-threaded CPU implementationis COPASI version 48 (build 35)
16 BioMed Research International
280
300
320
340
360
380
400
420
440
1e41e6
1e91e11 1e5
1e7
1e9
1e11
280 320 360 400 440
Clot
ting
time (
s)
k 58(M
minus1 sminus1 )
k27 (Mminus1sminus1)
Figure 12 Clotting time according to PSA-2D over reaction constants 11989627and 119896
58 The sweep ranges are [2 sdot 104 2 sdot 1010] and [1 sdot 105 2 sdot 1011]
respectively
100
200
300
400
Tissue factor (M) Factor VIII
(M)
Clot
ting
time (
s)
100
150
200
250
300
350
400
450
1eminus131eminus11
1eminus91eminus14
1eminus12
1eminus10
Figure 13 Clotting time at different initial concentrations of Tissue Factor and factor VIII The sweep ranges are [500 sdot 10minus14 500 sdot 10minus10]and [700 sdot 10minus14 500 sdot 10minus10] respectively
A direct comparison of the performances between thesetwo different architectures and implementations is not aneasy task since CPUs are optimized for single-thread exe-cution and exploit a large number of optimizations (eghigher clock frequency instruction caching pipelining outof order execution and branch prediction) whereas GPUsare optimized for graphics processing and parallel exe-cution of identical code relying only on multilevel datacaching For this reason in this paper we propose anempirical comparison of the performances based on therunning times of identical batches of simulations of the BCCmodel
332 Benchmark Results In Figure 15 we show the compar-ison of the running times required to run several batchesof simulations executed to carry out the PSA-1D over thereaction constant 119896
27 The choice of comparing the perfor-
mances of the GPU and CPU implementations by executingbatches of simulations that are related to a PSA instead ofrunning 119899 independent but identical simulations (ie allcharacterized by the same parameterization of the model)is due to the fact that these results represent a more realisticscenario in the computational analysis of biological systemswhereby it is useful to investigate large search spaces ofparameters corresponding to different perturbed conditionsof themodel [26]Moreover for the evaluation of the running
time the execution of a batch of 119899 identical deterministicsimulations would be futile The figure clearly shows thatcoagSODA always performs better than the CPU counter-part In particular while the CPU performance increaseslinearly with the number of simulations the running timesare strongly reduced on the GPU in this case the overallrunning time roughly corresponds to the time required forthe execution of the slowest simulations This is due to thefact that different parameterizations may require differenttime steps for LSODA to converge and the execution ofa block terminates as long as all the threads it containsterminate In turn the execution of a kernel terminates whenall blocks terminate for this reason a single simulationwhose dynamics requires more steps than the others mayaffect the overall running time
In Table 3 we report the running times of all batchesof simulations along with the speedup achieved on theGPU In particular these results highlight that the advantageof exploiting the GPUs for the simulation of the BCCmodel becomes more evident as the number of simula-tions increases with a 181times speedup when a PSA with10000 different parameterizations is executed Therefore theGPU accelerated analysis of the BCC model with coag-SODA represents a novel relevant computational means toinvestigate the behavior of this complex biological systemunder nonphysiological conditions and could be exploited
BioMed Research International 17
0
200
400
600
Tissue factor (M) Factor IX (M
)
Clot
ting
time (
s)
0
100
200
300
400
500
600
700
1eminus12
1eminus10
1eminus8
1eminus131eminus11
1eminus9
Figure 14 Clotting time at different initial concentrations of Tissue Factor and factor IXThe sweep ranges are [500 sdot 10minus14 500 sdot 10minus10] and[900 sdot 10
minus12 900 sdot 10
minus8] respectively
5 10 50 100 500 1000 5000 10000
Runn
ing
time (
s)
Number of simulations
CPUGPU
1e2
1e3
1e4
1e5
1e6
1e7
Figure 15 Comparison of the computational time required to execute an increasing number of simulations of the BCCmodel on CPU (IntelCore i5 23 GHz) and GPU (Nvidia Tesla K20c GPU clock 706MHz) The computational time is expressed in seconds The time valuesrelated to 1000 5000 and 10000 simulations on the CPU were estimated by regression (see also Table 3)
Table 3 CPU versus GPU performance comparison
Number ofsimulations CPU time (sec) GPU time (sec) Speedup
5 3808 1278 298times10 10604 2329 456times50 51704 8058 642times100 116528 10977 1061times500 536058 17394 3081times1000 1073582lowast 19982 5373times5000 5363507lowast 30960 1732times10000 10724975lowast 58951 1819timeslowastValues estimated by regression
to efficiently determine the response of the BCC to differenttherapeutical drugs
4 Conclusions
Thanks to their high-performance computing capabilitiesand the very low costs GPUs nowadays represent a suitable
technology for the development and the application of par-allel computational methods for in silico analysis of complexbiological systems However the implementation of efficientcomputational tools able to fully exploit the large potentialityof GPUs is still challenging since good programming skillsare required to implement GPU-based algorithmic methodsand to handle specific features as an optimal usage ofmemoryor the communication bandwidth between GPU and CPUMoreover algorithms cannot be directly ported on the GPUbecause of the limited programming capabilities allowed byGPUkernels as amatter of fact they need to be redesigned tomeet the requirements of the underlying SIMD architecture
In this work we presented coagSODA a GPU-poweredsimulator specifically developed to carry out fast parallelsimulations of the BCC model coagSODA was designed tooffer a black-box solution usable by any final user in aneasy way It relies on cupSODA [24] a numerical integratorfor ODEs that we previously implemented for the GPUarchitecture based on the LSODA algorithm and capableof automatically translating a reaction-based model into aset of coupled ODEs In addition to mass-action kinet-ics coagSODA implements specific functions to compute
18 BioMed Research International
the kinetics of Hill function based reactions such as thoseinvolved in the platelets activity of the BCC model coag-SODA exploits the massive parallel capabilities of modernGPUs and our results demonstrated that it can achieve arelevant reduction of the computational time required toexecutemany concurrent and independent simulations of theBCC dynamics The mutual independence of the simulationsallows fully exploiting the underlying SIMD architecturemoreover coagSODA benefits from an additional speedupthanks to our choice of storing the state of the systeminto the low-latency shared memory (a solution that wasalready implemented in cupSODA) Since the BCC modelis large (96 reactions 71 chemical species) a large amountof shared memory was assigned to each thread stronglyreducing the theoretical occupancy of the GPU that is theratio of active warps with respect to the maximum numberof warps supported by each streaming multiprocessor of theGPU However the results of the analysis presented in thiswork performed on the BCC model show that coagSODAachieves a relevant boost with respect to a reference CPUimplementation For instance in the case of 10000 simula-tions we achieved a noticeable 181times speedup Interestinglythe performances of coagSODA are better than COPASI evenfor small batches of simulations These results indicate thatfor biological models consisting in many reactions and manyspecies our GPU implementation of LSODA becomes moreprofitable than the CPU counterpart as the number of con-current simulations increases making it suitable especiallywhen performing demanding computational analysis suchas for example parameter sweep parameter estimation orsensitivity analysis
As a test case in this paper we presented several parame-ter sweep analyses over reaction constants and initial concen-trations of factors involved in the BCC model Other com-putational analyses on mathematical models of this pathwaywere previously presented For instance in [58] a parameterestimation of the reaction constants of different modelsinvolving the activation of factor X factor V prothrombinand the inactivation of factors was performed The aim ofthis analysis was to discriminate between different modelsto unravel the mechanisms on the basis of the BCC Similaranalyses were executed on a model describing thrombingeneration in plasma since the reaction mechanism thereaction constants and initial concentrations were unknown[59] In [34] a sensitivity analysis of a model consisting of44 species over 34 chemical species was presented reactionconstants were varied between 10 and 1000 of theirreference value to the aim of identifying the most influentialfactors of the BCC
All these analyses were performed bymeans of sequentialsimulations of the models under investigation thereforein general only small batches of simulations could be run(for instance in [34] only 836 simulations were executed toexecute the sensitivity analysis of the model) On the otherhand in this paper we presented the results of different PSAefficiently executed bymeans of coagSODA overall providinguseful information regarding unknownparameters and inter-esting insights into the functioning of the BCC A thoroughsensitivity analysis of the whole BCC model consisting in
around 5 sdot 105 simulations is currently in progress on GPUsby our group exploiting the great computational efficiency ofcoagSODA
The results of our computational analyses should benow validated by means of specifically designed laboratoryexperiments In particular we identified a plausible valuefor the constant of the reaction describing the autoactivationof factor XII (since no reference values can be found inliterature) moreover the results of the PSA over factor IXsuggest that a deficiency of this factor is not enough to causesevere bleeding disorders as haemophilia B but an alterationof other factors seems to be necessary for the occurrence ofsuch condition (eg the lack of Tissue Factor as suggestedby the PSA-2D over Tissue Factor and factor IX) Finallythe PSA over factors VIII and IX showed in a situationcharacterized by very high concentrations of these factorsa counterintuitive behavior in which the clotting time isincreased with respect to the value obtained in physiologicalconditions Indeed it would be interesting to design ad hoclaboratory experiments to verify if the BCCmodel is actuallypredictive in such extreme situations
The coagSODA software and the SBML version of theBCC model used in this work are available from the authorsupon request
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This work is partially supported by SYSBIO Centre forSystems Biology
References
[1] B B Aldridge JM BurkeDA Lauffenburger andP K SorgerldquoPhysicochemicalmodelling of cell signalling pathwaysrdquoNatureCell Biology vol 8 no 11 pp 1195ndash1203 2006
[2] I-C Chou and E O Voit ldquoRecent developments in parameterestimation and structure identification of biochemical andgenomic systemsrdquoMathematical Biosciences vol 219 no 2 pp57ndash83 2009
[3] D Besozzi P Cazzaniga D Pescini G Mauri S Colomboand EMartegani ldquoThe role of feedback control mechanisms onthe establishment of oscillatory regimes in the RascAMPPKApathway in S cerevisiaerdquo EURASIP Journal on Bioinformaticsand Systems Biology vol 2012 no 10 2012
[4] A Saltelli M Ratto T Andres et al Global Sensitivity AnalysisThe Primer Wiley-Interscience 2008
[5] A Raue C Kreutz T Maiwald et al ldquoStructural and practicalidentifiability analysis of partially observed dynamical modelsby exploiting the profile likelihoodrdquo Bioinformatics vol 25 no15 pp 1923ndash1929 2009
[6] A Drager M Kronfeld M J Ziller et al ldquoModeling metabolicnetworks in C glutamicum a comparison of rate laws incombination with various parameter optimization strategiesrdquoBMC Systems Biology vol 3 article 5 2009
BioMed Research International 19
[7] D Besozzi P Cazzaniga G Mauri D Pescini and L Van-neschi ldquoA comparison of genetic algorithms and particle swarmoptimization for parameter estimation in stochastic biochem-ical systemsrdquo in Proceedings of the 7th European ConferenceEvolutionary Computation Machine Learning and Data Miningin Bioinformatics (EvoBIO rsquo09) C Pizzuti M Ritchie and MGiacobini Eds vol 5483 of Lecture Notes in Computer Sciencepp 116ndash127 Springer 2009
[8] M S Nobile D Besozzi P Cazzaniga GMauri andD PescinildquocupSODA aGPU-basedmulti-swarmPSOmethod for param-eter estimation in stochastic biological systems exploitingdiscrete-time target seriesrdquo in Proceedings of the 10th EuropeanConference Evolutionary Computation Machine Learning andData Mining in Bioinformatics (EvoBIO rsquo12) M Giacobini LVanneschi and W Bush Eds vol 7246 of Lecture Notes inComputer Science pp 74ndash85 Springer
[9] M S Nobile D Besozzi P Cazzaniga D Pescini and GMaurildquoReverse engineering of kinetic reaction networks by meansof Cartesian Genetic Programming and Particle Swarm Opti-mizationrdquo in Proceedings of the IEEE Congress on EvolutionaryComputation (CEC rsquo13) pp 1594ndash1601 June 2013
[10] M S Nobile ldquoEvolutionary inference of biochemical inter-action networks accelerated on Graphics Processing Unitsrdquoin Proceedings of the 11th International Conference on HighPerformance Computingamp Simulation 2013 (HPCS rsquo13) pp 668ndash670 Helsinki Finland 2013
[11] K Kentzoglanakis and M Poole ldquoA swarm intelligence frame-work for reconstructing gene networks searching for bio-logically plausible architecturesrdquo IEEEACM Transactions onComputational Biology and Bioinformatics vol 9 no 2 pp 358ndash371 2012
[12] J R Koza W Mydlowec G Lanza Y Jessen and M A KeaneldquoAutomatic computational discovery of chemical reaction net-works using genetic programmingrdquo inComputationalDiscoveryof Scientific Knowledge S Dzeroski and L Todorovski Edsvol 4660 of Lecture Notes in Computer Science pp 205ndash227Springer 2007
[13] S Ando E Sakamoto and H Iba ldquoEvolutionary modeling andinference of gene networkrdquo Information Sciences vol 145 no3-4 pp 237ndash259 2002
[14] L Dematte and D Prandi ldquoGPU computing for systemsbiologyrdquo Briefings in Bioinformatics vol 11 no 3 Article IDbbq006 pp 323ndash333 2010
[15] J L Payne N A Sinnott-Armstrong and J HMoore ldquoExploit-ing graphics processing units for computational biology andbioinformaticsrdquo Interdisciplinary Sciences Computational LifeSciences vol 2 no 3 pp 213ndash220 2010
[16] M J Harvey and G De Fabritiis ldquoA survey of computationalmolecular science using graphics processing unitsrdquoWiley Inter-disciplinary Reviews Computational Molecular Science vol 2no 5 pp 734ndash742 2012
[17] Y Zhou J Liepe X Sheng M P H Stumpf and C BarnesldquoGPU accelerated biochemical network simulationrdquo Bioinfor-matics vol 27 no 6 Article ID btr015 pp 874ndash876 2011
[18] M Vigelius A Lane and B Meyer ldquoAccelerating reaction-diffusion simulations with general-purpose graphics processingunitsrdquo Bioinformatics vol 27 no 2 Article ID btq622 pp 288ndash290 2011
[19] D J Wilkinson ldquoStochastic modelling for quantitative descrip-tion of heterogeneous biological systemsrdquo Nature ReviewsGenetics vol 10 no 2 pp 122ndash133 2009
[20] J C Butcher ldquoTrees and numerical methods for ordinarydifferential equationsrdquo Numerical Algorithms vol 53 no 2-3pp 153ndash170 2010
[21] D T Gillespie ldquoStochastic simulation of chemical kineticsrdquoAnnual Review of Physical Chemistry vol 58 pp 35ndash55 2007
[22] L Petzold ldquoAutomatic selection of methods for solving stiffand nonstiff systems of ordinary differential equationsrdquo SIAMJournal of Scientific and Statistical Computing vol 4 no 1 pp136ndash148 1983
[23] Y Cao D T Gillespie and L R Petzold ldquoEfficient step sizeselection for the tau-leaping simulation methodrdquo The Journalof Chemical Physics vol 124 no 4 2006
[24] M S Nobile D Besozzi P Cazzaniga GMauri andD PescinildquocupSODA a CUDA-powered simulator of mass-action kinet-icsrdquo in Proceedings of the 12th International Conference onParallel Computing Technologies (PaCT rsquo13) V Malyshkin Edvol 7979 of Lecture Notes in Computer Science pp 344ndash357
[25] M S Nobile P Cazzaniga D Besozzi and G Mauri ldquoGPU-accelerated simulations of mass-action kinetics models withcupSODArdquoThe Journal of Supercomputing 2014
[26] M S Nobile P Cazzaniga D Besozzi D Pescini and GMauri ldquocuTauLeaping a GPU-powered tau-leaping stochasticsimulator for massive parallel analyses of biological systemsrdquoPLoS ONE vol 9 no 3 2014
[27] K Rogers Blood Physiology and CirculationTheHuman BodyBritannica Educational Publishing 2010
[28] A D Michelson Platelets Elsevier 2nd edition 2007[29] R W Colman J Hirsh V J Marder A W Clowes and J
N George Hemostasis and Thrombosis Basic Principles andClinical Practice Lippincott Williams ampWilkins 2006
[30] M S Chatterjee W S Denney H Jing and S L DiamondldquoSystems biology of coagulation initiation Kinetics of thrombingeneration in resting and activated human bloodrdquo PLoS Com-putational Biology vol 6 no 9 Article ID e1000950 2010
[31] M Bellini D Besozzi P Cazzaniga GMauri andM S NobileldquoSimulation and analysis of the blood coagulation cascadeaccelerated on GPUrdquo in Proceedings of the 22nd EuromicroInternational Conference on Parallel Distributed and Network-Based Processing (PDP) pp 590ndash593 IEEE Computer SocietyConference Publishing Services 2014
[32] T Wajima G K Isbister and S B Duffull ldquoA comprehensivemodel for the humoral coagulation network in humansrdquo Clini-cal Pharmacology and Therapeutics vol 86 no 3 pp 290ndash2982009
[33] K Lo W S Denney and S L Diamond ldquoStochastic modelingof blood coagulation initiationrdquo Pathophysiology of Haemostasisand Thrombosis vol 34 no 2-3 pp 80ndash90 2006
[34] C M Danforth T Orfeo K G Mann K E Brummel-Ziedins and S J Everse ldquoThe impact of uncertainty in a bloodcoagulation modelrdquo Mathematical Medicine and Biology vol26 no 4 Article ID dqp011 pp 323ndash336 2009
[35] S D Lewis P P Shields and J A Shafer ldquoCharacterizationof the kinetic pathway for liberation of fibrinopeptides duringassembly of fibrinrdquoThe Journal of Biological Chemistry vol 260no 18 pp 10192ndash10199 1985
[36] M F Hockin K C Jones S J Everse and K G Mann ldquoAmodel for the stoichiometric regulation of blood coagulationrdquoThe Journal of Biological Chemistry vol 277 no 21 pp 18322ndash18333 2002
[37] K Brummel-Ziedins C YVossen F R Rosendaal KUmezakiand K G Mann ldquoThe plasma hemostatic proteome thrombin
20 BioMed Research International
generation in healthy individualsrdquo Journal of Thrombosis andHaemostasis vol 3 no 7 pp 1472ndash1481 2005
[38] K E Brummel-Ziedins C Y Vossen S Butenas K G Mannand F R Rosendaal ldquoThrombin generation profiles in deepvenous thrombosisrdquo Journal of Thrombosis and Haemostasisvol 3 no 11 pp 2497ndash2505 2005
[39] S D Bungay P A Gentry and R D Gentry ldquoA mathematicalmodel of lipid-mediated thrombin generationrdquo MathematicalMedicine and Biology vol 20 no 1 pp 105ndash129 2003
[40] S Butenas T Orfeo M T Gissel K E Brummel and K GMann ldquoThe significance of circulating factor IXa in bloodrdquoThe Journal of Biological Chemistry vol 279 no 22 pp 22875ndash22882 2004
[41] D Voet and J G Voet Biochemistry Wiley 2011[42] A C Guyton and J E Hall Textbook of Medical Physiology
Saunders 2010[43] M F M James and A M Roche ldquoDose-response relationship
between plasma ionized calcium concentration and thrombe-lastographyrdquo Journal of Cardiothoracic and Vascular Anesthesiavol 18 no 5 pp 581ndash586 2004
[44] D L Nelson and M M Cox Lehninger Principles of Biochem-istry W H Freeman Company 2004
[45] R M Farber ldquoTopical perspective on massive threading andparallelismrdquo Journal of Molecular Graphics and Modelling vol30 pp 82ndash89 2011
[46] OWolkenhauerMUllahWKolch andK-HCho ldquoModelingand simulation of intracellular dynamics choosing an appropri-ate frameworkrdquo IEEETransactions onNanobioscience vol 3 no3 pp 200ndash207 2004
[47] B H Estridge A P Reynolds and N J Walters Basic MedicalLaboratory Techniques Cengage Learning 4th edition 2000
[48] T Renne A H Schmaier K F Nickel M Blomback and CMaas ldquoIn vivo roles of factor XIIrdquo Blood vol 120 no 22 pp4296ndash4303 2012
[49] P V Jenkins O Rawley O P Smith and J S ODonnellldquoElevated factor VIII levels and risk of venous thrombosisrdquoBritish Journal of Haematology vol 157 no 6 pp 653ndash663 2012
[50] C E Walsh and K M Batt ldquoHemophilia clinical gene therapybrief reviewrdquo Translational Research vol 161 no 4 pp 307ndash3122013
[51] A Van Hylckama Vlieg I K Van Der Linden R M Bertinaand F R Rosendaal ldquoHigh levels of factor IX increase the risk ofvenous thrombosisrdquo Blood vol 95 no 12 pp 3678ndash3682 2000
[52] K Brummel-Ziedins ldquoModels for thrombin generation and riskof diseaserdquo Journal ofThrombosis and Haemostasis vol 11 no 1pp 212ndash223 2013
[53] U Seligsohn and A Lubetsky ldquoGenetic susceptibility to venousthrombosisrdquoTheNew England Journal of Medicine vol 344 no16 pp 1222ndash1231 2001
[54] I Martinelli P Bucciarelli and P M Mannucci ldquoThromboticrisk factors basic pathophysiologyrdquo Critical Care Medicine vol38 no 2 pp S3ndashS9 2010
[55] K Brummel-Ziedins A Undas T Orfeo et al ldquoThrombingeneration in acute coronary syndrome and stable coronaryartery disease dependence on plasma factor compositionrdquoJournal ofThrombosis andHaemostasis vol 6 no 1 pp 104ndash1102008
[56] S Hoops R Gauges C Lee et al ldquoCOPASImdasha complexpathway simulatorrdquo Bioinformatics vol 22 no 24 pp 3067ndash3074 2006
[57] Nvidia NVIDIArsquos Next Generation CUDA Compute Architec-ture Kepler GK110 2012
[58] W Stortelder W Hemker and H Hemker ldquoMathematicalmodelling in blood coagulation simulation and parameterestimationrdquoModelling Analysis and Simulation vol 20 pp 1ndash111997
[59] S M C Orfao G Jank and K Mottaghy ldquoA new mathematicalapproach to modelling thrombin generationrdquo InternationalJournal of Artificial Organs vol 29 no 7 pp 703ndash708 2006
[60] Nvidia Nvidia CUDA C Programming Guide v50 2012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Anatomy Research International
PeptidesInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
International Journal of
Volume 2014
Zoology
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Molecular Biology International
GenomicsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
BioinformaticsAdvances in
Marine BiologyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Signal TransductionJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
BioMed Research International
Evolutionary BiologyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Biochemistry Research International
ArchaeaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Genetics Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in
Virolog y
Hindawi Publishing Corporationhttpwwwhindawicom
Nucleic AcidsJournal of
Volume 2014
Stem CellsInternational
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Enzyme Research
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Microbiology
BioMed Research International 3
2 Materials and Methods
21 A Mechanistic Model of the Blood Coagulation CascadeBlood is an essential component in human life whoseprimary functions are to feed cells by delivering a multitudeof nutrients such as oxygen and to carry away the cellularwastes such as carbon dioxide Specialized cells and fluidsin blood perform many physiological functions and can beisolated and analyzed through specific laboratory tests givingthe opportunity to settle a personrsquos health condition Allblood components are kept within appropriate concentrationranges by means of fine-tuned regulatory mechanisms ruledby several feedback controls the constancy of blood com-position is maintained thanks to the circulation through anintricate network of vessels In particular humans evolved acomplex hemostatic system that under physiological condi-tions maintains blood in a fluid state however in responseto any vascular injury this system is able to rapidly react andseal the defects in the vesselsrsquo wall in order to stop the bloodleakage [29] Indeed the circulatory system is self-sealingotherwise a continuous blood flow from even the smallestwound would become a threat for the individualrsquos life
To allow blood coagulation in humans there exist 13blood clotting proteins called coagulation factors which areusually designated by Roman numerals I through XIII Asa consequence of a vascular injury platelets become activeand the Tissue Factor (TF also called factor III) is exposedin the subendothelial tissue starting the blood coagulationcascade (BCC) The ultimate goal of the BCC is to convertprothrombin (factor II) into thrombin (factor IIamdashie theactive factor II) the enzyme that catalyzes the formation ofa clot Traditionally the BCC is divided into the extrinsic andintrinsic pathways both leading to the activation of factor X[41] The last part of the cascade downstream of this factoris called the common pathway and leads to the formationof fibrin monomers whose polymers finally constitute thebackbone of the clot
Excluding thrombin all the enzymes involved in bloodclotting are characterized by a low activity which increasesupon binding to a specific protein cofactor (eg factors Vand VIII) or to appropriate phospholipid surfaces (eg theplasmamembranes of active platelets) [41] Even calcium ionshave a central role in coagulation since they are essentialto start and enhance numerous reactions without calciumions the blood coagulation cannot occur [42] In the BCCpathways the activity of the various active proteases is limitedby several inhibitory factors which allow regulating thewhole cascade When the hemostatic system is unregulatedthrombosis (ie the formation of a blood clot obstructing theblood flow in vessels) may occur due to impairment in theinhibitory pathway or because the functioning of the naturalanticoagulant processes is overwhelmed by the strength of thehemostatic stimulus [29]
The BCC model we consider in this work is a slightlyreduced version of the ldquoPlatelet-Plasmardquo deterministic modeldefined in [30] built upon a previous model [36] whichdescribes all parts of blood coagulation the plateletsactivation and aggregation the extrinsic intrinsic andcommon pathways (with the exception of factor XIII)
the action of several inhibitory molecules (Tissue Fac-tor Pathway Inhibitor antithrombin III C1-inhibitor 1205721-antitrypsin and 1205722-antiplasmin) In addition to simulate thecoagulation process in vitro Chatterjee et al also modeledthe action of corn trypsin inhibitor (CTI) which inhibits theactivation of the so-called contact system as well as the actionof the fluorogenic substrate Boc-VPR-MCA widely used inlaboratories for thrombin titration [30] The role of calciumwas not explicitly included in the model but considered asa nonlimiting factor as in living organisms this ion veryrarely drops to such low levels able to alter the kinetics of theformation of the fibrin clot [43]
Since the aim of this work is the investigation of bloodcoagulation in vivo we exclude a small set of reactions givenin [30] that have no effect on the clotting time To be moreprecise we do not consider the reactions occurring in vitro(namely entries 28 and 35 in Table 1 in [30]) as well asthe reactions downstream the fibrinogen conversion that isthe interactions between the fibrin polymers thrombin andantithrombin III (namely entries 55 56 and 57 in Table 1 in[30])
For the sake of completeness we describe in Table 1 theBCC model considered in this work overall consisting in 96reactions among 71 molecular species A graphical sketch ofthemainmolecular interactions among the BCC componentsis given in Figure 1
The model can be partitioned into four functional mod-ules
(1) Thefirstmodule corresponds to the extrinsic pathwaywhich consists in
(i) the formation of a complex between TissueFactor and factor VII (modeled by reactions1199031 119903
4)
(ii) the activation of factor VII by the complexbetween Tissue Factor and factor VIIa (reaction1199035)
(iii) the activation of factor IX by the complexbetween Tissue Factor and factor VIIa (reac-tions 119903
13 11990314 11990315) and by factor VIIa (reactions
11990378 11990379 11990380)
(iv) the activation of factor X by the complexbetween Tissue Factor and factor VIIa(1199038 119903
12) and by factor VIIa (119903
81 11990382 11990383)
(2) The second module corresponds to the intrinsic path-way which consists in
(i) the formation of a complex between factor VIIIaand factor IXa (modeled by reactions 119903
18and
11990319)
(ii) the activation of factor X by the complexbetween factor VIIIa and factor IXa (reactions11990320 11990321 11990322) and by factor IXa (reactions 119903
72 11990373
11990374)
(iii) the activation of factor XII by factor XII itself(reaction 119903
44) by factor XIIa (reactions 119903
45 11990346
11990347) and by kallikrein (reactions 119903
51 11990352 11990353)
4 BioMed Research International
Table 1 Reaction-based model of the blood coagulation cascade (reduced version of the ldquoPlatelet-Plasmardquo model described in [30]) Themodel consists in 96 reactions among 71 molecular species With the exception of reaction 119903
44(see Section 31) the values of all reaction
constants were taken from [30]
119899 Reactants Products Constant (119896)1199031
TF + fVII TF-fVII 320 sdot 106Mminus1 sminus1
1199032
TF-fVII TF + fVII 310 sdot 10minus2 sminus1
1199033
TF + fVIIa TF-fVIIa 230 sdot 107Mminus1 sminus1
1199034
TF-fVIIa TF + fVIIa 310 sdot 10minus5 sminus1
1199035
TF-fVIIa + fVII TF-fVIIa + fVIIa 440 sdot 105Mminus1 sminus1
1199036
fXa + fVII fXa + fVIIa 130 sdot 107Mminus1 sminus1
1199037
fIIa + fVII fIIa + fVIIa 230 sdot 104Mminus1 sminus1
1199038
TF-fVIIa + fX TF-fVIIa-fX 250 sdot 107Mminus1 sminus1
1199039
TF-fVIIa-fX TF-fVIIa + fX 105 sdot 10minus2 sminus1
11990310
TF-fVIIa-fX TF-fVIIa-fXa 600 sminus1
11990311
TF-fVIIa-fXa TF-fVIIa + fXa 1900 sminus1
11990312
TF-fVIIa + fXa TF-fVIIa-fXa 220 sdot 107Mminus1 sminus1
11990313
TF-fVIIa + fIX TF-fVIIa-fIX 100 sdot 107Mminus1 sminus1
11990314
TF-fVIIa-fIX TF-fVIIa + fIX 240 sminus1
11990315
TF-fVIIa-fIX TF-fVIIa + fIXa 180 sminus1
11990316
fII + fXa fIIa + fXa 750 sdot 103Mminus1 sminus1
11990317
fIIa + fVIII fIIa + fVIIIa 200 sdot 107Mminus1 sminus1
11990318
fVIIIa + fIXa fIXa-fVIIIa 100 sdot 107Mminus1 sminus1
11990319
fIXa-fVIIIa fVIIIa + fIXa 100 sdot 10minus4 sminus1
11990320
fIXa-fVIIIa + fX fIXa-fVIIIa-fX 100 sdot 108Mminus1 sminus1
11990321
fIXa-fVIIIa-fX fIXa-fVIIIa + fX 100 sdot 10minus5 sminus1
11990322
fIXa-fVIIIa-fX fIXa-fVIIIa + fXa 820 sminus1
11990323
fVIIIa fVIIIa1-L + fVIIIa2 600 sdot 10minus5 sminus1
11990324
fVIIIa1-L + fVIIIa2 fVIIIa 220 sdot 104 Mminus1 sminus1
11990325
fIXa-fVIIIa-fX fVIIIa1-L + fVIIIa2 + fX + fIXa 100 sdot 10minus3 sminus1
11990326
fIXa-fVIIIa fVIIIa1-L + fVIIIa2 + fIXa 100 sdot 10minus3 sminus1
11990327
fIIa + fV fIIa + fVa 200 sdot 107Mminus1 sminus1
11990328
fXa + fVa fXa-fVa 400 sdot 108Mminus1 sminus1
11990329
fXa-fVa fXa + fVa 02 sminus1
11990330
fXa-fVa + fII fXa-fVa-fII 100 sdot 108Mminus1 sminus1
11990331
fXa-fVa-fII fXa-fVa + fII 10300 sminus1
11990332
fXa-fVa-fII fXa-fVa + fmIIa 6350 sminus1
11990333
fXa-fVa + fmIIa fXa-fVa + fIIa 150 sdot 107Mminus1 sminus1
11990334
fXa + TFPI fXa-TFPI 900 sdot 105Mminus1 sminus1
11990335
fXa-TFPI fXa + TFPI 360 sdot 10minus4 sminus1
11990336
TF-fVIIa-fXa + TFPI TF-fVIIa-fXa-TFPI 320 sdot 108 Mminus1 sminus1
11990337
TF-fVIIa-fXa-TFPI TF-fVIIa-fXa + TFPI 110 sdot 10minus2 sminus1
11990338
TF-fVIIa + fXa-TFPI TF-fVIIa-fXa-TFPI 500 sdot 107Mminus1 sminus1
11990339
fXa + ATIII fXa-ATIII 150 sdot 103Mminus1 sminus1
11990340
fmIIa + ATIII fmIIa-ATIII 710 sdot 103Mminus1 sminus1
11990341
fIXa + ATIII fIXa-ATIII 490 sdot 102Mminus1 sminus1
11990342
fIIa + ATIII fIIa-ATIII 710 sdot 103Mminus1 sminus1
11990343
TF-fVIIa + ATIII TF-fVIIa-ATIII 230 sdot 102Mminus1 sminus1
11990344
fXII fXIIa 500 sdot 10minus4 sminus1
11990345
fXIIa + fXII fXIIa-fXII 100 sdot 108Mminus1 sminus1
11990346
fXIIa-fXII fXIIa + fXII 75000 sminus1
11990347
fXIIa-fXII fXIIa + fXIIa 330 sdot 10minus2 sminus1
11990348
fXIIa + PKal fXIIa-PKal 100 sdot 108Mminus1 sminus1
11990349
fXIIa-PKal fXIIa + PKal 360 sdot 103 sminus1
BioMed Research International 5
Table 1 Continued
119899 Reactants Products Constant (119896)11990350
fXIIa-PKal fXIIa + Kal 4000 sminus1
11990351
fXII + Kal fXII-Kal 100 sdot 108Mminus1 sminus1
11990352
fXII-Kal fXII + Kal 4530 sminus1
11990353
fXII-Kal fXIIa + Kal 570 sminus1
11990354
PKal + Kal Kal + Kal 270 sdot 104Mminus1 sminus1
11990355
Kal Kal119894
110 sdot 10minus2 sminus1
11990356
fXIIa + C1inh fXIIa-C1inh 360 sdot 103Mminus1 sminus1
11990357
fXIIa + ATIII fXIIa-ATIII 2160Mminus1 sminus1
11990358
fXI + fIIa fXI-fIIa 100 sdot 108Mminus1 sminus1
11990359
fXI-fIIa fXI + fIIa 500 sminus1
11990360
fXI-fIIa fXIa + fIIa 130 sdot 10minus4 sminus1
11990361
fXIIa + fXI fXIIa-fXI 100 sdot 108Mminus1 sminus1
11990362
fXIIa-fXI fXIIa + fXI 200 sminus1
11990363
fXIIa-fXI fXIIa + fXIa 570 sdot 10minus4 sminus1
11990364
fXIa + fXI fXIa + fXIa 319 sdot 106Mminus1 sminus1
11990365
fXIa + ATIII fXIa-ATIII 320 sdot 102Mminus1 sminus1
11990366
fXIa + C1inh fXIa-C1inh 180 sdot 103Mminus1 sminus1
11990367
fXIa + A1AT fXIa-A1AT 100 sdot 102Mminus1 sminus1
11990368
fXIa + A2AP fXIa-A2AP 43 sdot 103Mminus1 sminus1
11990369
fXIa + fIX fXIa-fIX 100 sdot 108Mminus1 sminus1
11990370
fXIa-fIX fXIa + fIX 4100 sminus1
11990371
fXIa-fIX fXIa + fIXa 770 sminus1
11990372
fIXa + fX fIXa-fX 100 sdot 108Mminus1 sminus1
11990373
fIXa-fX fIXa + fX 064 sminus1
11990374
fIXa-fX fIXa + fXa 700 sdot 10minus4 sminus1
11990375
fXa + fVIII fXa-fVIII 100 sdot 108Mminus1 sminus1
11990376
fXa-fVIII fXa + fVIII 210 sminus1
11990377
fXa-fVIII fXa + fVIIIa 0023 sminus1
11990378
fVIIa + fIX fVIIa-fIX 100 sdot 108Mminus1 sminus1
11990379
fVIIa-fIX fVIIa + fIX 090 sminus1
11990380
fVIIa-fIX fVIIa + fIXa 360 sdot 10minus5 sminus1
11990381
fVIIa + fX fVIIa-fX 100 sdot 108Mminus1 sminus1
11990382
fVIIa-fX fVIIa + fX 21000 sminus1
11990383
fVIIa-fX fVIIa + fXa 160 sdot 10minus6 sminus1
11990384
Fbg + fIIa Fbg-fIIa 100 sdot 108Mminus1 sminus1
11990385
Fbg-fIIa Fbg + fIIa 63600 sminus1
11990386
Fbg-fIIa Fbn1 + fIIa + FPA 8400 sminus1
11990387
Fbn1 + fIIa Fbn1-fIIa 100 sdot 108Mminus1 sminus1
11990388
Fbn1-fIIa Fbn1 + fIIa 74260 sminus1
11990389
Fbn1-fIIa Fbn2 + fIIa + FPB 740 sminus1
11990390
Fbn1 + Fbn1 (Fbn1)2 100 sdot 106Mminus1 sminus1
11990391
(Fbn1)2 2Fbn1 640 sdot 10minus2 sminus1
11990392
(Fbn1)2 + fIIa (Fbn1)2-fIIa 100 sdot 108Mminus1 sminus1
11990393
(Fbn1)2-fIIa (Fbn1)2 + fIIa 70100 sminus1
11990394
(Fbn1)2-fIIa (Fbn2)2 + fIIa + FPB 4900 sminus1
11990395
Fbn2 + fIIa Fbn2-fIIa 100 sdot 108Mminus1 sminus1
11990396
Fbn2-fIIa Fbn2 + fIIa 100 sdot 103 sminus1
6 BioMed Research International
fXIITF-fVII
TFfVIIa
fVII
TF-fVIIa
Fbg
FPA
Fbn1
FPB
Fbn2
Fbn1
(Fbn1)2
(Fbn2)2 FPB
fXa
fIIa
TF-fVIIa-fXa
fIX
fIXa
fX
fmIIa
fII
PKal
KalifXI
fXIa
fIX
fIXa
fVIII
fVIIIa
fIXa-fVIIIa
fVIIIa1 -L
fVIIIa2
fVIIIa2
fIXa
fV
fVa
fXa-fVa
fXa-TFPI
TFPI
TF-fVIIa-fXa-TFPI
fXa-AT
AT
AT
fXIa-AT
C1inh
fXIIa-C1inh
A1AT
A2AP
fXIa-A2APfXIa-A1AT
TF-fVIIa
fXIIa Kal
fVIIIa1 -L
Figure 1 Graphical representation of the blood coagulation cascade model considered in this work Legend Blue box coagulation factor redbox inhibitor and related complexes Black arrow complex formation green arrow catalytic activation violet arrow activation red arrowinhibition The reaction is reversible if the arrow tail consists in a dot
BioMed Research International 7
(iv) the activation of prekallikrein by factor XIIa(reactions 119903
48 11990349 11990350) and by kallikrein (reac-
tion 11990354)
(v) the activation of factor XI by factor XIIa (reac-tions 119903
61 11990362 11990363) and by factor XIa (reaction
11990364)
(vi) the activation of factor IX by factor XIa (reac-tions 119903
69 11990370 11990371)
(vii) the dissociation of free factor VIIIa (reactions11990323and 11990324)
(viii) the dissociation of factor VIIIa in complex withother factors (reactions 119903
25and 11990326)
(3) The third module corresponds to the common path-way which consists in
(i) the activation of factor II by factor Xa (modeledby reaction 119903
16) and by the complex between
factor Xa and factor Va through the formationof the intermediate meizothrombin (reactions11990330 119903
33)
(ii) the activation of factor VII by factor Xa (reac-tion 1199036) and by factor IIa (reaction 119903
7)
(iii) the activation of factor VIII by factor Xa (reac-tions 119903
75 11990376 11990377) and by factor IIa (reaction 119903
17)
(iv) the formation of a complex between factor Xaand factor Va (reactions 119903
28and 11990329)
(v) the activation of fibrinogen by factor IIa (reac-tions 119903
84 119903
96)
(vi) the activation of factor V by factor IIa (reaction11990327)
(vii) the activation of factor XI by factor IIa (reac-tions 119903
58 11990359 11990360)
(4) The fourth module describes the inhibition of coagu-lation carried out through the main inhibitors of theBCC (antithrombin Tissue Factor Pathway InhibitorC1-inihibitor 1205721-antitrypsin and 1205722-antiplasmin)These are tight binding inhibitors belonging to theserpin superfamily which form irreversible com-plexes This module consists in
(i) the inhibition of factor Xa by Tissue FactorPathway Inhibitor (modeled by reactions 119903
34and
11990335)
(ii) the inhibition of the complex between TissueFactor factor VIIa and factor Xa by TissueFactor Pathway Inhibitor (reactions 119903
36 11990337 11990338)
(iii) the inhibition of factor IIa by antithrombin(reactions 119903
40and 11990342)
(iv) the inhibition of factor Xa by antithrombin(reaction 119903
39)
(v) the inhibition of factor IXa by antithrombin(reaction 119903
41)
(vi) the inhibition of factor XIa by antithrombin(reaction 119903
65) and by C1-inhibitor (reaction 119903
66)
Table 2 Initial concentrations of molecular species in the bloodcoagulation cascade model (values taken from [30])
Species Symbol Concentration (M)1205721-Antitrypsin A1AT 450 sdot 10
minus5
1205722-Antiplasmin A2AP 100 sdot 10minus6
Antithrombin III ATIII 340 sdot 10minus6
C1-inhibitor C1inh 250 sdot 10minus6
Fibrinogen Fbg 900 sdot 10minus6
Factor II fII 140 sdot 10minus6
Factor V fV 200 sdot 10minus8
Factor VII fVII 100 sdot 10minus8
Active factor VII fVIIa 100 sdot 10minus10
Factor VIII fVIII 700 sdot 10minus10
Factor IX fIX 900 sdot 10minus8
Factor X fX 16 sdot 10minus7
Factor XI fXI 310 sdot 10minus8
Factor XII fXII 340 sdot 10minus7
Prekallikrein Pkal 450 sdot 10minus7
Tissue Factor TF 500 sdot 10minus12
Tissue Factor Pathway Inhibitor TFPI 250 sdot 10minus9
(vii) the inhibition of factor XIIa by C1-inhibitor(reaction 119903
56) by antithrombin (reaction 119903
57)
by 1205721-antitrypsin (reaction 11990367) and by 1205722-
antiplasmin (reaction 11990368)
(viii) the inhibition of the complex between TissueFactor and factor VIIa by antithrombin (reac-tion 11990343)
(ix) the inhibition of kallikrein (reaction 11990355)
The values of the initial concentrations of the molecularspecies occurring in the BCC model are given in Table 2According to [30] the concentrations of complexes and activefactors were set to 0 except for the active factor VII which isphysiologically present in the blood circulation even in theabsence of damage in a concentration that is approximatelyequal to 1 of the corresponding inactive factor [42]
The system of ordinary differential equations (ODEs)needed to carry out the simulations and the parametersweep analysis presented in Section 3 was derived fromthe reactions given in Table 1 according to the mass-actionlaw with the exception of 14 reactions belonging to the set119878120576= 11990319 11990321 11990323 11990329 11990331 11990335 11990346 11990349 11990352 11990362 11990370 11990373 11990376 11990382
The mass-action law is the fundamental empirical law thatgoverns biochemical reaction rates which states that in adilute solution the rate of an elementary reaction is propor-tional to the product of the concentration of its reactantsraised to the power of the corresponding stoichiometriccoefficient [44] The reactions in 119878
120576are instead influenced by
a specifically defined variable 120576 depending on aHill functionfit against experimental data which quantify the plateletactivation status which is used to model the physiologicallevels of thrombin concentration as a function of plateletactivation as thoroughly described in [30] More preciselythe value of 120576 influences the formation of some complexes
8 BioMed Research International
0 0 100 200 300 400 500 600 700
Fact
or II
a con
cent
ratio
n (M
)
Time (s)
1eminus7
2eminus7
3eminus7
4eminus7
5eminus7
6eminus7
7eminus7
8eminus7
9eminus7
Figure 2 Dynamics of thrombin (factor IIa) in physiological condition
occurring on the plateletsrsquo surface by modifying the activityof reactions of the form
119860 + 119861
1198961
999445999468
1198962
119860119861 (1)
Namely 120576 intervenes with the dissociation constant of thesereactions so that the corresponding standard ODEs arechanged to yield new equations of the form
119889 [119860119861]
119889119905
= 1198961sdot [119860] sdot [119861] minus
1198962sdot [119860] sdot [119861]
120576
(2)
The value of 120576 in (2) depends on the following HillfunctionH which quantifies the state of platelets activationaccording to the thrombin concentration (here denoted as[fIIa]) that is the factor catalyzing the formation of the fibrinclot
H ([fIIalowast (119905)]) =[fIIalowast (119905)]16123
[fIIalowast (119905)]16123 + (24279 sdot 10minus9)16123
(3)
where [fIIalowast(119905)] = max1199051015840isin[0119905][fIIa(1199051015840)] for a chosen time
interval [0 119905] of simulationThe value [fIIalowast(119905)] represents the maximum transient
thrombin concentration and is needed to simulate the factthat in physiological conditions the thrombin concentrationstarts decreasing after rising to a peak (Figure 2) So doing[fIIalowast(119905)] never decreases once it reaches its maximummagnitude [fIIalowast(119905)] is equivalent to [fIIa(119905)] until the con-centration of factor IIa reaches the peak while thereafter itremains constant at that value which is the maximum in theconsidered time interval [0 119905] FunctionH allows simulatingthe physiological condition whereby platelets remain activealso when the thrombin concentration decreases
For a given concentration of factor IIa the maximumplatelets activation state 120576max is defined as
120576max = 120576max0
+ (1 minus 120576max0
) sdotH ([fIIalowast (119905)]) (4)
where 120576max0
defines the basal activation state of the plateletsat simulation time 119905 = 0 The value 120576max
0
is initially set to 001assuming a basal 1 binding strength of coagulation factorsto the resting plateletsrsquo surface When the full activationof platelets is reached 120576max is equal to 1 and the complexdissociation constants are minimized (see [30] for moredetails)
22 CUDA Architecture and the coagSODA Simulator Intro-duced by Nvidia in 2006 Compute Unified Device Architec-ture (CUDA) is a parallel computing platform and program-ming model that provides programmers with a frameworkto exploit GPUs in general-purpose computational tasks(GPGPU computing) GPGPU computing is a low-cost andenergy-wise alternative to the traditional high-performancecomputing infrastructures (eg clusters of machines) whichgives access to the tera-scale computing on common work-stations of mid-range price However due to the innovativearchitecture and the intrinsic limitations of GPUs a directporting of sequential code on the GPU is most of the timesunfeasible therefore making it challenging to fully exploit itscomputational power and massive parallelism [45]
CUDA combines the single instruction multiple data(SIMD) architecture with multithreading which automat-ically handles the conditional divergence between threadsThe drawback of such flexibility is that any divergence ofthe execution flow between threads causes a serialization ofthe execution which affects the overall performances UnderCUDArsquos naming conventions the programmer implementsthe kernel that is a CC++ function which is loaded fromthe host (the CPU) to the devices (one or more GPUs)and replicated in many copies named threads Threads canbe organized in three-dimensional structures named blockswhich in turn are contained in three-dimensional grids(a schematic description is given in Figure 3 left side)Whenever the host runs a kernel the GPU creates thecorresponding grid and automatically schedules each blockon one free streaming multiprocessor available on the GPUallowing a transparent scaling of performances on different
BioMed Research International 9
Host
Kernel 1
Kernel 2
Device
Grid 1
Grid 2
Block(0 0)
Block(1 0)
Block(2 0)
Block(0 1)
Block(1 1)
Block(2 1)
Block (1 1)
Thread(0 0)
Thread(1 0)
Thread(2 0)
Thread(3 0)
Thread(4 0)
Thread(0 1)
Thread(1 1)
Thread(2 1)
Thread(3 1)
Thread(4 1)
Thread(0 2)
Thread(1 2)
Thread(2 2)
Thread(3 2)
Thread(4 2)
Global memory
Texture memory
Constant memory
Shared memory Shared memory
Block (0 0) Block (1 0)
Localmemory
Localmemory
Localmemory
Localmemory
Thread(0 0)
Thread(1 0)
Registers Registers Registers Registers
Thread(0 0)
Thread(1 0)
Grid
Figure 3 Schematic description of CUDArsquos architecture in terms of threads and memory hierarchy Left Side Threads organization a singlekernel is launched from the host (the CPU) and is executed in multiple threads on the device (the GPU) Threads can be organized in three-dimensional structures named blocks which can be in turn organized in three-dimensional grids The dimensions of blocks and grids areexplicitly defined by the programmer Right Side Memory hierarchy threads can access data from many different memories with differentscopes Registers and local memories are private for each thread Shared memory lets threads belonging to the same block communicate andhas low access latency All threads can access the global memory which suffers high latencies but it is cached since the introduction of theFermi architecture Texture and constant memory can be read from any thread and are equipped with a cache as well in this work we exploitthe constant memory Figures are taken from Nvidiarsquos CUDA programming guide [60]
devices Threads within a block are executed in groups of 32threads named warps
The GPU is equipped with different kinds of memoryIn this work we exploit the global memory (accessible fromall threads) the shared memory (accessible from threadsof the same block) the local memory (registers and arraysaccessible from owner thread) and the constant memory(cached and not modifiable) A schematic representation ofthis memory hierarchy is shown in Figure 3 right side Toachieve the best performances the sharedmemory should beexploited as much as possible however it is very limited (ie49152 bytes for each multiprocessor since the introductionof the Fermi architecture) and it introduces restrictions onthe blocksrsquo size On the other hand the global memory isvery large (thousands ofMBs) but suffers from high latenciesA solution to this problem was implemented on the Fermiarchitecture where the global memory is equipped with aL2 cache This architecture also introduced the possibility tobalance 64KB of fast on-chip memory between the sharedmemory and L1 cache using two possible configurations48KB for the shared memory and 16KB for L1 cache or16 KB for the shared memory and 48 for L1 cacheThe Keplerarchitecture used in this paper allows a third and perfectly
balanced configuration where shared memory and L1 cacheobtain the same amount of memory (32KB) See Figure 4 fora schematization of the memory architecture
The systematic analysis of models of biological systemsoften consists in the execution of large batches of simulationsOne of the standard analyses that can be executed on suchkind of models regards an intensive search within the param-eters space which requires large numbers of independentsimulations In order to reduce the computational burdenwe previously implemented on the CUDA architecture oneof the most efficient numerical integration algorithms forODEs LSODA that is able to automatically recognize stiffand nonstiff systems and to dynamically select between themost appropriate integration procedure (ie Adams methodin the absence of stiffness and the Backward DifferentiationFormulae otherwise) [22] This GPU-powered tool is calledcupSODA [24 25] and exploits CUDArsquos massive parallelismto execute different and independent simulations in eachthread thus reducing the computational time required by astandard CPU counterpart of LSODA
Besides being very efficient for the simulation of manyindependent simulations cupSODA is also user-friendlyLSODA was originally designed to solve ODEs systems
10 BioMed Research International
Thread
Shared memory L1 cache Read-onlydata cache
L2 cache
DRAM
Figure 4 Schematic description of memory hierarchies in Fermi and Kepler architectures GPUs relying on these architectures are equippedwith a two-level data cache and a read-only data cache Shared memory and L1 cache share the same on-chip 64KB memory banks theamount of memory can be reconfigured by the user according to the specific needs of the application Figure taken from Nvidiarsquos KeplerGK110 whitepaper [57]
written in the canonical form but the user is supposed tospecify the system of ODEs by implementing a custom Cfunction that is passed to the algorithmmoreover in order tospeed up the computation when dealing with stiff systems inLSODA the Jacobianmatrix associatedwith theODEs systemmust be implemented as a custom C function as well On thecontrary cupSODA was conceived as a black-box simulatorthat can be easily used without any programming skillscupSODA consists in a tool to automatically convert a genericmechanistic reaction-based model of a biological system intothe corresponding set of ODEs to comply with the mass-action kinetics [46] and to directly encode the obtainedsystem along with the corresponding Jacobian matrix as Carrays
This fully automaticmethodology can be exploited for thesimulation of models whose chemical kinetics is based on themass-action law The BCC model described in Section 21though includes a set 119878
120576of reactions that do not follow
the mass-action kinetics [30] the platelets activity is notexplicitly modeled by biochemical reactions but it is realizedby modulating the rate of the dissociation of the complexesformed on a plateletrsquos surface by means of the variable120576 which is calculated with a special equation during theintegration steps For this reason cupSODA was used asa starting point for the development of a new tool ableto compute at run-time the specific kinetics of this set ofreactions
This new CUDA-powered simulation tool named coag-SODA is specifically tailored for the simulation of the BCC
model developed in [30] In particular coagSODA realizesthe run-time calculation of the 120576 value required to correctlysimulate the activity of platelets which is determined accord-ing to the equations described in Section 21 The plateletsactivation state 120576 is calculated at each time instant by solvingthe following differential equation
119889120576
119889119905
= 119896 (120576max minus 120576) (5)
where the constant 119896 is inversely proportional to the timescale of platelets activation and is set to 0005 This isconsistent with the fact that platelets do not instantly achievetheir maximum attainable activation state (120576max) but theyreach it on a physiologically relevant timescale [30]
Dealing with the 14 reactions in the set 119878120576that are
influenced by the Hill functionH (see Section 21) the valueof [fIIalowast(119905)]must be stored on the GPU because during eachintegration step coagSODA recalculates (4) which exploitsthe value [fIIalowast(119905)] to determine a new 120576 value Since CUDArsquosarchitecture does not offer static variables the informationfor each thread has to be memorized in the global memoryThe accesses to the global memory and the computationalcosts due to these additional calculations slow down theintegration process with respect to the integration of ODEsperformed according to strictly mass-action based kinetics(as in the cupSODA simulator) Nevertheless in Section 33we show that our parallel implementation largely outper-forms a sequential counterpart of the LSODA algorithm
BioMed Research International 11
0 100 200 300 400 500 600 700
Fibr
inog
en co
ncen
trat
ion
(M)
Time (s)
30 of fibrinogen
0
1eminus6
2eminus6
3eminus6
4eminus6
5eminus6
6eminus6
7eminus6
8eminus6
9eminus6
Figure 5 Dynamics of fibrinogen in physiological condition The clotting time is defined as the time necessary to convert the 70 of thefibrinogen into fibrin
As in the case of cupSODA [24 25] coagSODA exploitsthe shared memory to improve performances by storing thecurrent state and time of the simulations in these low-latencymemory banks Despite the improvement of performancesensured by this solution it strongly affects the occupancyof GPUrsquos multiprocessors and therefore it represents thelimiting factor for the number of blocks that can be executedsimultaneously as a matter of fact coagSODA is limited to2 blocks per streaming multiprocessor on GPUs based onthe Fermi architecture with a reduced exploitation of theGPU with respect to the theoretical 8 blocks allowed by thisarchitecture
3 Results and Discussion
In this section we discuss the results of the parameter sweepanalysis (PSA) carried out on the BCC model to the aimof investigating either the prolongation or the reduction ofthe clotting time in response to perturbed values of somereaction constants and of the initial concentration of somemolecular species chosen according to their meaning withinthe whole pathway PSA was performed by generating aset of different initial conditions corresponding to differentparameterizations of the model and then automaticallyexecuting the deterministic simulations with coagSODATheuse ofGPU technology is fundamental in this type of analysisespecially for large biological systems as the BCC is becauseit drastically reduces the computational time
The sweep analysis for single parameters (PSA-1D) wasperformed considering a logarithmic sampling of numericalvalues of each parameter under investigation (reaction con-stant or initial molecular concentration) within a specifiedrange with respect to its physiological reference value Thesweep analysis over pairs of parameters (PSA-2D) was per-formed by simultaneously varying the values of two param-eters within a specified range considering a logarithmicsampling on the resulting lattice The logarithmic samplingallows uniformly spanning different orders of magnitude ofthe parameters value using a reduced and fine-grained set
of samples therefore efficiently analyzing the response of thesystem in a broad range of conditions
To determine the response of the BCC to perturbed con-ditions we chose the clotting time (CT) as output of the PSAThe CT is defined as the time necessary to convert the 70of the fibrinogen into fibrin (see Figure 5) conventionallyassumed to correspond to the time required to form the clot[47] and it is generally used in laboratory tests formonitoringthe therapywith anticoagulant drugs According to themodeldefined in [30] the reference value of CT is around 300seconds in physiological conditions We investigate herethe response of the BCC by evaluating the CT in variousconditions corresponding to different values of the reactionconstants varying over six orders of magnitude with respectto their physiological values (ie three below and three abovethe reference values if not otherwise specified) as well asto different values of the initial molecular concentrationsand varying over twelve orders of magnitude with respect totheir physiological values (ie six below and six above thereference values if not otherwise specified)
The total number of parallel simulations executed tocarry out these analyses was 100 for PSA-1D over reactionconstants 200 for PSA-1D over initial concentrations and1600 for PSA-2D
Finally we present the comparison of the performanceof the CPU and GPU to run an increasing number ofsimulations of the BCC model to prove the efficiency ofcoagSODA
31 PSA-1D of the BCC Model
311 Reaction 11990344 Thefirst PSA was performed to determine
the value of the kinetic constant for the autoactivation offactor XII (reaction 119903
44in Table 1) which corresponds to an
upstream process in the intrinsic pathway This analysis wasmotivated by two considerations Firstly by using the fullparameterization given in [30] the action of the intrinsicpathway turns out to be fundamental for the BCC in vivowhich is in contrast to experimental observations which
12 BioMed Research International
324
326
328
330
332
334
336
338
340
Clot
ting
time (
s)
1eminus
10
1eminus
9
1eminus
8
1eminus
7
1eminus
6
1eminus
5
1eminus
4
1eminus
3
k44 (sminus1)
Figure 6 Clotting time according to PSA-1D over constant 11989644of reaction 119903
44 XII rarr XIIaThe reference value of 119896
44used in [30] is 5sdot10minus4sminus1
the sweep range is [5 sdot 10minus10 5 sdot 10minus4]
280
300
320
340
360
380
400
1e4 1e5 1e6 1e7 1e8 1e9 1e10 1e11
Clot
ting
time (
s)
k27 (Mminus1sminus1)
Figure 7 Clotting time according to PSA-1D over constant 11989627of reaction 119903
27 IIa + V rarr IIa + Va The reference value is 2 sdot 107Mminus1 sminus1 the
sweep range is [2 sdot 104 2 sdot 1010]
indicate that the extrinsic pathway is the main responsibleof clot formation Secondly in [30] all constants values havea reference to experimental measurements except for theconstant of this reaction (which corresponds to entry 29 inTable 1 in [30])
Figure 6 shows the results of this PSA-1D where 11989644was
varied over six orders of magnitude considering the valuegiven in [30] as the upper limit of the sweep interval We canobserve that by decreasing the value of 119896
44with respect to
the value considered in [30] the CT increases with respect toits reference value however for values of this constant lowerthan 100 sdot 10minus6 sminus1 the CT remains unaltered and this canbe intuitively explained by the fact that in this situation thefibrinogen is mainly activated by the extrinsic pathway
Consequently we assigned the value 700 sdot 10minus6 sminus1 to11989644 achieving a CT that is comparable to the experimental
observations of the BCC in vivo This new value was used inall PSA discussed in what follows
312 Reactions 11990327
and 11990358 In the next PSA we investigated
the effect of the perturbation of the kinetics of two pivotal
reactions of the BCC model Reaction 11990327 which describes
the catalytic activation of factor V by factor IIa was chosenbecause it represents the main positive feedback within thecommon pathway reaction 119903
58 which is involved in the
activation of factor XI by binding to factor IIa was chosenbecause it represents the main positive feedback in theintrinsic pathway (Figure 1) Moreover preliminary PSA overall reaction constants of the BCC model given in Table 1evidenced that these two reactions are among the mostrelevant steps of the coagulation network
The PSA-1D over 11989627(Figure 7) shows that the CT is very
sensitive to the perturbation of the rate of this reaction whenthe reference value of its constant (ie 119896
27= 2 sdot 10
7Mminus1 sminus1)is either increased or decreased in particular when 119896
27is
very low a plateau in CT is reached since the strength ofthe positive feedback exerted by factor IIa is largely reduceda condition where the contribution of the amplification ofthe hemostatic stimulus (due by the common pathway) tothe formation of the clot is basically not effective On theother side the PSA-1D over 119896
58(Figure 8) shows that while
a decrease of the constant with respect to its reference value
BioMed Research International 13
310
320
330
340
350
360
370
1e5 1e6 1e7 1e8 1e9 1e10 1e11
Clot
ting
time (
s)
k58 (Mminus1sminus1)
Figure 8 Clotting time according to PSA-1D over constant 11989658of reaction 119903
58 IIa + XI rarr IIa-XI The reference value is 1 sdot 108Mminus1 sminus1 the
sweep range is [1 sdot 105 2 sdot 1011]
220
240
260
280
300
320
340
360
380
400
Clot
ting
time (
s)
Factor VIII (M)
Seve
re h
aem
ophi
lia
Mild
hae
mop
hilia
Prot
hrom
botic
stat
e
1eminus
16
1eminus
14
1eminus
12
1eminus
10
1eminus
8
1eminus
6
1eminus
4
1eminus
2
Figure 9 Clotting time at different initial concentrations of factor VIII The reference value is 700 sdot 10minus10M (dashed black line) the sweeprange is [700 sdot 10minus16 700 sdot 10minus4] The blue area indicates a condition in which factor VIII concentration is less than 1 of its physiologicalconcentration corresponding to a situation of severe haemophilia the red area indicates a condition in which factor VIII concentration isbetween 1 and 30 of its physiological concentration corresponding to a situation of mild haemophilia the green area indicates a conditionin which factor VIII concentration is greater than 130 of its physiological concentration corresponding to a situation of hypercoagulability
(ie 11989658= 1 sdot 10
8Mminus1 sminus1) does not have any substantialeffect an increase of its value leads to a progressive increasein the CT This increase is due to the fact that factor IIa issequestered in the formation of a complex with factor XI andhence it is no longer available as a free component in bloodto participate in other reactions especially those reactionsof the extrinsic pathway which principally lead to the clotformation in vivo This behavior highlights that the intrinsicpathway has a secondary role in blood coagulation in vivocompared with the extrinsic pathway as also evidenced byvarious experimental observations [48]
313 Factors VIII IX and II The next set of PSA-1D wasrealized by varying the initial concentrations of factors VIIIIX and II These factors were selected since both an excess
and a deficiency of their concentrations lead to diseasesrelated to blood clotting
The PSA-1D over factor VIII (Figure 9) shows thatincreasing the initial concentration of this factor resultsin decreasing the CT suggesting the possible presence ofhypercoagulable states in these perturbed conditions As amatter of fact high levels of factor VIII cause an increasedrisk of deep vein thrombosis and pulmonary embolism[49] On the other hand individuals with less than 1of the average concentration of factor VIII show a severehaemophilia A characterized by higher CT (Figure 9) andrequire infusions of plasma containing the deficient factorotherwise frequent spontaneous bleeding would occur [50]When the concentration of this factor is between 5 and 30of the average concentration individuals still risk bleeding incase of trauma [50]
14 BioMed Research International
250
300
350
400
450
500
550
600
650
700
Clot
ting
time (
s)
Factor IX (M)
1eminus
14
1eminus
12
1eminus
10
1eminus
8
1eminus
6
1eminus
4
1eminus
2
1e0
Figure 10 Clotting time at different initial concentrations of factor IXThe reference value is 900sdot10minus8M the sweep range is [900sdot10minus14 900sdot10minus2]
0
100
200
300
400
500
600
700
Clot
ting
time (
s)
Factor II (M)
1eminus
12
1eminus
10
1eminus
8
1eminus
6
1eminus
4
1eminus
2
1e0
1e2
Figure 11 Clotting time at different initial concentrations of thrombinThe reference value is 140sdot10minus6M the sweep range is [140sdot10minus12 140sdot100]
The PSA-1D over factor IX (Figure 10) shows only aslight decrease of the CT as the initial concentration offIX increases this is in contrast to recent studies thatdemonstrated how the excess of factor IX leads to anincreased risk of deep vein thrombosis [51] This result canbe explained by considering that (i) a high concentrationof factor IX is not sufficient to bring about coagulationproblems though when the concentration of other factorsis above the average value (yet not at pathological levels)prothrombotic states can be observed (ii) in this model weconsider average values as initial concentrations of factorshowever individuals are characterized by different (balanced)combinations of procoagulant and anticoagulant factor levelsthat altogether contribute to define a unique coagulationphenotype that reflects the developmental environmentalgenetic nutritional and pharmacological influences of eachindividual [52] On the contrary the lack of factor IX causes
haemophilia B characterized by higher CT with respect tothe reference value (Figure 10)
Furthermore by comparing the PSA-1D of factors VIIIand IX (Figures 9 and 10) it is clear that haemophilia A ismore serious than haemophilia B since the CT achieved inconditions of factor VIII deficiency is higher than the CTobtained in the case of factor IX deficiency
In both PSA-1D over factors VIII and IX we observedafter the initial decrease of the CT an unexpected increaseof the CT as the factor concentration increasesThis counter-intuitive behavior arises at very high concentrations of thesefactors (with respect to the average physiological levels) andto the best of our knowledge it was never observed in vivoNonetheless it would be interesting to verify by means of adhoc laboratory experiments if the model correctly describesthe behavior of the BCC even in these conditions or on thecontrary it is not predictive in these extreme situations
BioMed Research International 15
The PSA-1D over factor II (Figure 11) shows a dramaticdecrease of the CT as the initial concentration of this factorincreases (with respect to the average physiological level)This behavior resembles the effects of hypercoagulability (orthrombophilia) a disease caused by mutation G20210A inthe prothrombin gene [53] that causes an increase of theprothrombin level (factor II) in the blood flow resulting inan excessive formation of the active form of this factor thusheightening venous thrombosis risks [54] Hypercoagulabil-ity is usually treated with warfarin therapy or with otheranticoagulants with a similar effect These drugs decrease thecapacity of coagulation factors to become active preventingthe formation of unwanted thrombi
On the other hand when the initial concentration offactor II is low we achieved the effects of prothrombindeficiency a rare autosomal recessive disease that causes atendency to severe bleeding [29 55] As shown in Figure 11 aconcentration equal to 10of the physiological value of factorII (ie 14 sdot 10minus6 M) leads to clotting effects similar to severehaemophilia A
32 PSA-2D of the BCCModel We present here the results ofthe PSA-2D on the BCCmodel where couples of parameterswere varied to analyze the possible effects arising from thecombined perturbation of their values
321 Reactions 11990327
and 11990358 Figure 12 shows the effect of
the simultaneous variation of constants 11989627
and 11989658
(overthe same sweep ranges considered in the two PSA-1DSection 31) This result remarks that reaction 119903
27 involved
in the common pathway has a stronger influence on theBCC and that there is a synergic interplay between thesetwo reactions In particular when the value of 119896
27is low and
the value of 11989658
is high the CT is higher than the valuesachieved when only a single constant is changed because inthis condition both the intrinsic and the common pathwaysare simultaneously inhibited
322 Factor VIII Factor IX and Tissue Factor In the last twoPSA-2Dwe varied the initial concentration of factor VIII andTissue Factor and the initial concentration of factor IX andTissue Factor respectively
The initial concentrations of factors VIII and IX were var-ied over four orders of magnitude using their physiologicalvalues as upper limit for the sweep ranges the concentrationof Tissue Factor was varied over four orders of magnitudetwo above and two below its reference value (see Table 2)The rationale behind this choice is to observe how the BCCmodel in conditions corresponding to different states ofhaemophilia (obtained by decreasing the concentrations offactors VIII and IX) behaves with different initial concentra-tions of the Tissue Factor which is the upstream factor of theextrinsic pathway that is the most important element of theBCC
The results of these PSA-2D show that with respect tothe condition of haemophilia B in the case of haemophiliaA the amount of Tissue Factor (below its reference value)has a negligible influence on the CT as indicated by the
presence of a plateau in Figure 13 on the contrary concerninghaemophilia B a deficiency of Tissue Factor leads to anincrease of the CT especially when factor IX is present in lowconcentrations (Figure 14)
The different results achieved in the two PSA-2D are dueto the presence in the BCC model of a direct interactionbetweenTissue Factor and factor IX bymeans of theTF-fVIIacomplex (see reactions 119903
13 119903
15in Table 1) Indeed the lack
of Tissue Factor directly affects the concentration of activefactor IX which results in a strong alteration of the CT withrespect to physiological conditions
33 CPU versus GPU Performance Comparison In orderto show the relevant speedup achieved by coagSODA wepresent here the comparison of the computational effortrequired by GPU and CPU for the simulation of the BCCmodel The performances of our GPU simulator were com-pared with those obtained using the LSODA algorithmimplemented in the software COPASI [56] executing onthe CPU the same set of simulations that were run on theGPU COPASI is single-threaded and does not exploit thephysical and logical cores of the CPU therefore it representsa good benchmark as a single-node CPU-bound simulator ofbiological systems
In all simulations executed on both GPU and CPU westored 100 samples uniformly distributed in the time intervalconsidered for each simulation that is [0 700] seconds of thedynamics of all chemical species involved in the BCCmodelThe settings for the LSODA algorithm were the followingrelative error 1 sdot 10minus7 absolute error 1 sdot 10minus13 and maximumnumber of internal steps set to 20000
331 Benchmark Details The GPU used for the simulationsis a Nvidia Tesla K20c equipped with 2496 cores organized in13 streamingmultiprocessors GPU clock 706MHz and 5GBofDDR5RAM In all tests coagSODAwas compiled and exe-cuted with version 55 of CUDA libraries Even though thisGPU has compute capability 35 and is based on the GK110Kepler architecture currently coagSODA does not exploitany of the new features (eg Dynamic Parallelism Hyper-QRemote DMA) with the exception of the new ISA encodingwhich allows threads to exploit an increased number ofregisters (255 instead of 63) reducing register spilling intoglobal memory and increasing performances [57] Moreoveras described in Section 22 the Kepler architecture offers thepossibility to reconfigure the 64KB on-chip cache balancingbetween L1 cache and shared memory Since coagSODAexploits the shared memory to reduce the latencies duringthe access to the concentration values of the BCC model weopted for the following configuration 48KB for the sharedmemory 16 KB for the L1 cache
The performance of the GPU was compared against apersonal computer equippedwith a dual-core CPU Intel Corei5 frequency 23 GHz 4GB of DDR3 RAM running theoperating systemMacOSXLion 1075The software used as acoagSODA-equivalent single-threaded CPU implementationis COPASI version 48 (build 35)
16 BioMed Research International
280
300
320
340
360
380
400
420
440
1e41e6
1e91e11 1e5
1e7
1e9
1e11
280 320 360 400 440
Clot
ting
time (
s)
k 58(M
minus1 sminus1 )
k27 (Mminus1sminus1)
Figure 12 Clotting time according to PSA-2D over reaction constants 11989627and 119896
58 The sweep ranges are [2 sdot 104 2 sdot 1010] and [1 sdot 105 2 sdot 1011]
respectively
100
200
300
400
Tissue factor (M) Factor VIII
(M)
Clot
ting
time (
s)
100
150
200
250
300
350
400
450
1eminus131eminus11
1eminus91eminus14
1eminus12
1eminus10
Figure 13 Clotting time at different initial concentrations of Tissue Factor and factor VIII The sweep ranges are [500 sdot 10minus14 500 sdot 10minus10]and [700 sdot 10minus14 500 sdot 10minus10] respectively
A direct comparison of the performances between thesetwo different architectures and implementations is not aneasy task since CPUs are optimized for single-thread exe-cution and exploit a large number of optimizations (eghigher clock frequency instruction caching pipelining outof order execution and branch prediction) whereas GPUsare optimized for graphics processing and parallel exe-cution of identical code relying only on multilevel datacaching For this reason in this paper we propose anempirical comparison of the performances based on therunning times of identical batches of simulations of the BCCmodel
332 Benchmark Results In Figure 15 we show the compar-ison of the running times required to run several batchesof simulations executed to carry out the PSA-1D over thereaction constant 119896
27 The choice of comparing the perfor-
mances of the GPU and CPU implementations by executingbatches of simulations that are related to a PSA instead ofrunning 119899 independent but identical simulations (ie allcharacterized by the same parameterization of the model)is due to the fact that these results represent a more realisticscenario in the computational analysis of biological systemswhereby it is useful to investigate large search spaces ofparameters corresponding to different perturbed conditionsof themodel [26]Moreover for the evaluation of the running
time the execution of a batch of 119899 identical deterministicsimulations would be futile The figure clearly shows thatcoagSODA always performs better than the CPU counter-part In particular while the CPU performance increaseslinearly with the number of simulations the running timesare strongly reduced on the GPU in this case the overallrunning time roughly corresponds to the time required forthe execution of the slowest simulations This is due to thefact that different parameterizations may require differenttime steps for LSODA to converge and the execution ofa block terminates as long as all the threads it containsterminate In turn the execution of a kernel terminates whenall blocks terminate for this reason a single simulationwhose dynamics requires more steps than the others mayaffect the overall running time
In Table 3 we report the running times of all batchesof simulations along with the speedup achieved on theGPU In particular these results highlight that the advantageof exploiting the GPUs for the simulation of the BCCmodel becomes more evident as the number of simula-tions increases with a 181times speedup when a PSA with10000 different parameterizations is executed Therefore theGPU accelerated analysis of the BCC model with coag-SODA represents a novel relevant computational means toinvestigate the behavior of this complex biological systemunder nonphysiological conditions and could be exploited
BioMed Research International 17
0
200
400
600
Tissue factor (M) Factor IX (M
)
Clot
ting
time (
s)
0
100
200
300
400
500
600
700
1eminus12
1eminus10
1eminus8
1eminus131eminus11
1eminus9
Figure 14 Clotting time at different initial concentrations of Tissue Factor and factor IXThe sweep ranges are [500 sdot 10minus14 500 sdot 10minus10] and[900 sdot 10
minus12 900 sdot 10
minus8] respectively
5 10 50 100 500 1000 5000 10000
Runn
ing
time (
s)
Number of simulations
CPUGPU
1e2
1e3
1e4
1e5
1e6
1e7
Figure 15 Comparison of the computational time required to execute an increasing number of simulations of the BCCmodel on CPU (IntelCore i5 23 GHz) and GPU (Nvidia Tesla K20c GPU clock 706MHz) The computational time is expressed in seconds The time valuesrelated to 1000 5000 and 10000 simulations on the CPU were estimated by regression (see also Table 3)
Table 3 CPU versus GPU performance comparison
Number ofsimulations CPU time (sec) GPU time (sec) Speedup
5 3808 1278 298times10 10604 2329 456times50 51704 8058 642times100 116528 10977 1061times500 536058 17394 3081times1000 1073582lowast 19982 5373times5000 5363507lowast 30960 1732times10000 10724975lowast 58951 1819timeslowastValues estimated by regression
to efficiently determine the response of the BCC to differenttherapeutical drugs
4 Conclusions
Thanks to their high-performance computing capabilitiesand the very low costs GPUs nowadays represent a suitable
technology for the development and the application of par-allel computational methods for in silico analysis of complexbiological systems However the implementation of efficientcomputational tools able to fully exploit the large potentialityof GPUs is still challenging since good programming skillsare required to implement GPU-based algorithmic methodsand to handle specific features as an optimal usage ofmemoryor the communication bandwidth between GPU and CPUMoreover algorithms cannot be directly ported on the GPUbecause of the limited programming capabilities allowed byGPUkernels as amatter of fact they need to be redesigned tomeet the requirements of the underlying SIMD architecture
In this work we presented coagSODA a GPU-poweredsimulator specifically developed to carry out fast parallelsimulations of the BCC model coagSODA was designed tooffer a black-box solution usable by any final user in aneasy way It relies on cupSODA [24] a numerical integratorfor ODEs that we previously implemented for the GPUarchitecture based on the LSODA algorithm and capableof automatically translating a reaction-based model into aset of coupled ODEs In addition to mass-action kinet-ics coagSODA implements specific functions to compute
18 BioMed Research International
the kinetics of Hill function based reactions such as thoseinvolved in the platelets activity of the BCC model coag-SODA exploits the massive parallel capabilities of modernGPUs and our results demonstrated that it can achieve arelevant reduction of the computational time required toexecutemany concurrent and independent simulations of theBCC dynamics The mutual independence of the simulationsallows fully exploiting the underlying SIMD architecturemoreover coagSODA benefits from an additional speedupthanks to our choice of storing the state of the systeminto the low-latency shared memory (a solution that wasalready implemented in cupSODA) Since the BCC modelis large (96 reactions 71 chemical species) a large amountof shared memory was assigned to each thread stronglyreducing the theoretical occupancy of the GPU that is theratio of active warps with respect to the maximum numberof warps supported by each streaming multiprocessor of theGPU However the results of the analysis presented in thiswork performed on the BCC model show that coagSODAachieves a relevant boost with respect to a reference CPUimplementation For instance in the case of 10000 simula-tions we achieved a noticeable 181times speedup Interestinglythe performances of coagSODA are better than COPASI evenfor small batches of simulations These results indicate thatfor biological models consisting in many reactions and manyspecies our GPU implementation of LSODA becomes moreprofitable than the CPU counterpart as the number of con-current simulations increases making it suitable especiallywhen performing demanding computational analysis suchas for example parameter sweep parameter estimation orsensitivity analysis
As a test case in this paper we presented several parame-ter sweep analyses over reaction constants and initial concen-trations of factors involved in the BCC model Other com-putational analyses on mathematical models of this pathwaywere previously presented For instance in [58] a parameterestimation of the reaction constants of different modelsinvolving the activation of factor X factor V prothrombinand the inactivation of factors was performed The aim ofthis analysis was to discriminate between different modelsto unravel the mechanisms on the basis of the BCC Similaranalyses were executed on a model describing thrombingeneration in plasma since the reaction mechanism thereaction constants and initial concentrations were unknown[59] In [34] a sensitivity analysis of a model consisting of44 species over 34 chemical species was presented reactionconstants were varied between 10 and 1000 of theirreference value to the aim of identifying the most influentialfactors of the BCC
All these analyses were performed bymeans of sequentialsimulations of the models under investigation thereforein general only small batches of simulations could be run(for instance in [34] only 836 simulations were executed toexecute the sensitivity analysis of the model) On the otherhand in this paper we presented the results of different PSAefficiently executed bymeans of coagSODA overall providinguseful information regarding unknownparameters and inter-esting insights into the functioning of the BCC A thoroughsensitivity analysis of the whole BCC model consisting in
around 5 sdot 105 simulations is currently in progress on GPUsby our group exploiting the great computational efficiency ofcoagSODA
The results of our computational analyses should benow validated by means of specifically designed laboratoryexperiments In particular we identified a plausible valuefor the constant of the reaction describing the autoactivationof factor XII (since no reference values can be found inliterature) moreover the results of the PSA over factor IXsuggest that a deficiency of this factor is not enough to causesevere bleeding disorders as haemophilia B but an alterationof other factors seems to be necessary for the occurrence ofsuch condition (eg the lack of Tissue Factor as suggestedby the PSA-2D over Tissue Factor and factor IX) Finallythe PSA over factors VIII and IX showed in a situationcharacterized by very high concentrations of these factorsa counterintuitive behavior in which the clotting time isincreased with respect to the value obtained in physiologicalconditions Indeed it would be interesting to design ad hoclaboratory experiments to verify if the BCCmodel is actuallypredictive in such extreme situations
The coagSODA software and the SBML version of theBCC model used in this work are available from the authorsupon request
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This work is partially supported by SYSBIO Centre forSystems Biology
References
[1] B B Aldridge JM BurkeDA Lauffenburger andP K SorgerldquoPhysicochemicalmodelling of cell signalling pathwaysrdquoNatureCell Biology vol 8 no 11 pp 1195ndash1203 2006
[2] I-C Chou and E O Voit ldquoRecent developments in parameterestimation and structure identification of biochemical andgenomic systemsrdquoMathematical Biosciences vol 219 no 2 pp57ndash83 2009
[3] D Besozzi P Cazzaniga D Pescini G Mauri S Colomboand EMartegani ldquoThe role of feedback control mechanisms onthe establishment of oscillatory regimes in the RascAMPPKApathway in S cerevisiaerdquo EURASIP Journal on Bioinformaticsand Systems Biology vol 2012 no 10 2012
[4] A Saltelli M Ratto T Andres et al Global Sensitivity AnalysisThe Primer Wiley-Interscience 2008
[5] A Raue C Kreutz T Maiwald et al ldquoStructural and practicalidentifiability analysis of partially observed dynamical modelsby exploiting the profile likelihoodrdquo Bioinformatics vol 25 no15 pp 1923ndash1929 2009
[6] A Drager M Kronfeld M J Ziller et al ldquoModeling metabolicnetworks in C glutamicum a comparison of rate laws incombination with various parameter optimization strategiesrdquoBMC Systems Biology vol 3 article 5 2009
BioMed Research International 19
[7] D Besozzi P Cazzaniga G Mauri D Pescini and L Van-neschi ldquoA comparison of genetic algorithms and particle swarmoptimization for parameter estimation in stochastic biochem-ical systemsrdquo in Proceedings of the 7th European ConferenceEvolutionary Computation Machine Learning and Data Miningin Bioinformatics (EvoBIO rsquo09) C Pizzuti M Ritchie and MGiacobini Eds vol 5483 of Lecture Notes in Computer Sciencepp 116ndash127 Springer 2009
[8] M S Nobile D Besozzi P Cazzaniga GMauri andD PescinildquocupSODA aGPU-basedmulti-swarmPSOmethod for param-eter estimation in stochastic biological systems exploitingdiscrete-time target seriesrdquo in Proceedings of the 10th EuropeanConference Evolutionary Computation Machine Learning andData Mining in Bioinformatics (EvoBIO rsquo12) M Giacobini LVanneschi and W Bush Eds vol 7246 of Lecture Notes inComputer Science pp 74ndash85 Springer
[9] M S Nobile D Besozzi P Cazzaniga D Pescini and GMaurildquoReverse engineering of kinetic reaction networks by meansof Cartesian Genetic Programming and Particle Swarm Opti-mizationrdquo in Proceedings of the IEEE Congress on EvolutionaryComputation (CEC rsquo13) pp 1594ndash1601 June 2013
[10] M S Nobile ldquoEvolutionary inference of biochemical inter-action networks accelerated on Graphics Processing Unitsrdquoin Proceedings of the 11th International Conference on HighPerformance Computingamp Simulation 2013 (HPCS rsquo13) pp 668ndash670 Helsinki Finland 2013
[11] K Kentzoglanakis and M Poole ldquoA swarm intelligence frame-work for reconstructing gene networks searching for bio-logically plausible architecturesrdquo IEEEACM Transactions onComputational Biology and Bioinformatics vol 9 no 2 pp 358ndash371 2012
[12] J R Koza W Mydlowec G Lanza Y Jessen and M A KeaneldquoAutomatic computational discovery of chemical reaction net-works using genetic programmingrdquo inComputationalDiscoveryof Scientific Knowledge S Dzeroski and L Todorovski Edsvol 4660 of Lecture Notes in Computer Science pp 205ndash227Springer 2007
[13] S Ando E Sakamoto and H Iba ldquoEvolutionary modeling andinference of gene networkrdquo Information Sciences vol 145 no3-4 pp 237ndash259 2002
[14] L Dematte and D Prandi ldquoGPU computing for systemsbiologyrdquo Briefings in Bioinformatics vol 11 no 3 Article IDbbq006 pp 323ndash333 2010
[15] J L Payne N A Sinnott-Armstrong and J HMoore ldquoExploit-ing graphics processing units for computational biology andbioinformaticsrdquo Interdisciplinary Sciences Computational LifeSciences vol 2 no 3 pp 213ndash220 2010
[16] M J Harvey and G De Fabritiis ldquoA survey of computationalmolecular science using graphics processing unitsrdquoWiley Inter-disciplinary Reviews Computational Molecular Science vol 2no 5 pp 734ndash742 2012
[17] Y Zhou J Liepe X Sheng M P H Stumpf and C BarnesldquoGPU accelerated biochemical network simulationrdquo Bioinfor-matics vol 27 no 6 Article ID btr015 pp 874ndash876 2011
[18] M Vigelius A Lane and B Meyer ldquoAccelerating reaction-diffusion simulations with general-purpose graphics processingunitsrdquo Bioinformatics vol 27 no 2 Article ID btq622 pp 288ndash290 2011
[19] D J Wilkinson ldquoStochastic modelling for quantitative descrip-tion of heterogeneous biological systemsrdquo Nature ReviewsGenetics vol 10 no 2 pp 122ndash133 2009
[20] J C Butcher ldquoTrees and numerical methods for ordinarydifferential equationsrdquo Numerical Algorithms vol 53 no 2-3pp 153ndash170 2010
[21] D T Gillespie ldquoStochastic simulation of chemical kineticsrdquoAnnual Review of Physical Chemistry vol 58 pp 35ndash55 2007
[22] L Petzold ldquoAutomatic selection of methods for solving stiffand nonstiff systems of ordinary differential equationsrdquo SIAMJournal of Scientific and Statistical Computing vol 4 no 1 pp136ndash148 1983
[23] Y Cao D T Gillespie and L R Petzold ldquoEfficient step sizeselection for the tau-leaping simulation methodrdquo The Journalof Chemical Physics vol 124 no 4 2006
[24] M S Nobile D Besozzi P Cazzaniga GMauri andD PescinildquocupSODA a CUDA-powered simulator of mass-action kinet-icsrdquo in Proceedings of the 12th International Conference onParallel Computing Technologies (PaCT rsquo13) V Malyshkin Edvol 7979 of Lecture Notes in Computer Science pp 344ndash357
[25] M S Nobile P Cazzaniga D Besozzi and G Mauri ldquoGPU-accelerated simulations of mass-action kinetics models withcupSODArdquoThe Journal of Supercomputing 2014
[26] M S Nobile P Cazzaniga D Besozzi D Pescini and GMauri ldquocuTauLeaping a GPU-powered tau-leaping stochasticsimulator for massive parallel analyses of biological systemsrdquoPLoS ONE vol 9 no 3 2014
[27] K Rogers Blood Physiology and CirculationTheHuman BodyBritannica Educational Publishing 2010
[28] A D Michelson Platelets Elsevier 2nd edition 2007[29] R W Colman J Hirsh V J Marder A W Clowes and J
N George Hemostasis and Thrombosis Basic Principles andClinical Practice Lippincott Williams ampWilkins 2006
[30] M S Chatterjee W S Denney H Jing and S L DiamondldquoSystems biology of coagulation initiation Kinetics of thrombingeneration in resting and activated human bloodrdquo PLoS Com-putational Biology vol 6 no 9 Article ID e1000950 2010
[31] M Bellini D Besozzi P Cazzaniga GMauri andM S NobileldquoSimulation and analysis of the blood coagulation cascadeaccelerated on GPUrdquo in Proceedings of the 22nd EuromicroInternational Conference on Parallel Distributed and Network-Based Processing (PDP) pp 590ndash593 IEEE Computer SocietyConference Publishing Services 2014
[32] T Wajima G K Isbister and S B Duffull ldquoA comprehensivemodel for the humoral coagulation network in humansrdquo Clini-cal Pharmacology and Therapeutics vol 86 no 3 pp 290ndash2982009
[33] K Lo W S Denney and S L Diamond ldquoStochastic modelingof blood coagulation initiationrdquo Pathophysiology of Haemostasisand Thrombosis vol 34 no 2-3 pp 80ndash90 2006
[34] C M Danforth T Orfeo K G Mann K E Brummel-Ziedins and S J Everse ldquoThe impact of uncertainty in a bloodcoagulation modelrdquo Mathematical Medicine and Biology vol26 no 4 Article ID dqp011 pp 323ndash336 2009
[35] S D Lewis P P Shields and J A Shafer ldquoCharacterizationof the kinetic pathway for liberation of fibrinopeptides duringassembly of fibrinrdquoThe Journal of Biological Chemistry vol 260no 18 pp 10192ndash10199 1985
[36] M F Hockin K C Jones S J Everse and K G Mann ldquoAmodel for the stoichiometric regulation of blood coagulationrdquoThe Journal of Biological Chemistry vol 277 no 21 pp 18322ndash18333 2002
[37] K Brummel-Ziedins C YVossen F R Rosendaal KUmezakiand K G Mann ldquoThe plasma hemostatic proteome thrombin
20 BioMed Research International
generation in healthy individualsrdquo Journal of Thrombosis andHaemostasis vol 3 no 7 pp 1472ndash1481 2005
[38] K E Brummel-Ziedins C Y Vossen S Butenas K G Mannand F R Rosendaal ldquoThrombin generation profiles in deepvenous thrombosisrdquo Journal of Thrombosis and Haemostasisvol 3 no 11 pp 2497ndash2505 2005
[39] S D Bungay P A Gentry and R D Gentry ldquoA mathematicalmodel of lipid-mediated thrombin generationrdquo MathematicalMedicine and Biology vol 20 no 1 pp 105ndash129 2003
[40] S Butenas T Orfeo M T Gissel K E Brummel and K GMann ldquoThe significance of circulating factor IXa in bloodrdquoThe Journal of Biological Chemistry vol 279 no 22 pp 22875ndash22882 2004
[41] D Voet and J G Voet Biochemistry Wiley 2011[42] A C Guyton and J E Hall Textbook of Medical Physiology
Saunders 2010[43] M F M James and A M Roche ldquoDose-response relationship
between plasma ionized calcium concentration and thrombe-lastographyrdquo Journal of Cardiothoracic and Vascular Anesthesiavol 18 no 5 pp 581ndash586 2004
[44] D L Nelson and M M Cox Lehninger Principles of Biochem-istry W H Freeman Company 2004
[45] R M Farber ldquoTopical perspective on massive threading andparallelismrdquo Journal of Molecular Graphics and Modelling vol30 pp 82ndash89 2011
[46] OWolkenhauerMUllahWKolch andK-HCho ldquoModelingand simulation of intracellular dynamics choosing an appropri-ate frameworkrdquo IEEETransactions onNanobioscience vol 3 no3 pp 200ndash207 2004
[47] B H Estridge A P Reynolds and N J Walters Basic MedicalLaboratory Techniques Cengage Learning 4th edition 2000
[48] T Renne A H Schmaier K F Nickel M Blomback and CMaas ldquoIn vivo roles of factor XIIrdquo Blood vol 120 no 22 pp4296ndash4303 2012
[49] P V Jenkins O Rawley O P Smith and J S ODonnellldquoElevated factor VIII levels and risk of venous thrombosisrdquoBritish Journal of Haematology vol 157 no 6 pp 653ndash663 2012
[50] C E Walsh and K M Batt ldquoHemophilia clinical gene therapybrief reviewrdquo Translational Research vol 161 no 4 pp 307ndash3122013
[51] A Van Hylckama Vlieg I K Van Der Linden R M Bertinaand F R Rosendaal ldquoHigh levels of factor IX increase the risk ofvenous thrombosisrdquo Blood vol 95 no 12 pp 3678ndash3682 2000
[52] K Brummel-Ziedins ldquoModels for thrombin generation and riskof diseaserdquo Journal ofThrombosis and Haemostasis vol 11 no 1pp 212ndash223 2013
[53] U Seligsohn and A Lubetsky ldquoGenetic susceptibility to venousthrombosisrdquoTheNew England Journal of Medicine vol 344 no16 pp 1222ndash1231 2001
[54] I Martinelli P Bucciarelli and P M Mannucci ldquoThromboticrisk factors basic pathophysiologyrdquo Critical Care Medicine vol38 no 2 pp S3ndashS9 2010
[55] K Brummel-Ziedins A Undas T Orfeo et al ldquoThrombingeneration in acute coronary syndrome and stable coronaryartery disease dependence on plasma factor compositionrdquoJournal ofThrombosis andHaemostasis vol 6 no 1 pp 104ndash1102008
[56] S Hoops R Gauges C Lee et al ldquoCOPASImdasha complexpathway simulatorrdquo Bioinformatics vol 22 no 24 pp 3067ndash3074 2006
[57] Nvidia NVIDIArsquos Next Generation CUDA Compute Architec-ture Kepler GK110 2012
[58] W Stortelder W Hemker and H Hemker ldquoMathematicalmodelling in blood coagulation simulation and parameterestimationrdquoModelling Analysis and Simulation vol 20 pp 1ndash111997
[59] S M C Orfao G Jank and K Mottaghy ldquoA new mathematicalapproach to modelling thrombin generationrdquo InternationalJournal of Artificial Organs vol 29 no 7 pp 703ndash708 2006
[60] Nvidia Nvidia CUDA C Programming Guide v50 2012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Anatomy Research International
PeptidesInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
International Journal of
Volume 2014
Zoology
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Molecular Biology International
GenomicsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
BioinformaticsAdvances in
Marine BiologyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Signal TransductionJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
BioMed Research International
Evolutionary BiologyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Biochemistry Research International
ArchaeaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Genetics Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in
Virolog y
Hindawi Publishing Corporationhttpwwwhindawicom
Nucleic AcidsJournal of
Volume 2014
Stem CellsInternational
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Enzyme Research
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Microbiology
4 BioMed Research International
Table 1 Reaction-based model of the blood coagulation cascade (reduced version of the ldquoPlatelet-Plasmardquo model described in [30]) Themodel consists in 96 reactions among 71 molecular species With the exception of reaction 119903
44(see Section 31) the values of all reaction
constants were taken from [30]
119899 Reactants Products Constant (119896)1199031
TF + fVII TF-fVII 320 sdot 106Mminus1 sminus1
1199032
TF-fVII TF + fVII 310 sdot 10minus2 sminus1
1199033
TF + fVIIa TF-fVIIa 230 sdot 107Mminus1 sminus1
1199034
TF-fVIIa TF + fVIIa 310 sdot 10minus5 sminus1
1199035
TF-fVIIa + fVII TF-fVIIa + fVIIa 440 sdot 105Mminus1 sminus1
1199036
fXa + fVII fXa + fVIIa 130 sdot 107Mminus1 sminus1
1199037
fIIa + fVII fIIa + fVIIa 230 sdot 104Mminus1 sminus1
1199038
TF-fVIIa + fX TF-fVIIa-fX 250 sdot 107Mminus1 sminus1
1199039
TF-fVIIa-fX TF-fVIIa + fX 105 sdot 10minus2 sminus1
11990310
TF-fVIIa-fX TF-fVIIa-fXa 600 sminus1
11990311
TF-fVIIa-fXa TF-fVIIa + fXa 1900 sminus1
11990312
TF-fVIIa + fXa TF-fVIIa-fXa 220 sdot 107Mminus1 sminus1
11990313
TF-fVIIa + fIX TF-fVIIa-fIX 100 sdot 107Mminus1 sminus1
11990314
TF-fVIIa-fIX TF-fVIIa + fIX 240 sminus1
11990315
TF-fVIIa-fIX TF-fVIIa + fIXa 180 sminus1
11990316
fII + fXa fIIa + fXa 750 sdot 103Mminus1 sminus1
11990317
fIIa + fVIII fIIa + fVIIIa 200 sdot 107Mminus1 sminus1
11990318
fVIIIa + fIXa fIXa-fVIIIa 100 sdot 107Mminus1 sminus1
11990319
fIXa-fVIIIa fVIIIa + fIXa 100 sdot 10minus4 sminus1
11990320
fIXa-fVIIIa + fX fIXa-fVIIIa-fX 100 sdot 108Mminus1 sminus1
11990321
fIXa-fVIIIa-fX fIXa-fVIIIa + fX 100 sdot 10minus5 sminus1
11990322
fIXa-fVIIIa-fX fIXa-fVIIIa + fXa 820 sminus1
11990323
fVIIIa fVIIIa1-L + fVIIIa2 600 sdot 10minus5 sminus1
11990324
fVIIIa1-L + fVIIIa2 fVIIIa 220 sdot 104 Mminus1 sminus1
11990325
fIXa-fVIIIa-fX fVIIIa1-L + fVIIIa2 + fX + fIXa 100 sdot 10minus3 sminus1
11990326
fIXa-fVIIIa fVIIIa1-L + fVIIIa2 + fIXa 100 sdot 10minus3 sminus1
11990327
fIIa + fV fIIa + fVa 200 sdot 107Mminus1 sminus1
11990328
fXa + fVa fXa-fVa 400 sdot 108Mminus1 sminus1
11990329
fXa-fVa fXa + fVa 02 sminus1
11990330
fXa-fVa + fII fXa-fVa-fII 100 sdot 108Mminus1 sminus1
11990331
fXa-fVa-fII fXa-fVa + fII 10300 sminus1
11990332
fXa-fVa-fII fXa-fVa + fmIIa 6350 sminus1
11990333
fXa-fVa + fmIIa fXa-fVa + fIIa 150 sdot 107Mminus1 sminus1
11990334
fXa + TFPI fXa-TFPI 900 sdot 105Mminus1 sminus1
11990335
fXa-TFPI fXa + TFPI 360 sdot 10minus4 sminus1
11990336
TF-fVIIa-fXa + TFPI TF-fVIIa-fXa-TFPI 320 sdot 108 Mminus1 sminus1
11990337
TF-fVIIa-fXa-TFPI TF-fVIIa-fXa + TFPI 110 sdot 10minus2 sminus1
11990338
TF-fVIIa + fXa-TFPI TF-fVIIa-fXa-TFPI 500 sdot 107Mminus1 sminus1
11990339
fXa + ATIII fXa-ATIII 150 sdot 103Mminus1 sminus1
11990340
fmIIa + ATIII fmIIa-ATIII 710 sdot 103Mminus1 sminus1
11990341
fIXa + ATIII fIXa-ATIII 490 sdot 102Mminus1 sminus1
11990342
fIIa + ATIII fIIa-ATIII 710 sdot 103Mminus1 sminus1
11990343
TF-fVIIa + ATIII TF-fVIIa-ATIII 230 sdot 102Mminus1 sminus1
11990344
fXII fXIIa 500 sdot 10minus4 sminus1
11990345
fXIIa + fXII fXIIa-fXII 100 sdot 108Mminus1 sminus1
11990346
fXIIa-fXII fXIIa + fXII 75000 sminus1
11990347
fXIIa-fXII fXIIa + fXIIa 330 sdot 10minus2 sminus1
11990348
fXIIa + PKal fXIIa-PKal 100 sdot 108Mminus1 sminus1
11990349
fXIIa-PKal fXIIa + PKal 360 sdot 103 sminus1
BioMed Research International 5
Table 1 Continued
119899 Reactants Products Constant (119896)11990350
fXIIa-PKal fXIIa + Kal 4000 sminus1
11990351
fXII + Kal fXII-Kal 100 sdot 108Mminus1 sminus1
11990352
fXII-Kal fXII + Kal 4530 sminus1
11990353
fXII-Kal fXIIa + Kal 570 sminus1
11990354
PKal + Kal Kal + Kal 270 sdot 104Mminus1 sminus1
11990355
Kal Kal119894
110 sdot 10minus2 sminus1
11990356
fXIIa + C1inh fXIIa-C1inh 360 sdot 103Mminus1 sminus1
11990357
fXIIa + ATIII fXIIa-ATIII 2160Mminus1 sminus1
11990358
fXI + fIIa fXI-fIIa 100 sdot 108Mminus1 sminus1
11990359
fXI-fIIa fXI + fIIa 500 sminus1
11990360
fXI-fIIa fXIa + fIIa 130 sdot 10minus4 sminus1
11990361
fXIIa + fXI fXIIa-fXI 100 sdot 108Mminus1 sminus1
11990362
fXIIa-fXI fXIIa + fXI 200 sminus1
11990363
fXIIa-fXI fXIIa + fXIa 570 sdot 10minus4 sminus1
11990364
fXIa + fXI fXIa + fXIa 319 sdot 106Mminus1 sminus1
11990365
fXIa + ATIII fXIa-ATIII 320 sdot 102Mminus1 sminus1
11990366
fXIa + C1inh fXIa-C1inh 180 sdot 103Mminus1 sminus1
11990367
fXIa + A1AT fXIa-A1AT 100 sdot 102Mminus1 sminus1
11990368
fXIa + A2AP fXIa-A2AP 43 sdot 103Mminus1 sminus1
11990369
fXIa + fIX fXIa-fIX 100 sdot 108Mminus1 sminus1
11990370
fXIa-fIX fXIa + fIX 4100 sminus1
11990371
fXIa-fIX fXIa + fIXa 770 sminus1
11990372
fIXa + fX fIXa-fX 100 sdot 108Mminus1 sminus1
11990373
fIXa-fX fIXa + fX 064 sminus1
11990374
fIXa-fX fIXa + fXa 700 sdot 10minus4 sminus1
11990375
fXa + fVIII fXa-fVIII 100 sdot 108Mminus1 sminus1
11990376
fXa-fVIII fXa + fVIII 210 sminus1
11990377
fXa-fVIII fXa + fVIIIa 0023 sminus1
11990378
fVIIa + fIX fVIIa-fIX 100 sdot 108Mminus1 sminus1
11990379
fVIIa-fIX fVIIa + fIX 090 sminus1
11990380
fVIIa-fIX fVIIa + fIXa 360 sdot 10minus5 sminus1
11990381
fVIIa + fX fVIIa-fX 100 sdot 108Mminus1 sminus1
11990382
fVIIa-fX fVIIa + fX 21000 sminus1
11990383
fVIIa-fX fVIIa + fXa 160 sdot 10minus6 sminus1
11990384
Fbg + fIIa Fbg-fIIa 100 sdot 108Mminus1 sminus1
11990385
Fbg-fIIa Fbg + fIIa 63600 sminus1
11990386
Fbg-fIIa Fbn1 + fIIa + FPA 8400 sminus1
11990387
Fbn1 + fIIa Fbn1-fIIa 100 sdot 108Mminus1 sminus1
11990388
Fbn1-fIIa Fbn1 + fIIa 74260 sminus1
11990389
Fbn1-fIIa Fbn2 + fIIa + FPB 740 sminus1
11990390
Fbn1 + Fbn1 (Fbn1)2 100 sdot 106Mminus1 sminus1
11990391
(Fbn1)2 2Fbn1 640 sdot 10minus2 sminus1
11990392
(Fbn1)2 + fIIa (Fbn1)2-fIIa 100 sdot 108Mminus1 sminus1
11990393
(Fbn1)2-fIIa (Fbn1)2 + fIIa 70100 sminus1
11990394
(Fbn1)2-fIIa (Fbn2)2 + fIIa + FPB 4900 sminus1
11990395
Fbn2 + fIIa Fbn2-fIIa 100 sdot 108Mminus1 sminus1
11990396
Fbn2-fIIa Fbn2 + fIIa 100 sdot 103 sminus1
6 BioMed Research International
fXIITF-fVII
TFfVIIa
fVII
TF-fVIIa
Fbg
FPA
Fbn1
FPB
Fbn2
Fbn1
(Fbn1)2
(Fbn2)2 FPB
fXa
fIIa
TF-fVIIa-fXa
fIX
fIXa
fX
fmIIa
fII
PKal
KalifXI
fXIa
fIX
fIXa
fVIII
fVIIIa
fIXa-fVIIIa
fVIIIa1 -L
fVIIIa2
fVIIIa2
fIXa
fV
fVa
fXa-fVa
fXa-TFPI
TFPI
TF-fVIIa-fXa-TFPI
fXa-AT
AT
AT
fXIa-AT
C1inh
fXIIa-C1inh
A1AT
A2AP
fXIa-A2APfXIa-A1AT
TF-fVIIa
fXIIa Kal
fVIIIa1 -L
Figure 1 Graphical representation of the blood coagulation cascade model considered in this work Legend Blue box coagulation factor redbox inhibitor and related complexes Black arrow complex formation green arrow catalytic activation violet arrow activation red arrowinhibition The reaction is reversible if the arrow tail consists in a dot
BioMed Research International 7
(iv) the activation of prekallikrein by factor XIIa(reactions 119903
48 11990349 11990350) and by kallikrein (reac-
tion 11990354)
(v) the activation of factor XI by factor XIIa (reac-tions 119903
61 11990362 11990363) and by factor XIa (reaction
11990364)
(vi) the activation of factor IX by factor XIa (reac-tions 119903
69 11990370 11990371)
(vii) the dissociation of free factor VIIIa (reactions11990323and 11990324)
(viii) the dissociation of factor VIIIa in complex withother factors (reactions 119903
25and 11990326)
(3) The third module corresponds to the common path-way which consists in
(i) the activation of factor II by factor Xa (modeledby reaction 119903
16) and by the complex between
factor Xa and factor Va through the formationof the intermediate meizothrombin (reactions11990330 119903
33)
(ii) the activation of factor VII by factor Xa (reac-tion 1199036) and by factor IIa (reaction 119903
7)
(iii) the activation of factor VIII by factor Xa (reac-tions 119903
75 11990376 11990377) and by factor IIa (reaction 119903
17)
(iv) the formation of a complex between factor Xaand factor Va (reactions 119903
28and 11990329)
(v) the activation of fibrinogen by factor IIa (reac-tions 119903
84 119903
96)
(vi) the activation of factor V by factor IIa (reaction11990327)
(vii) the activation of factor XI by factor IIa (reac-tions 119903
58 11990359 11990360)
(4) The fourth module describes the inhibition of coagu-lation carried out through the main inhibitors of theBCC (antithrombin Tissue Factor Pathway InhibitorC1-inihibitor 1205721-antitrypsin and 1205722-antiplasmin)These are tight binding inhibitors belonging to theserpin superfamily which form irreversible com-plexes This module consists in
(i) the inhibition of factor Xa by Tissue FactorPathway Inhibitor (modeled by reactions 119903
34and
11990335)
(ii) the inhibition of the complex between TissueFactor factor VIIa and factor Xa by TissueFactor Pathway Inhibitor (reactions 119903
36 11990337 11990338)
(iii) the inhibition of factor IIa by antithrombin(reactions 119903
40and 11990342)
(iv) the inhibition of factor Xa by antithrombin(reaction 119903
39)
(v) the inhibition of factor IXa by antithrombin(reaction 119903
41)
(vi) the inhibition of factor XIa by antithrombin(reaction 119903
65) and by C1-inhibitor (reaction 119903
66)
Table 2 Initial concentrations of molecular species in the bloodcoagulation cascade model (values taken from [30])
Species Symbol Concentration (M)1205721-Antitrypsin A1AT 450 sdot 10
minus5
1205722-Antiplasmin A2AP 100 sdot 10minus6
Antithrombin III ATIII 340 sdot 10minus6
C1-inhibitor C1inh 250 sdot 10minus6
Fibrinogen Fbg 900 sdot 10minus6
Factor II fII 140 sdot 10minus6
Factor V fV 200 sdot 10minus8
Factor VII fVII 100 sdot 10minus8
Active factor VII fVIIa 100 sdot 10minus10
Factor VIII fVIII 700 sdot 10minus10
Factor IX fIX 900 sdot 10minus8
Factor X fX 16 sdot 10minus7
Factor XI fXI 310 sdot 10minus8
Factor XII fXII 340 sdot 10minus7
Prekallikrein Pkal 450 sdot 10minus7
Tissue Factor TF 500 sdot 10minus12
Tissue Factor Pathway Inhibitor TFPI 250 sdot 10minus9
(vii) the inhibition of factor XIIa by C1-inhibitor(reaction 119903
56) by antithrombin (reaction 119903
57)
by 1205721-antitrypsin (reaction 11990367) and by 1205722-
antiplasmin (reaction 11990368)
(viii) the inhibition of the complex between TissueFactor and factor VIIa by antithrombin (reac-tion 11990343)
(ix) the inhibition of kallikrein (reaction 11990355)
The values of the initial concentrations of the molecularspecies occurring in the BCC model are given in Table 2According to [30] the concentrations of complexes and activefactors were set to 0 except for the active factor VII which isphysiologically present in the blood circulation even in theabsence of damage in a concentration that is approximatelyequal to 1 of the corresponding inactive factor [42]
The system of ordinary differential equations (ODEs)needed to carry out the simulations and the parametersweep analysis presented in Section 3 was derived fromthe reactions given in Table 1 according to the mass-actionlaw with the exception of 14 reactions belonging to the set119878120576= 11990319 11990321 11990323 11990329 11990331 11990335 11990346 11990349 11990352 11990362 11990370 11990373 11990376 11990382
The mass-action law is the fundamental empirical law thatgoverns biochemical reaction rates which states that in adilute solution the rate of an elementary reaction is propor-tional to the product of the concentration of its reactantsraised to the power of the corresponding stoichiometriccoefficient [44] The reactions in 119878
120576are instead influenced by
a specifically defined variable 120576 depending on aHill functionfit against experimental data which quantify the plateletactivation status which is used to model the physiologicallevels of thrombin concentration as a function of plateletactivation as thoroughly described in [30] More preciselythe value of 120576 influences the formation of some complexes
8 BioMed Research International
0 0 100 200 300 400 500 600 700
Fact
or II
a con
cent
ratio
n (M
)
Time (s)
1eminus7
2eminus7
3eminus7
4eminus7
5eminus7
6eminus7
7eminus7
8eminus7
9eminus7
Figure 2 Dynamics of thrombin (factor IIa) in physiological condition
occurring on the plateletsrsquo surface by modifying the activityof reactions of the form
119860 + 119861
1198961
999445999468
1198962
119860119861 (1)
Namely 120576 intervenes with the dissociation constant of thesereactions so that the corresponding standard ODEs arechanged to yield new equations of the form
119889 [119860119861]
119889119905
= 1198961sdot [119860] sdot [119861] minus
1198962sdot [119860] sdot [119861]
120576
(2)
The value of 120576 in (2) depends on the following HillfunctionH which quantifies the state of platelets activationaccording to the thrombin concentration (here denoted as[fIIa]) that is the factor catalyzing the formation of the fibrinclot
H ([fIIalowast (119905)]) =[fIIalowast (119905)]16123
[fIIalowast (119905)]16123 + (24279 sdot 10minus9)16123
(3)
where [fIIalowast(119905)] = max1199051015840isin[0119905][fIIa(1199051015840)] for a chosen time
interval [0 119905] of simulationThe value [fIIalowast(119905)] represents the maximum transient
thrombin concentration and is needed to simulate the factthat in physiological conditions the thrombin concentrationstarts decreasing after rising to a peak (Figure 2) So doing[fIIalowast(119905)] never decreases once it reaches its maximummagnitude [fIIalowast(119905)] is equivalent to [fIIa(119905)] until the con-centration of factor IIa reaches the peak while thereafter itremains constant at that value which is the maximum in theconsidered time interval [0 119905] FunctionH allows simulatingthe physiological condition whereby platelets remain activealso when the thrombin concentration decreases
For a given concentration of factor IIa the maximumplatelets activation state 120576max is defined as
120576max = 120576max0
+ (1 minus 120576max0
) sdotH ([fIIalowast (119905)]) (4)
where 120576max0
defines the basal activation state of the plateletsat simulation time 119905 = 0 The value 120576max
0
is initially set to 001assuming a basal 1 binding strength of coagulation factorsto the resting plateletsrsquo surface When the full activationof platelets is reached 120576max is equal to 1 and the complexdissociation constants are minimized (see [30] for moredetails)
22 CUDA Architecture and the coagSODA Simulator Intro-duced by Nvidia in 2006 Compute Unified Device Architec-ture (CUDA) is a parallel computing platform and program-ming model that provides programmers with a frameworkto exploit GPUs in general-purpose computational tasks(GPGPU computing) GPGPU computing is a low-cost andenergy-wise alternative to the traditional high-performancecomputing infrastructures (eg clusters of machines) whichgives access to the tera-scale computing on common work-stations of mid-range price However due to the innovativearchitecture and the intrinsic limitations of GPUs a directporting of sequential code on the GPU is most of the timesunfeasible therefore making it challenging to fully exploit itscomputational power and massive parallelism [45]
CUDA combines the single instruction multiple data(SIMD) architecture with multithreading which automat-ically handles the conditional divergence between threadsThe drawback of such flexibility is that any divergence ofthe execution flow between threads causes a serialization ofthe execution which affects the overall performances UnderCUDArsquos naming conventions the programmer implementsthe kernel that is a CC++ function which is loaded fromthe host (the CPU) to the devices (one or more GPUs)and replicated in many copies named threads Threads canbe organized in three-dimensional structures named blockswhich in turn are contained in three-dimensional grids(a schematic description is given in Figure 3 left side)Whenever the host runs a kernel the GPU creates thecorresponding grid and automatically schedules each blockon one free streaming multiprocessor available on the GPUallowing a transparent scaling of performances on different
BioMed Research International 9
Host
Kernel 1
Kernel 2
Device
Grid 1
Grid 2
Block(0 0)
Block(1 0)
Block(2 0)
Block(0 1)
Block(1 1)
Block(2 1)
Block (1 1)
Thread(0 0)
Thread(1 0)
Thread(2 0)
Thread(3 0)
Thread(4 0)
Thread(0 1)
Thread(1 1)
Thread(2 1)
Thread(3 1)
Thread(4 1)
Thread(0 2)
Thread(1 2)
Thread(2 2)
Thread(3 2)
Thread(4 2)
Global memory
Texture memory
Constant memory
Shared memory Shared memory
Block (0 0) Block (1 0)
Localmemory
Localmemory
Localmemory
Localmemory
Thread(0 0)
Thread(1 0)
Registers Registers Registers Registers
Thread(0 0)
Thread(1 0)
Grid
Figure 3 Schematic description of CUDArsquos architecture in terms of threads and memory hierarchy Left Side Threads organization a singlekernel is launched from the host (the CPU) and is executed in multiple threads on the device (the GPU) Threads can be organized in three-dimensional structures named blocks which can be in turn organized in three-dimensional grids The dimensions of blocks and grids areexplicitly defined by the programmer Right Side Memory hierarchy threads can access data from many different memories with differentscopes Registers and local memories are private for each thread Shared memory lets threads belonging to the same block communicate andhas low access latency All threads can access the global memory which suffers high latencies but it is cached since the introduction of theFermi architecture Texture and constant memory can be read from any thread and are equipped with a cache as well in this work we exploitthe constant memory Figures are taken from Nvidiarsquos CUDA programming guide [60]
devices Threads within a block are executed in groups of 32threads named warps
The GPU is equipped with different kinds of memoryIn this work we exploit the global memory (accessible fromall threads) the shared memory (accessible from threadsof the same block) the local memory (registers and arraysaccessible from owner thread) and the constant memory(cached and not modifiable) A schematic representation ofthis memory hierarchy is shown in Figure 3 right side Toachieve the best performances the sharedmemory should beexploited as much as possible however it is very limited (ie49152 bytes for each multiprocessor since the introductionof the Fermi architecture) and it introduces restrictions onthe blocksrsquo size On the other hand the global memory isvery large (thousands ofMBs) but suffers from high latenciesA solution to this problem was implemented on the Fermiarchitecture where the global memory is equipped with aL2 cache This architecture also introduced the possibility tobalance 64KB of fast on-chip memory between the sharedmemory and L1 cache using two possible configurations48KB for the shared memory and 16KB for L1 cache or16 KB for the shared memory and 48 for L1 cacheThe Keplerarchitecture used in this paper allows a third and perfectly
balanced configuration where shared memory and L1 cacheobtain the same amount of memory (32KB) See Figure 4 fora schematization of the memory architecture
The systematic analysis of models of biological systemsoften consists in the execution of large batches of simulationsOne of the standard analyses that can be executed on suchkind of models regards an intensive search within the param-eters space which requires large numbers of independentsimulations In order to reduce the computational burdenwe previously implemented on the CUDA architecture oneof the most efficient numerical integration algorithms forODEs LSODA that is able to automatically recognize stiffand nonstiff systems and to dynamically select between themost appropriate integration procedure (ie Adams methodin the absence of stiffness and the Backward DifferentiationFormulae otherwise) [22] This GPU-powered tool is calledcupSODA [24 25] and exploits CUDArsquos massive parallelismto execute different and independent simulations in eachthread thus reducing the computational time required by astandard CPU counterpart of LSODA
Besides being very efficient for the simulation of manyindependent simulations cupSODA is also user-friendlyLSODA was originally designed to solve ODEs systems
10 BioMed Research International
Thread
Shared memory L1 cache Read-onlydata cache
L2 cache
DRAM
Figure 4 Schematic description of memory hierarchies in Fermi and Kepler architectures GPUs relying on these architectures are equippedwith a two-level data cache and a read-only data cache Shared memory and L1 cache share the same on-chip 64KB memory banks theamount of memory can be reconfigured by the user according to the specific needs of the application Figure taken from Nvidiarsquos KeplerGK110 whitepaper [57]
written in the canonical form but the user is supposed tospecify the system of ODEs by implementing a custom Cfunction that is passed to the algorithmmoreover in order tospeed up the computation when dealing with stiff systems inLSODA the Jacobianmatrix associatedwith theODEs systemmust be implemented as a custom C function as well On thecontrary cupSODA was conceived as a black-box simulatorthat can be easily used without any programming skillscupSODA consists in a tool to automatically convert a genericmechanistic reaction-based model of a biological system intothe corresponding set of ODEs to comply with the mass-action kinetics [46] and to directly encode the obtainedsystem along with the corresponding Jacobian matrix as Carrays
This fully automaticmethodology can be exploited for thesimulation of models whose chemical kinetics is based on themass-action law The BCC model described in Section 21though includes a set 119878
120576of reactions that do not follow
the mass-action kinetics [30] the platelets activity is notexplicitly modeled by biochemical reactions but it is realizedby modulating the rate of the dissociation of the complexesformed on a plateletrsquos surface by means of the variable120576 which is calculated with a special equation during theintegration steps For this reason cupSODA was used asa starting point for the development of a new tool ableto compute at run-time the specific kinetics of this set ofreactions
This new CUDA-powered simulation tool named coag-SODA is specifically tailored for the simulation of the BCC
model developed in [30] In particular coagSODA realizesthe run-time calculation of the 120576 value required to correctlysimulate the activity of platelets which is determined accord-ing to the equations described in Section 21 The plateletsactivation state 120576 is calculated at each time instant by solvingthe following differential equation
119889120576
119889119905
= 119896 (120576max minus 120576) (5)
where the constant 119896 is inversely proportional to the timescale of platelets activation and is set to 0005 This isconsistent with the fact that platelets do not instantly achievetheir maximum attainable activation state (120576max) but theyreach it on a physiologically relevant timescale [30]
Dealing with the 14 reactions in the set 119878120576that are
influenced by the Hill functionH (see Section 21) the valueof [fIIalowast(119905)]must be stored on the GPU because during eachintegration step coagSODA recalculates (4) which exploitsthe value [fIIalowast(119905)] to determine a new 120576 value Since CUDArsquosarchitecture does not offer static variables the informationfor each thread has to be memorized in the global memoryThe accesses to the global memory and the computationalcosts due to these additional calculations slow down theintegration process with respect to the integration of ODEsperformed according to strictly mass-action based kinetics(as in the cupSODA simulator) Nevertheless in Section 33we show that our parallel implementation largely outper-forms a sequential counterpart of the LSODA algorithm
BioMed Research International 11
0 100 200 300 400 500 600 700
Fibr
inog
en co
ncen
trat
ion
(M)
Time (s)
30 of fibrinogen
0
1eminus6
2eminus6
3eminus6
4eminus6
5eminus6
6eminus6
7eminus6
8eminus6
9eminus6
Figure 5 Dynamics of fibrinogen in physiological condition The clotting time is defined as the time necessary to convert the 70 of thefibrinogen into fibrin
As in the case of cupSODA [24 25] coagSODA exploitsthe shared memory to improve performances by storing thecurrent state and time of the simulations in these low-latencymemory banks Despite the improvement of performancesensured by this solution it strongly affects the occupancyof GPUrsquos multiprocessors and therefore it represents thelimiting factor for the number of blocks that can be executedsimultaneously as a matter of fact coagSODA is limited to2 blocks per streaming multiprocessor on GPUs based onthe Fermi architecture with a reduced exploitation of theGPU with respect to the theoretical 8 blocks allowed by thisarchitecture
3 Results and Discussion
In this section we discuss the results of the parameter sweepanalysis (PSA) carried out on the BCC model to the aimof investigating either the prolongation or the reduction ofthe clotting time in response to perturbed values of somereaction constants and of the initial concentration of somemolecular species chosen according to their meaning withinthe whole pathway PSA was performed by generating aset of different initial conditions corresponding to differentparameterizations of the model and then automaticallyexecuting the deterministic simulations with coagSODATheuse ofGPU technology is fundamental in this type of analysisespecially for large biological systems as the BCC is becauseit drastically reduces the computational time
The sweep analysis for single parameters (PSA-1D) wasperformed considering a logarithmic sampling of numericalvalues of each parameter under investigation (reaction con-stant or initial molecular concentration) within a specifiedrange with respect to its physiological reference value Thesweep analysis over pairs of parameters (PSA-2D) was per-formed by simultaneously varying the values of two param-eters within a specified range considering a logarithmicsampling on the resulting lattice The logarithmic samplingallows uniformly spanning different orders of magnitude ofthe parameters value using a reduced and fine-grained set
of samples therefore efficiently analyzing the response of thesystem in a broad range of conditions
To determine the response of the BCC to perturbed con-ditions we chose the clotting time (CT) as output of the PSAThe CT is defined as the time necessary to convert the 70of the fibrinogen into fibrin (see Figure 5) conventionallyassumed to correspond to the time required to form the clot[47] and it is generally used in laboratory tests formonitoringthe therapywith anticoagulant drugs According to themodeldefined in [30] the reference value of CT is around 300seconds in physiological conditions We investigate herethe response of the BCC by evaluating the CT in variousconditions corresponding to different values of the reactionconstants varying over six orders of magnitude with respectto their physiological values (ie three below and three abovethe reference values if not otherwise specified) as well asto different values of the initial molecular concentrationsand varying over twelve orders of magnitude with respect totheir physiological values (ie six below and six above thereference values if not otherwise specified)
The total number of parallel simulations executed tocarry out these analyses was 100 for PSA-1D over reactionconstants 200 for PSA-1D over initial concentrations and1600 for PSA-2D
Finally we present the comparison of the performanceof the CPU and GPU to run an increasing number ofsimulations of the BCC model to prove the efficiency ofcoagSODA
31 PSA-1D of the BCC Model
311 Reaction 11990344 Thefirst PSA was performed to determine
the value of the kinetic constant for the autoactivation offactor XII (reaction 119903
44in Table 1) which corresponds to an
upstream process in the intrinsic pathway This analysis wasmotivated by two considerations Firstly by using the fullparameterization given in [30] the action of the intrinsicpathway turns out to be fundamental for the BCC in vivowhich is in contrast to experimental observations which
12 BioMed Research International
324
326
328
330
332
334
336
338
340
Clot
ting
time (
s)
1eminus
10
1eminus
9
1eminus
8
1eminus
7
1eminus
6
1eminus
5
1eminus
4
1eminus
3
k44 (sminus1)
Figure 6 Clotting time according to PSA-1D over constant 11989644of reaction 119903
44 XII rarr XIIaThe reference value of 119896
44used in [30] is 5sdot10minus4sminus1
the sweep range is [5 sdot 10minus10 5 sdot 10minus4]
280
300
320
340
360
380
400
1e4 1e5 1e6 1e7 1e8 1e9 1e10 1e11
Clot
ting
time (
s)
k27 (Mminus1sminus1)
Figure 7 Clotting time according to PSA-1D over constant 11989627of reaction 119903
27 IIa + V rarr IIa + Va The reference value is 2 sdot 107Mminus1 sminus1 the
sweep range is [2 sdot 104 2 sdot 1010]
indicate that the extrinsic pathway is the main responsibleof clot formation Secondly in [30] all constants values havea reference to experimental measurements except for theconstant of this reaction (which corresponds to entry 29 inTable 1 in [30])
Figure 6 shows the results of this PSA-1D where 11989644was
varied over six orders of magnitude considering the valuegiven in [30] as the upper limit of the sweep interval We canobserve that by decreasing the value of 119896
44with respect to
the value considered in [30] the CT increases with respect toits reference value however for values of this constant lowerthan 100 sdot 10minus6 sminus1 the CT remains unaltered and this canbe intuitively explained by the fact that in this situation thefibrinogen is mainly activated by the extrinsic pathway
Consequently we assigned the value 700 sdot 10minus6 sminus1 to11989644 achieving a CT that is comparable to the experimental
observations of the BCC in vivo This new value was used inall PSA discussed in what follows
312 Reactions 11990327
and 11990358 In the next PSA we investigated
the effect of the perturbation of the kinetics of two pivotal
reactions of the BCC model Reaction 11990327 which describes
the catalytic activation of factor V by factor IIa was chosenbecause it represents the main positive feedback within thecommon pathway reaction 119903
58 which is involved in the
activation of factor XI by binding to factor IIa was chosenbecause it represents the main positive feedback in theintrinsic pathway (Figure 1) Moreover preliminary PSA overall reaction constants of the BCC model given in Table 1evidenced that these two reactions are among the mostrelevant steps of the coagulation network
The PSA-1D over 11989627(Figure 7) shows that the CT is very
sensitive to the perturbation of the rate of this reaction whenthe reference value of its constant (ie 119896
27= 2 sdot 10
7Mminus1 sminus1)is either increased or decreased in particular when 119896
27is
very low a plateau in CT is reached since the strength ofthe positive feedback exerted by factor IIa is largely reduceda condition where the contribution of the amplification ofthe hemostatic stimulus (due by the common pathway) tothe formation of the clot is basically not effective On theother side the PSA-1D over 119896
58(Figure 8) shows that while
a decrease of the constant with respect to its reference value
BioMed Research International 13
310
320
330
340
350
360
370
1e5 1e6 1e7 1e8 1e9 1e10 1e11
Clot
ting
time (
s)
k58 (Mminus1sminus1)
Figure 8 Clotting time according to PSA-1D over constant 11989658of reaction 119903
58 IIa + XI rarr IIa-XI The reference value is 1 sdot 108Mminus1 sminus1 the
sweep range is [1 sdot 105 2 sdot 1011]
220
240
260
280
300
320
340
360
380
400
Clot
ting
time (
s)
Factor VIII (M)
Seve
re h
aem
ophi
lia
Mild
hae
mop
hilia
Prot
hrom
botic
stat
e
1eminus
16
1eminus
14
1eminus
12
1eminus
10
1eminus
8
1eminus
6
1eminus
4
1eminus
2
Figure 9 Clotting time at different initial concentrations of factor VIII The reference value is 700 sdot 10minus10M (dashed black line) the sweeprange is [700 sdot 10minus16 700 sdot 10minus4] The blue area indicates a condition in which factor VIII concentration is less than 1 of its physiologicalconcentration corresponding to a situation of severe haemophilia the red area indicates a condition in which factor VIII concentration isbetween 1 and 30 of its physiological concentration corresponding to a situation of mild haemophilia the green area indicates a conditionin which factor VIII concentration is greater than 130 of its physiological concentration corresponding to a situation of hypercoagulability
(ie 11989658= 1 sdot 10
8Mminus1 sminus1) does not have any substantialeffect an increase of its value leads to a progressive increasein the CT This increase is due to the fact that factor IIa issequestered in the formation of a complex with factor XI andhence it is no longer available as a free component in bloodto participate in other reactions especially those reactionsof the extrinsic pathway which principally lead to the clotformation in vivo This behavior highlights that the intrinsicpathway has a secondary role in blood coagulation in vivocompared with the extrinsic pathway as also evidenced byvarious experimental observations [48]
313 Factors VIII IX and II The next set of PSA-1D wasrealized by varying the initial concentrations of factors VIIIIX and II These factors were selected since both an excess
and a deficiency of their concentrations lead to diseasesrelated to blood clotting
The PSA-1D over factor VIII (Figure 9) shows thatincreasing the initial concentration of this factor resultsin decreasing the CT suggesting the possible presence ofhypercoagulable states in these perturbed conditions As amatter of fact high levels of factor VIII cause an increasedrisk of deep vein thrombosis and pulmonary embolism[49] On the other hand individuals with less than 1of the average concentration of factor VIII show a severehaemophilia A characterized by higher CT (Figure 9) andrequire infusions of plasma containing the deficient factorotherwise frequent spontaneous bleeding would occur [50]When the concentration of this factor is between 5 and 30of the average concentration individuals still risk bleeding incase of trauma [50]
14 BioMed Research International
250
300
350
400
450
500
550
600
650
700
Clot
ting
time (
s)
Factor IX (M)
1eminus
14
1eminus
12
1eminus
10
1eminus
8
1eminus
6
1eminus
4
1eminus
2
1e0
Figure 10 Clotting time at different initial concentrations of factor IXThe reference value is 900sdot10minus8M the sweep range is [900sdot10minus14 900sdot10minus2]
0
100
200
300
400
500
600
700
Clot
ting
time (
s)
Factor II (M)
1eminus
12
1eminus
10
1eminus
8
1eminus
6
1eminus
4
1eminus
2
1e0
1e2
Figure 11 Clotting time at different initial concentrations of thrombinThe reference value is 140sdot10minus6M the sweep range is [140sdot10minus12 140sdot100]
The PSA-1D over factor IX (Figure 10) shows only aslight decrease of the CT as the initial concentration offIX increases this is in contrast to recent studies thatdemonstrated how the excess of factor IX leads to anincreased risk of deep vein thrombosis [51] This result canbe explained by considering that (i) a high concentrationof factor IX is not sufficient to bring about coagulationproblems though when the concentration of other factorsis above the average value (yet not at pathological levels)prothrombotic states can be observed (ii) in this model weconsider average values as initial concentrations of factorshowever individuals are characterized by different (balanced)combinations of procoagulant and anticoagulant factor levelsthat altogether contribute to define a unique coagulationphenotype that reflects the developmental environmentalgenetic nutritional and pharmacological influences of eachindividual [52] On the contrary the lack of factor IX causes
haemophilia B characterized by higher CT with respect tothe reference value (Figure 10)
Furthermore by comparing the PSA-1D of factors VIIIand IX (Figures 9 and 10) it is clear that haemophilia A ismore serious than haemophilia B since the CT achieved inconditions of factor VIII deficiency is higher than the CTobtained in the case of factor IX deficiency
In both PSA-1D over factors VIII and IX we observedafter the initial decrease of the CT an unexpected increaseof the CT as the factor concentration increasesThis counter-intuitive behavior arises at very high concentrations of thesefactors (with respect to the average physiological levels) andto the best of our knowledge it was never observed in vivoNonetheless it would be interesting to verify by means of adhoc laboratory experiments if the model correctly describesthe behavior of the BCC even in these conditions or on thecontrary it is not predictive in these extreme situations
BioMed Research International 15
The PSA-1D over factor II (Figure 11) shows a dramaticdecrease of the CT as the initial concentration of this factorincreases (with respect to the average physiological level)This behavior resembles the effects of hypercoagulability (orthrombophilia) a disease caused by mutation G20210A inthe prothrombin gene [53] that causes an increase of theprothrombin level (factor II) in the blood flow resulting inan excessive formation of the active form of this factor thusheightening venous thrombosis risks [54] Hypercoagulabil-ity is usually treated with warfarin therapy or with otheranticoagulants with a similar effect These drugs decrease thecapacity of coagulation factors to become active preventingthe formation of unwanted thrombi
On the other hand when the initial concentration offactor II is low we achieved the effects of prothrombindeficiency a rare autosomal recessive disease that causes atendency to severe bleeding [29 55] As shown in Figure 11 aconcentration equal to 10of the physiological value of factorII (ie 14 sdot 10minus6 M) leads to clotting effects similar to severehaemophilia A
32 PSA-2D of the BCCModel We present here the results ofthe PSA-2D on the BCCmodel where couples of parameterswere varied to analyze the possible effects arising from thecombined perturbation of their values
321 Reactions 11990327
and 11990358 Figure 12 shows the effect of
the simultaneous variation of constants 11989627
and 11989658
(overthe same sweep ranges considered in the two PSA-1DSection 31) This result remarks that reaction 119903
27 involved
in the common pathway has a stronger influence on theBCC and that there is a synergic interplay between thesetwo reactions In particular when the value of 119896
27is low and
the value of 11989658
is high the CT is higher than the valuesachieved when only a single constant is changed because inthis condition both the intrinsic and the common pathwaysare simultaneously inhibited
322 Factor VIII Factor IX and Tissue Factor In the last twoPSA-2Dwe varied the initial concentration of factor VIII andTissue Factor and the initial concentration of factor IX andTissue Factor respectively
The initial concentrations of factors VIII and IX were var-ied over four orders of magnitude using their physiologicalvalues as upper limit for the sweep ranges the concentrationof Tissue Factor was varied over four orders of magnitudetwo above and two below its reference value (see Table 2)The rationale behind this choice is to observe how the BCCmodel in conditions corresponding to different states ofhaemophilia (obtained by decreasing the concentrations offactors VIII and IX) behaves with different initial concentra-tions of the Tissue Factor which is the upstream factor of theextrinsic pathway that is the most important element of theBCC
The results of these PSA-2D show that with respect tothe condition of haemophilia B in the case of haemophiliaA the amount of Tissue Factor (below its reference value)has a negligible influence on the CT as indicated by the
presence of a plateau in Figure 13 on the contrary concerninghaemophilia B a deficiency of Tissue Factor leads to anincrease of the CT especially when factor IX is present in lowconcentrations (Figure 14)
The different results achieved in the two PSA-2D are dueto the presence in the BCC model of a direct interactionbetweenTissue Factor and factor IX bymeans of theTF-fVIIacomplex (see reactions 119903
13 119903
15in Table 1) Indeed the lack
of Tissue Factor directly affects the concentration of activefactor IX which results in a strong alteration of the CT withrespect to physiological conditions
33 CPU versus GPU Performance Comparison In orderto show the relevant speedup achieved by coagSODA wepresent here the comparison of the computational effortrequired by GPU and CPU for the simulation of the BCCmodel The performances of our GPU simulator were com-pared with those obtained using the LSODA algorithmimplemented in the software COPASI [56] executing onthe CPU the same set of simulations that were run on theGPU COPASI is single-threaded and does not exploit thephysical and logical cores of the CPU therefore it representsa good benchmark as a single-node CPU-bound simulator ofbiological systems
In all simulations executed on both GPU and CPU westored 100 samples uniformly distributed in the time intervalconsidered for each simulation that is [0 700] seconds of thedynamics of all chemical species involved in the BCCmodelThe settings for the LSODA algorithm were the followingrelative error 1 sdot 10minus7 absolute error 1 sdot 10minus13 and maximumnumber of internal steps set to 20000
331 Benchmark Details The GPU used for the simulationsis a Nvidia Tesla K20c equipped with 2496 cores organized in13 streamingmultiprocessors GPU clock 706MHz and 5GBofDDR5RAM In all tests coagSODAwas compiled and exe-cuted with version 55 of CUDA libraries Even though thisGPU has compute capability 35 and is based on the GK110Kepler architecture currently coagSODA does not exploitany of the new features (eg Dynamic Parallelism Hyper-QRemote DMA) with the exception of the new ISA encodingwhich allows threads to exploit an increased number ofregisters (255 instead of 63) reducing register spilling intoglobal memory and increasing performances [57] Moreoveras described in Section 22 the Kepler architecture offers thepossibility to reconfigure the 64KB on-chip cache balancingbetween L1 cache and shared memory Since coagSODAexploits the shared memory to reduce the latencies duringthe access to the concentration values of the BCC model weopted for the following configuration 48KB for the sharedmemory 16 KB for the L1 cache
The performance of the GPU was compared against apersonal computer equippedwith a dual-core CPU Intel Corei5 frequency 23 GHz 4GB of DDR3 RAM running theoperating systemMacOSXLion 1075The software used as acoagSODA-equivalent single-threaded CPU implementationis COPASI version 48 (build 35)
16 BioMed Research International
280
300
320
340
360
380
400
420
440
1e41e6
1e91e11 1e5
1e7
1e9
1e11
280 320 360 400 440
Clot
ting
time (
s)
k 58(M
minus1 sminus1 )
k27 (Mminus1sminus1)
Figure 12 Clotting time according to PSA-2D over reaction constants 11989627and 119896
58 The sweep ranges are [2 sdot 104 2 sdot 1010] and [1 sdot 105 2 sdot 1011]
respectively
100
200
300
400
Tissue factor (M) Factor VIII
(M)
Clot
ting
time (
s)
100
150
200
250
300
350
400
450
1eminus131eminus11
1eminus91eminus14
1eminus12
1eminus10
Figure 13 Clotting time at different initial concentrations of Tissue Factor and factor VIII The sweep ranges are [500 sdot 10minus14 500 sdot 10minus10]and [700 sdot 10minus14 500 sdot 10minus10] respectively
A direct comparison of the performances between thesetwo different architectures and implementations is not aneasy task since CPUs are optimized for single-thread exe-cution and exploit a large number of optimizations (eghigher clock frequency instruction caching pipelining outof order execution and branch prediction) whereas GPUsare optimized for graphics processing and parallel exe-cution of identical code relying only on multilevel datacaching For this reason in this paper we propose anempirical comparison of the performances based on therunning times of identical batches of simulations of the BCCmodel
332 Benchmark Results In Figure 15 we show the compar-ison of the running times required to run several batchesof simulations executed to carry out the PSA-1D over thereaction constant 119896
27 The choice of comparing the perfor-
mances of the GPU and CPU implementations by executingbatches of simulations that are related to a PSA instead ofrunning 119899 independent but identical simulations (ie allcharacterized by the same parameterization of the model)is due to the fact that these results represent a more realisticscenario in the computational analysis of biological systemswhereby it is useful to investigate large search spaces ofparameters corresponding to different perturbed conditionsof themodel [26]Moreover for the evaluation of the running
time the execution of a batch of 119899 identical deterministicsimulations would be futile The figure clearly shows thatcoagSODA always performs better than the CPU counter-part In particular while the CPU performance increaseslinearly with the number of simulations the running timesare strongly reduced on the GPU in this case the overallrunning time roughly corresponds to the time required forthe execution of the slowest simulations This is due to thefact that different parameterizations may require differenttime steps for LSODA to converge and the execution ofa block terminates as long as all the threads it containsterminate In turn the execution of a kernel terminates whenall blocks terminate for this reason a single simulationwhose dynamics requires more steps than the others mayaffect the overall running time
In Table 3 we report the running times of all batchesof simulations along with the speedup achieved on theGPU In particular these results highlight that the advantageof exploiting the GPUs for the simulation of the BCCmodel becomes more evident as the number of simula-tions increases with a 181times speedup when a PSA with10000 different parameterizations is executed Therefore theGPU accelerated analysis of the BCC model with coag-SODA represents a novel relevant computational means toinvestigate the behavior of this complex biological systemunder nonphysiological conditions and could be exploited
BioMed Research International 17
0
200
400
600
Tissue factor (M) Factor IX (M
)
Clot
ting
time (
s)
0
100
200
300
400
500
600
700
1eminus12
1eminus10
1eminus8
1eminus131eminus11
1eminus9
Figure 14 Clotting time at different initial concentrations of Tissue Factor and factor IXThe sweep ranges are [500 sdot 10minus14 500 sdot 10minus10] and[900 sdot 10
minus12 900 sdot 10
minus8] respectively
5 10 50 100 500 1000 5000 10000
Runn
ing
time (
s)
Number of simulations
CPUGPU
1e2
1e3
1e4
1e5
1e6
1e7
Figure 15 Comparison of the computational time required to execute an increasing number of simulations of the BCCmodel on CPU (IntelCore i5 23 GHz) and GPU (Nvidia Tesla K20c GPU clock 706MHz) The computational time is expressed in seconds The time valuesrelated to 1000 5000 and 10000 simulations on the CPU were estimated by regression (see also Table 3)
Table 3 CPU versus GPU performance comparison
Number ofsimulations CPU time (sec) GPU time (sec) Speedup
5 3808 1278 298times10 10604 2329 456times50 51704 8058 642times100 116528 10977 1061times500 536058 17394 3081times1000 1073582lowast 19982 5373times5000 5363507lowast 30960 1732times10000 10724975lowast 58951 1819timeslowastValues estimated by regression
to efficiently determine the response of the BCC to differenttherapeutical drugs
4 Conclusions
Thanks to their high-performance computing capabilitiesand the very low costs GPUs nowadays represent a suitable
technology for the development and the application of par-allel computational methods for in silico analysis of complexbiological systems However the implementation of efficientcomputational tools able to fully exploit the large potentialityof GPUs is still challenging since good programming skillsare required to implement GPU-based algorithmic methodsand to handle specific features as an optimal usage ofmemoryor the communication bandwidth between GPU and CPUMoreover algorithms cannot be directly ported on the GPUbecause of the limited programming capabilities allowed byGPUkernels as amatter of fact they need to be redesigned tomeet the requirements of the underlying SIMD architecture
In this work we presented coagSODA a GPU-poweredsimulator specifically developed to carry out fast parallelsimulations of the BCC model coagSODA was designed tooffer a black-box solution usable by any final user in aneasy way It relies on cupSODA [24] a numerical integratorfor ODEs that we previously implemented for the GPUarchitecture based on the LSODA algorithm and capableof automatically translating a reaction-based model into aset of coupled ODEs In addition to mass-action kinet-ics coagSODA implements specific functions to compute
18 BioMed Research International
the kinetics of Hill function based reactions such as thoseinvolved in the platelets activity of the BCC model coag-SODA exploits the massive parallel capabilities of modernGPUs and our results demonstrated that it can achieve arelevant reduction of the computational time required toexecutemany concurrent and independent simulations of theBCC dynamics The mutual independence of the simulationsallows fully exploiting the underlying SIMD architecturemoreover coagSODA benefits from an additional speedupthanks to our choice of storing the state of the systeminto the low-latency shared memory (a solution that wasalready implemented in cupSODA) Since the BCC modelis large (96 reactions 71 chemical species) a large amountof shared memory was assigned to each thread stronglyreducing the theoretical occupancy of the GPU that is theratio of active warps with respect to the maximum numberof warps supported by each streaming multiprocessor of theGPU However the results of the analysis presented in thiswork performed on the BCC model show that coagSODAachieves a relevant boost with respect to a reference CPUimplementation For instance in the case of 10000 simula-tions we achieved a noticeable 181times speedup Interestinglythe performances of coagSODA are better than COPASI evenfor small batches of simulations These results indicate thatfor biological models consisting in many reactions and manyspecies our GPU implementation of LSODA becomes moreprofitable than the CPU counterpart as the number of con-current simulations increases making it suitable especiallywhen performing demanding computational analysis suchas for example parameter sweep parameter estimation orsensitivity analysis
As a test case in this paper we presented several parame-ter sweep analyses over reaction constants and initial concen-trations of factors involved in the BCC model Other com-putational analyses on mathematical models of this pathwaywere previously presented For instance in [58] a parameterestimation of the reaction constants of different modelsinvolving the activation of factor X factor V prothrombinand the inactivation of factors was performed The aim ofthis analysis was to discriminate between different modelsto unravel the mechanisms on the basis of the BCC Similaranalyses were executed on a model describing thrombingeneration in plasma since the reaction mechanism thereaction constants and initial concentrations were unknown[59] In [34] a sensitivity analysis of a model consisting of44 species over 34 chemical species was presented reactionconstants were varied between 10 and 1000 of theirreference value to the aim of identifying the most influentialfactors of the BCC
All these analyses were performed bymeans of sequentialsimulations of the models under investigation thereforein general only small batches of simulations could be run(for instance in [34] only 836 simulations were executed toexecute the sensitivity analysis of the model) On the otherhand in this paper we presented the results of different PSAefficiently executed bymeans of coagSODA overall providinguseful information regarding unknownparameters and inter-esting insights into the functioning of the BCC A thoroughsensitivity analysis of the whole BCC model consisting in
around 5 sdot 105 simulations is currently in progress on GPUsby our group exploiting the great computational efficiency ofcoagSODA
The results of our computational analyses should benow validated by means of specifically designed laboratoryexperiments In particular we identified a plausible valuefor the constant of the reaction describing the autoactivationof factor XII (since no reference values can be found inliterature) moreover the results of the PSA over factor IXsuggest that a deficiency of this factor is not enough to causesevere bleeding disorders as haemophilia B but an alterationof other factors seems to be necessary for the occurrence ofsuch condition (eg the lack of Tissue Factor as suggestedby the PSA-2D over Tissue Factor and factor IX) Finallythe PSA over factors VIII and IX showed in a situationcharacterized by very high concentrations of these factorsa counterintuitive behavior in which the clotting time isincreased with respect to the value obtained in physiologicalconditions Indeed it would be interesting to design ad hoclaboratory experiments to verify if the BCCmodel is actuallypredictive in such extreme situations
The coagSODA software and the SBML version of theBCC model used in this work are available from the authorsupon request
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This work is partially supported by SYSBIO Centre forSystems Biology
References
[1] B B Aldridge JM BurkeDA Lauffenburger andP K SorgerldquoPhysicochemicalmodelling of cell signalling pathwaysrdquoNatureCell Biology vol 8 no 11 pp 1195ndash1203 2006
[2] I-C Chou and E O Voit ldquoRecent developments in parameterestimation and structure identification of biochemical andgenomic systemsrdquoMathematical Biosciences vol 219 no 2 pp57ndash83 2009
[3] D Besozzi P Cazzaniga D Pescini G Mauri S Colomboand EMartegani ldquoThe role of feedback control mechanisms onthe establishment of oscillatory regimes in the RascAMPPKApathway in S cerevisiaerdquo EURASIP Journal on Bioinformaticsand Systems Biology vol 2012 no 10 2012
[4] A Saltelli M Ratto T Andres et al Global Sensitivity AnalysisThe Primer Wiley-Interscience 2008
[5] A Raue C Kreutz T Maiwald et al ldquoStructural and practicalidentifiability analysis of partially observed dynamical modelsby exploiting the profile likelihoodrdquo Bioinformatics vol 25 no15 pp 1923ndash1929 2009
[6] A Drager M Kronfeld M J Ziller et al ldquoModeling metabolicnetworks in C glutamicum a comparison of rate laws incombination with various parameter optimization strategiesrdquoBMC Systems Biology vol 3 article 5 2009
BioMed Research International 19
[7] D Besozzi P Cazzaniga G Mauri D Pescini and L Van-neschi ldquoA comparison of genetic algorithms and particle swarmoptimization for parameter estimation in stochastic biochem-ical systemsrdquo in Proceedings of the 7th European ConferenceEvolutionary Computation Machine Learning and Data Miningin Bioinformatics (EvoBIO rsquo09) C Pizzuti M Ritchie and MGiacobini Eds vol 5483 of Lecture Notes in Computer Sciencepp 116ndash127 Springer 2009
[8] M S Nobile D Besozzi P Cazzaniga GMauri andD PescinildquocupSODA aGPU-basedmulti-swarmPSOmethod for param-eter estimation in stochastic biological systems exploitingdiscrete-time target seriesrdquo in Proceedings of the 10th EuropeanConference Evolutionary Computation Machine Learning andData Mining in Bioinformatics (EvoBIO rsquo12) M Giacobini LVanneschi and W Bush Eds vol 7246 of Lecture Notes inComputer Science pp 74ndash85 Springer
[9] M S Nobile D Besozzi P Cazzaniga D Pescini and GMaurildquoReverse engineering of kinetic reaction networks by meansof Cartesian Genetic Programming and Particle Swarm Opti-mizationrdquo in Proceedings of the IEEE Congress on EvolutionaryComputation (CEC rsquo13) pp 1594ndash1601 June 2013
[10] M S Nobile ldquoEvolutionary inference of biochemical inter-action networks accelerated on Graphics Processing Unitsrdquoin Proceedings of the 11th International Conference on HighPerformance Computingamp Simulation 2013 (HPCS rsquo13) pp 668ndash670 Helsinki Finland 2013
[11] K Kentzoglanakis and M Poole ldquoA swarm intelligence frame-work for reconstructing gene networks searching for bio-logically plausible architecturesrdquo IEEEACM Transactions onComputational Biology and Bioinformatics vol 9 no 2 pp 358ndash371 2012
[12] J R Koza W Mydlowec G Lanza Y Jessen and M A KeaneldquoAutomatic computational discovery of chemical reaction net-works using genetic programmingrdquo inComputationalDiscoveryof Scientific Knowledge S Dzeroski and L Todorovski Edsvol 4660 of Lecture Notes in Computer Science pp 205ndash227Springer 2007
[13] S Ando E Sakamoto and H Iba ldquoEvolutionary modeling andinference of gene networkrdquo Information Sciences vol 145 no3-4 pp 237ndash259 2002
[14] L Dematte and D Prandi ldquoGPU computing for systemsbiologyrdquo Briefings in Bioinformatics vol 11 no 3 Article IDbbq006 pp 323ndash333 2010
[15] J L Payne N A Sinnott-Armstrong and J HMoore ldquoExploit-ing graphics processing units for computational biology andbioinformaticsrdquo Interdisciplinary Sciences Computational LifeSciences vol 2 no 3 pp 213ndash220 2010
[16] M J Harvey and G De Fabritiis ldquoA survey of computationalmolecular science using graphics processing unitsrdquoWiley Inter-disciplinary Reviews Computational Molecular Science vol 2no 5 pp 734ndash742 2012
[17] Y Zhou J Liepe X Sheng M P H Stumpf and C BarnesldquoGPU accelerated biochemical network simulationrdquo Bioinfor-matics vol 27 no 6 Article ID btr015 pp 874ndash876 2011
[18] M Vigelius A Lane and B Meyer ldquoAccelerating reaction-diffusion simulations with general-purpose graphics processingunitsrdquo Bioinformatics vol 27 no 2 Article ID btq622 pp 288ndash290 2011
[19] D J Wilkinson ldquoStochastic modelling for quantitative descrip-tion of heterogeneous biological systemsrdquo Nature ReviewsGenetics vol 10 no 2 pp 122ndash133 2009
[20] J C Butcher ldquoTrees and numerical methods for ordinarydifferential equationsrdquo Numerical Algorithms vol 53 no 2-3pp 153ndash170 2010
[21] D T Gillespie ldquoStochastic simulation of chemical kineticsrdquoAnnual Review of Physical Chemistry vol 58 pp 35ndash55 2007
[22] L Petzold ldquoAutomatic selection of methods for solving stiffand nonstiff systems of ordinary differential equationsrdquo SIAMJournal of Scientific and Statistical Computing vol 4 no 1 pp136ndash148 1983
[23] Y Cao D T Gillespie and L R Petzold ldquoEfficient step sizeselection for the tau-leaping simulation methodrdquo The Journalof Chemical Physics vol 124 no 4 2006
[24] M S Nobile D Besozzi P Cazzaniga GMauri andD PescinildquocupSODA a CUDA-powered simulator of mass-action kinet-icsrdquo in Proceedings of the 12th International Conference onParallel Computing Technologies (PaCT rsquo13) V Malyshkin Edvol 7979 of Lecture Notes in Computer Science pp 344ndash357
[25] M S Nobile P Cazzaniga D Besozzi and G Mauri ldquoGPU-accelerated simulations of mass-action kinetics models withcupSODArdquoThe Journal of Supercomputing 2014
[26] M S Nobile P Cazzaniga D Besozzi D Pescini and GMauri ldquocuTauLeaping a GPU-powered tau-leaping stochasticsimulator for massive parallel analyses of biological systemsrdquoPLoS ONE vol 9 no 3 2014
[27] K Rogers Blood Physiology and CirculationTheHuman BodyBritannica Educational Publishing 2010
[28] A D Michelson Platelets Elsevier 2nd edition 2007[29] R W Colman J Hirsh V J Marder A W Clowes and J
N George Hemostasis and Thrombosis Basic Principles andClinical Practice Lippincott Williams ampWilkins 2006
[30] M S Chatterjee W S Denney H Jing and S L DiamondldquoSystems biology of coagulation initiation Kinetics of thrombingeneration in resting and activated human bloodrdquo PLoS Com-putational Biology vol 6 no 9 Article ID e1000950 2010
[31] M Bellini D Besozzi P Cazzaniga GMauri andM S NobileldquoSimulation and analysis of the blood coagulation cascadeaccelerated on GPUrdquo in Proceedings of the 22nd EuromicroInternational Conference on Parallel Distributed and Network-Based Processing (PDP) pp 590ndash593 IEEE Computer SocietyConference Publishing Services 2014
[32] T Wajima G K Isbister and S B Duffull ldquoA comprehensivemodel for the humoral coagulation network in humansrdquo Clini-cal Pharmacology and Therapeutics vol 86 no 3 pp 290ndash2982009
[33] K Lo W S Denney and S L Diamond ldquoStochastic modelingof blood coagulation initiationrdquo Pathophysiology of Haemostasisand Thrombosis vol 34 no 2-3 pp 80ndash90 2006
[34] C M Danforth T Orfeo K G Mann K E Brummel-Ziedins and S J Everse ldquoThe impact of uncertainty in a bloodcoagulation modelrdquo Mathematical Medicine and Biology vol26 no 4 Article ID dqp011 pp 323ndash336 2009
[35] S D Lewis P P Shields and J A Shafer ldquoCharacterizationof the kinetic pathway for liberation of fibrinopeptides duringassembly of fibrinrdquoThe Journal of Biological Chemistry vol 260no 18 pp 10192ndash10199 1985
[36] M F Hockin K C Jones S J Everse and K G Mann ldquoAmodel for the stoichiometric regulation of blood coagulationrdquoThe Journal of Biological Chemistry vol 277 no 21 pp 18322ndash18333 2002
[37] K Brummel-Ziedins C YVossen F R Rosendaal KUmezakiand K G Mann ldquoThe plasma hemostatic proteome thrombin
20 BioMed Research International
generation in healthy individualsrdquo Journal of Thrombosis andHaemostasis vol 3 no 7 pp 1472ndash1481 2005
[38] K E Brummel-Ziedins C Y Vossen S Butenas K G Mannand F R Rosendaal ldquoThrombin generation profiles in deepvenous thrombosisrdquo Journal of Thrombosis and Haemostasisvol 3 no 11 pp 2497ndash2505 2005
[39] S D Bungay P A Gentry and R D Gentry ldquoA mathematicalmodel of lipid-mediated thrombin generationrdquo MathematicalMedicine and Biology vol 20 no 1 pp 105ndash129 2003
[40] S Butenas T Orfeo M T Gissel K E Brummel and K GMann ldquoThe significance of circulating factor IXa in bloodrdquoThe Journal of Biological Chemistry vol 279 no 22 pp 22875ndash22882 2004
[41] D Voet and J G Voet Biochemistry Wiley 2011[42] A C Guyton and J E Hall Textbook of Medical Physiology
Saunders 2010[43] M F M James and A M Roche ldquoDose-response relationship
between plasma ionized calcium concentration and thrombe-lastographyrdquo Journal of Cardiothoracic and Vascular Anesthesiavol 18 no 5 pp 581ndash586 2004
[44] D L Nelson and M M Cox Lehninger Principles of Biochem-istry W H Freeman Company 2004
[45] R M Farber ldquoTopical perspective on massive threading andparallelismrdquo Journal of Molecular Graphics and Modelling vol30 pp 82ndash89 2011
[46] OWolkenhauerMUllahWKolch andK-HCho ldquoModelingand simulation of intracellular dynamics choosing an appropri-ate frameworkrdquo IEEETransactions onNanobioscience vol 3 no3 pp 200ndash207 2004
[47] B H Estridge A P Reynolds and N J Walters Basic MedicalLaboratory Techniques Cengage Learning 4th edition 2000
[48] T Renne A H Schmaier K F Nickel M Blomback and CMaas ldquoIn vivo roles of factor XIIrdquo Blood vol 120 no 22 pp4296ndash4303 2012
[49] P V Jenkins O Rawley O P Smith and J S ODonnellldquoElevated factor VIII levels and risk of venous thrombosisrdquoBritish Journal of Haematology vol 157 no 6 pp 653ndash663 2012
[50] C E Walsh and K M Batt ldquoHemophilia clinical gene therapybrief reviewrdquo Translational Research vol 161 no 4 pp 307ndash3122013
[51] A Van Hylckama Vlieg I K Van Der Linden R M Bertinaand F R Rosendaal ldquoHigh levels of factor IX increase the risk ofvenous thrombosisrdquo Blood vol 95 no 12 pp 3678ndash3682 2000
[52] K Brummel-Ziedins ldquoModels for thrombin generation and riskof diseaserdquo Journal ofThrombosis and Haemostasis vol 11 no 1pp 212ndash223 2013
[53] U Seligsohn and A Lubetsky ldquoGenetic susceptibility to venousthrombosisrdquoTheNew England Journal of Medicine vol 344 no16 pp 1222ndash1231 2001
[54] I Martinelli P Bucciarelli and P M Mannucci ldquoThromboticrisk factors basic pathophysiologyrdquo Critical Care Medicine vol38 no 2 pp S3ndashS9 2010
[55] K Brummel-Ziedins A Undas T Orfeo et al ldquoThrombingeneration in acute coronary syndrome and stable coronaryartery disease dependence on plasma factor compositionrdquoJournal ofThrombosis andHaemostasis vol 6 no 1 pp 104ndash1102008
[56] S Hoops R Gauges C Lee et al ldquoCOPASImdasha complexpathway simulatorrdquo Bioinformatics vol 22 no 24 pp 3067ndash3074 2006
[57] Nvidia NVIDIArsquos Next Generation CUDA Compute Architec-ture Kepler GK110 2012
[58] W Stortelder W Hemker and H Hemker ldquoMathematicalmodelling in blood coagulation simulation and parameterestimationrdquoModelling Analysis and Simulation vol 20 pp 1ndash111997
[59] S M C Orfao G Jank and K Mottaghy ldquoA new mathematicalapproach to modelling thrombin generationrdquo InternationalJournal of Artificial Organs vol 29 no 7 pp 703ndash708 2006
[60] Nvidia Nvidia CUDA C Programming Guide v50 2012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Anatomy Research International
PeptidesInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
International Journal of
Volume 2014
Zoology
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Molecular Biology International
GenomicsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
BioinformaticsAdvances in
Marine BiologyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Signal TransductionJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
BioMed Research International
Evolutionary BiologyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Biochemistry Research International
ArchaeaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Genetics Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in
Virolog y
Hindawi Publishing Corporationhttpwwwhindawicom
Nucleic AcidsJournal of
Volume 2014
Stem CellsInternational
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Enzyme Research
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Microbiology
BioMed Research International 5
Table 1 Continued
119899 Reactants Products Constant (119896)11990350
fXIIa-PKal fXIIa + Kal 4000 sminus1
11990351
fXII + Kal fXII-Kal 100 sdot 108Mminus1 sminus1
11990352
fXII-Kal fXII + Kal 4530 sminus1
11990353
fXII-Kal fXIIa + Kal 570 sminus1
11990354
PKal + Kal Kal + Kal 270 sdot 104Mminus1 sminus1
11990355
Kal Kal119894
110 sdot 10minus2 sminus1
11990356
fXIIa + C1inh fXIIa-C1inh 360 sdot 103Mminus1 sminus1
11990357
fXIIa + ATIII fXIIa-ATIII 2160Mminus1 sminus1
11990358
fXI + fIIa fXI-fIIa 100 sdot 108Mminus1 sminus1
11990359
fXI-fIIa fXI + fIIa 500 sminus1
11990360
fXI-fIIa fXIa + fIIa 130 sdot 10minus4 sminus1
11990361
fXIIa + fXI fXIIa-fXI 100 sdot 108Mminus1 sminus1
11990362
fXIIa-fXI fXIIa + fXI 200 sminus1
11990363
fXIIa-fXI fXIIa + fXIa 570 sdot 10minus4 sminus1
11990364
fXIa + fXI fXIa + fXIa 319 sdot 106Mminus1 sminus1
11990365
fXIa + ATIII fXIa-ATIII 320 sdot 102Mminus1 sminus1
11990366
fXIa + C1inh fXIa-C1inh 180 sdot 103Mminus1 sminus1
11990367
fXIa + A1AT fXIa-A1AT 100 sdot 102Mminus1 sminus1
11990368
fXIa + A2AP fXIa-A2AP 43 sdot 103Mminus1 sminus1
11990369
fXIa + fIX fXIa-fIX 100 sdot 108Mminus1 sminus1
11990370
fXIa-fIX fXIa + fIX 4100 sminus1
11990371
fXIa-fIX fXIa + fIXa 770 sminus1
11990372
fIXa + fX fIXa-fX 100 sdot 108Mminus1 sminus1
11990373
fIXa-fX fIXa + fX 064 sminus1
11990374
fIXa-fX fIXa + fXa 700 sdot 10minus4 sminus1
11990375
fXa + fVIII fXa-fVIII 100 sdot 108Mminus1 sminus1
11990376
fXa-fVIII fXa + fVIII 210 sminus1
11990377
fXa-fVIII fXa + fVIIIa 0023 sminus1
11990378
fVIIa + fIX fVIIa-fIX 100 sdot 108Mminus1 sminus1
11990379
fVIIa-fIX fVIIa + fIX 090 sminus1
11990380
fVIIa-fIX fVIIa + fIXa 360 sdot 10minus5 sminus1
11990381
fVIIa + fX fVIIa-fX 100 sdot 108Mminus1 sminus1
11990382
fVIIa-fX fVIIa + fX 21000 sminus1
11990383
fVIIa-fX fVIIa + fXa 160 sdot 10minus6 sminus1
11990384
Fbg + fIIa Fbg-fIIa 100 sdot 108Mminus1 sminus1
11990385
Fbg-fIIa Fbg + fIIa 63600 sminus1
11990386
Fbg-fIIa Fbn1 + fIIa + FPA 8400 sminus1
11990387
Fbn1 + fIIa Fbn1-fIIa 100 sdot 108Mminus1 sminus1
11990388
Fbn1-fIIa Fbn1 + fIIa 74260 sminus1
11990389
Fbn1-fIIa Fbn2 + fIIa + FPB 740 sminus1
11990390
Fbn1 + Fbn1 (Fbn1)2 100 sdot 106Mminus1 sminus1
11990391
(Fbn1)2 2Fbn1 640 sdot 10minus2 sminus1
11990392
(Fbn1)2 + fIIa (Fbn1)2-fIIa 100 sdot 108Mminus1 sminus1
11990393
(Fbn1)2-fIIa (Fbn1)2 + fIIa 70100 sminus1
11990394
(Fbn1)2-fIIa (Fbn2)2 + fIIa + FPB 4900 sminus1
11990395
Fbn2 + fIIa Fbn2-fIIa 100 sdot 108Mminus1 sminus1
11990396
Fbn2-fIIa Fbn2 + fIIa 100 sdot 103 sminus1
6 BioMed Research International
fXIITF-fVII
TFfVIIa
fVII
TF-fVIIa
Fbg
FPA
Fbn1
FPB
Fbn2
Fbn1
(Fbn1)2
(Fbn2)2 FPB
fXa
fIIa
TF-fVIIa-fXa
fIX
fIXa
fX
fmIIa
fII
PKal
KalifXI
fXIa
fIX
fIXa
fVIII
fVIIIa
fIXa-fVIIIa
fVIIIa1 -L
fVIIIa2
fVIIIa2
fIXa
fV
fVa
fXa-fVa
fXa-TFPI
TFPI
TF-fVIIa-fXa-TFPI
fXa-AT
AT
AT
fXIa-AT
C1inh
fXIIa-C1inh
A1AT
A2AP
fXIa-A2APfXIa-A1AT
TF-fVIIa
fXIIa Kal
fVIIIa1 -L
Figure 1 Graphical representation of the blood coagulation cascade model considered in this work Legend Blue box coagulation factor redbox inhibitor and related complexes Black arrow complex formation green arrow catalytic activation violet arrow activation red arrowinhibition The reaction is reversible if the arrow tail consists in a dot
BioMed Research International 7
(iv) the activation of prekallikrein by factor XIIa(reactions 119903
48 11990349 11990350) and by kallikrein (reac-
tion 11990354)
(v) the activation of factor XI by factor XIIa (reac-tions 119903
61 11990362 11990363) and by factor XIa (reaction
11990364)
(vi) the activation of factor IX by factor XIa (reac-tions 119903
69 11990370 11990371)
(vii) the dissociation of free factor VIIIa (reactions11990323and 11990324)
(viii) the dissociation of factor VIIIa in complex withother factors (reactions 119903
25and 11990326)
(3) The third module corresponds to the common path-way which consists in
(i) the activation of factor II by factor Xa (modeledby reaction 119903
16) and by the complex between
factor Xa and factor Va through the formationof the intermediate meizothrombin (reactions11990330 119903
33)
(ii) the activation of factor VII by factor Xa (reac-tion 1199036) and by factor IIa (reaction 119903
7)
(iii) the activation of factor VIII by factor Xa (reac-tions 119903
75 11990376 11990377) and by factor IIa (reaction 119903
17)
(iv) the formation of a complex between factor Xaand factor Va (reactions 119903
28and 11990329)
(v) the activation of fibrinogen by factor IIa (reac-tions 119903
84 119903
96)
(vi) the activation of factor V by factor IIa (reaction11990327)
(vii) the activation of factor XI by factor IIa (reac-tions 119903
58 11990359 11990360)
(4) The fourth module describes the inhibition of coagu-lation carried out through the main inhibitors of theBCC (antithrombin Tissue Factor Pathway InhibitorC1-inihibitor 1205721-antitrypsin and 1205722-antiplasmin)These are tight binding inhibitors belonging to theserpin superfamily which form irreversible com-plexes This module consists in
(i) the inhibition of factor Xa by Tissue FactorPathway Inhibitor (modeled by reactions 119903
34and
11990335)
(ii) the inhibition of the complex between TissueFactor factor VIIa and factor Xa by TissueFactor Pathway Inhibitor (reactions 119903
36 11990337 11990338)
(iii) the inhibition of factor IIa by antithrombin(reactions 119903
40and 11990342)
(iv) the inhibition of factor Xa by antithrombin(reaction 119903
39)
(v) the inhibition of factor IXa by antithrombin(reaction 119903
41)
(vi) the inhibition of factor XIa by antithrombin(reaction 119903
65) and by C1-inhibitor (reaction 119903
66)
Table 2 Initial concentrations of molecular species in the bloodcoagulation cascade model (values taken from [30])
Species Symbol Concentration (M)1205721-Antitrypsin A1AT 450 sdot 10
minus5
1205722-Antiplasmin A2AP 100 sdot 10minus6
Antithrombin III ATIII 340 sdot 10minus6
C1-inhibitor C1inh 250 sdot 10minus6
Fibrinogen Fbg 900 sdot 10minus6
Factor II fII 140 sdot 10minus6
Factor V fV 200 sdot 10minus8
Factor VII fVII 100 sdot 10minus8
Active factor VII fVIIa 100 sdot 10minus10
Factor VIII fVIII 700 sdot 10minus10
Factor IX fIX 900 sdot 10minus8
Factor X fX 16 sdot 10minus7
Factor XI fXI 310 sdot 10minus8
Factor XII fXII 340 sdot 10minus7
Prekallikrein Pkal 450 sdot 10minus7
Tissue Factor TF 500 sdot 10minus12
Tissue Factor Pathway Inhibitor TFPI 250 sdot 10minus9
(vii) the inhibition of factor XIIa by C1-inhibitor(reaction 119903
56) by antithrombin (reaction 119903
57)
by 1205721-antitrypsin (reaction 11990367) and by 1205722-
antiplasmin (reaction 11990368)
(viii) the inhibition of the complex between TissueFactor and factor VIIa by antithrombin (reac-tion 11990343)
(ix) the inhibition of kallikrein (reaction 11990355)
The values of the initial concentrations of the molecularspecies occurring in the BCC model are given in Table 2According to [30] the concentrations of complexes and activefactors were set to 0 except for the active factor VII which isphysiologically present in the blood circulation even in theabsence of damage in a concentration that is approximatelyequal to 1 of the corresponding inactive factor [42]
The system of ordinary differential equations (ODEs)needed to carry out the simulations and the parametersweep analysis presented in Section 3 was derived fromthe reactions given in Table 1 according to the mass-actionlaw with the exception of 14 reactions belonging to the set119878120576= 11990319 11990321 11990323 11990329 11990331 11990335 11990346 11990349 11990352 11990362 11990370 11990373 11990376 11990382
The mass-action law is the fundamental empirical law thatgoverns biochemical reaction rates which states that in adilute solution the rate of an elementary reaction is propor-tional to the product of the concentration of its reactantsraised to the power of the corresponding stoichiometriccoefficient [44] The reactions in 119878
120576are instead influenced by
a specifically defined variable 120576 depending on aHill functionfit against experimental data which quantify the plateletactivation status which is used to model the physiologicallevels of thrombin concentration as a function of plateletactivation as thoroughly described in [30] More preciselythe value of 120576 influences the formation of some complexes
8 BioMed Research International
0 0 100 200 300 400 500 600 700
Fact
or II
a con
cent
ratio
n (M
)
Time (s)
1eminus7
2eminus7
3eminus7
4eminus7
5eminus7
6eminus7
7eminus7
8eminus7
9eminus7
Figure 2 Dynamics of thrombin (factor IIa) in physiological condition
occurring on the plateletsrsquo surface by modifying the activityof reactions of the form
119860 + 119861
1198961
999445999468
1198962
119860119861 (1)
Namely 120576 intervenes with the dissociation constant of thesereactions so that the corresponding standard ODEs arechanged to yield new equations of the form
119889 [119860119861]
119889119905
= 1198961sdot [119860] sdot [119861] minus
1198962sdot [119860] sdot [119861]
120576
(2)
The value of 120576 in (2) depends on the following HillfunctionH which quantifies the state of platelets activationaccording to the thrombin concentration (here denoted as[fIIa]) that is the factor catalyzing the formation of the fibrinclot
H ([fIIalowast (119905)]) =[fIIalowast (119905)]16123
[fIIalowast (119905)]16123 + (24279 sdot 10minus9)16123
(3)
where [fIIalowast(119905)] = max1199051015840isin[0119905][fIIa(1199051015840)] for a chosen time
interval [0 119905] of simulationThe value [fIIalowast(119905)] represents the maximum transient
thrombin concentration and is needed to simulate the factthat in physiological conditions the thrombin concentrationstarts decreasing after rising to a peak (Figure 2) So doing[fIIalowast(119905)] never decreases once it reaches its maximummagnitude [fIIalowast(119905)] is equivalent to [fIIa(119905)] until the con-centration of factor IIa reaches the peak while thereafter itremains constant at that value which is the maximum in theconsidered time interval [0 119905] FunctionH allows simulatingthe physiological condition whereby platelets remain activealso when the thrombin concentration decreases
For a given concentration of factor IIa the maximumplatelets activation state 120576max is defined as
120576max = 120576max0
+ (1 minus 120576max0
) sdotH ([fIIalowast (119905)]) (4)
where 120576max0
defines the basal activation state of the plateletsat simulation time 119905 = 0 The value 120576max
0
is initially set to 001assuming a basal 1 binding strength of coagulation factorsto the resting plateletsrsquo surface When the full activationof platelets is reached 120576max is equal to 1 and the complexdissociation constants are minimized (see [30] for moredetails)
22 CUDA Architecture and the coagSODA Simulator Intro-duced by Nvidia in 2006 Compute Unified Device Architec-ture (CUDA) is a parallel computing platform and program-ming model that provides programmers with a frameworkto exploit GPUs in general-purpose computational tasks(GPGPU computing) GPGPU computing is a low-cost andenergy-wise alternative to the traditional high-performancecomputing infrastructures (eg clusters of machines) whichgives access to the tera-scale computing on common work-stations of mid-range price However due to the innovativearchitecture and the intrinsic limitations of GPUs a directporting of sequential code on the GPU is most of the timesunfeasible therefore making it challenging to fully exploit itscomputational power and massive parallelism [45]
CUDA combines the single instruction multiple data(SIMD) architecture with multithreading which automat-ically handles the conditional divergence between threadsThe drawback of such flexibility is that any divergence ofthe execution flow between threads causes a serialization ofthe execution which affects the overall performances UnderCUDArsquos naming conventions the programmer implementsthe kernel that is a CC++ function which is loaded fromthe host (the CPU) to the devices (one or more GPUs)and replicated in many copies named threads Threads canbe organized in three-dimensional structures named blockswhich in turn are contained in three-dimensional grids(a schematic description is given in Figure 3 left side)Whenever the host runs a kernel the GPU creates thecorresponding grid and automatically schedules each blockon one free streaming multiprocessor available on the GPUallowing a transparent scaling of performances on different
BioMed Research International 9
Host
Kernel 1
Kernel 2
Device
Grid 1
Grid 2
Block(0 0)
Block(1 0)
Block(2 0)
Block(0 1)
Block(1 1)
Block(2 1)
Block (1 1)
Thread(0 0)
Thread(1 0)
Thread(2 0)
Thread(3 0)
Thread(4 0)
Thread(0 1)
Thread(1 1)
Thread(2 1)
Thread(3 1)
Thread(4 1)
Thread(0 2)
Thread(1 2)
Thread(2 2)
Thread(3 2)
Thread(4 2)
Global memory
Texture memory
Constant memory
Shared memory Shared memory
Block (0 0) Block (1 0)
Localmemory
Localmemory
Localmemory
Localmemory
Thread(0 0)
Thread(1 0)
Registers Registers Registers Registers
Thread(0 0)
Thread(1 0)
Grid
Figure 3 Schematic description of CUDArsquos architecture in terms of threads and memory hierarchy Left Side Threads organization a singlekernel is launched from the host (the CPU) and is executed in multiple threads on the device (the GPU) Threads can be organized in three-dimensional structures named blocks which can be in turn organized in three-dimensional grids The dimensions of blocks and grids areexplicitly defined by the programmer Right Side Memory hierarchy threads can access data from many different memories with differentscopes Registers and local memories are private for each thread Shared memory lets threads belonging to the same block communicate andhas low access latency All threads can access the global memory which suffers high latencies but it is cached since the introduction of theFermi architecture Texture and constant memory can be read from any thread and are equipped with a cache as well in this work we exploitthe constant memory Figures are taken from Nvidiarsquos CUDA programming guide [60]
devices Threads within a block are executed in groups of 32threads named warps
The GPU is equipped with different kinds of memoryIn this work we exploit the global memory (accessible fromall threads) the shared memory (accessible from threadsof the same block) the local memory (registers and arraysaccessible from owner thread) and the constant memory(cached and not modifiable) A schematic representation ofthis memory hierarchy is shown in Figure 3 right side Toachieve the best performances the sharedmemory should beexploited as much as possible however it is very limited (ie49152 bytes for each multiprocessor since the introductionof the Fermi architecture) and it introduces restrictions onthe blocksrsquo size On the other hand the global memory isvery large (thousands ofMBs) but suffers from high latenciesA solution to this problem was implemented on the Fermiarchitecture where the global memory is equipped with aL2 cache This architecture also introduced the possibility tobalance 64KB of fast on-chip memory between the sharedmemory and L1 cache using two possible configurations48KB for the shared memory and 16KB for L1 cache or16 KB for the shared memory and 48 for L1 cacheThe Keplerarchitecture used in this paper allows a third and perfectly
balanced configuration where shared memory and L1 cacheobtain the same amount of memory (32KB) See Figure 4 fora schematization of the memory architecture
The systematic analysis of models of biological systemsoften consists in the execution of large batches of simulationsOne of the standard analyses that can be executed on suchkind of models regards an intensive search within the param-eters space which requires large numbers of independentsimulations In order to reduce the computational burdenwe previously implemented on the CUDA architecture oneof the most efficient numerical integration algorithms forODEs LSODA that is able to automatically recognize stiffand nonstiff systems and to dynamically select between themost appropriate integration procedure (ie Adams methodin the absence of stiffness and the Backward DifferentiationFormulae otherwise) [22] This GPU-powered tool is calledcupSODA [24 25] and exploits CUDArsquos massive parallelismto execute different and independent simulations in eachthread thus reducing the computational time required by astandard CPU counterpart of LSODA
Besides being very efficient for the simulation of manyindependent simulations cupSODA is also user-friendlyLSODA was originally designed to solve ODEs systems
10 BioMed Research International
Thread
Shared memory L1 cache Read-onlydata cache
L2 cache
DRAM
Figure 4 Schematic description of memory hierarchies in Fermi and Kepler architectures GPUs relying on these architectures are equippedwith a two-level data cache and a read-only data cache Shared memory and L1 cache share the same on-chip 64KB memory banks theamount of memory can be reconfigured by the user according to the specific needs of the application Figure taken from Nvidiarsquos KeplerGK110 whitepaper [57]
written in the canonical form but the user is supposed tospecify the system of ODEs by implementing a custom Cfunction that is passed to the algorithmmoreover in order tospeed up the computation when dealing with stiff systems inLSODA the Jacobianmatrix associatedwith theODEs systemmust be implemented as a custom C function as well On thecontrary cupSODA was conceived as a black-box simulatorthat can be easily used without any programming skillscupSODA consists in a tool to automatically convert a genericmechanistic reaction-based model of a biological system intothe corresponding set of ODEs to comply with the mass-action kinetics [46] and to directly encode the obtainedsystem along with the corresponding Jacobian matrix as Carrays
This fully automaticmethodology can be exploited for thesimulation of models whose chemical kinetics is based on themass-action law The BCC model described in Section 21though includes a set 119878
120576of reactions that do not follow
the mass-action kinetics [30] the platelets activity is notexplicitly modeled by biochemical reactions but it is realizedby modulating the rate of the dissociation of the complexesformed on a plateletrsquos surface by means of the variable120576 which is calculated with a special equation during theintegration steps For this reason cupSODA was used asa starting point for the development of a new tool ableto compute at run-time the specific kinetics of this set ofreactions
This new CUDA-powered simulation tool named coag-SODA is specifically tailored for the simulation of the BCC
model developed in [30] In particular coagSODA realizesthe run-time calculation of the 120576 value required to correctlysimulate the activity of platelets which is determined accord-ing to the equations described in Section 21 The plateletsactivation state 120576 is calculated at each time instant by solvingthe following differential equation
119889120576
119889119905
= 119896 (120576max minus 120576) (5)
where the constant 119896 is inversely proportional to the timescale of platelets activation and is set to 0005 This isconsistent with the fact that platelets do not instantly achievetheir maximum attainable activation state (120576max) but theyreach it on a physiologically relevant timescale [30]
Dealing with the 14 reactions in the set 119878120576that are
influenced by the Hill functionH (see Section 21) the valueof [fIIalowast(119905)]must be stored on the GPU because during eachintegration step coagSODA recalculates (4) which exploitsthe value [fIIalowast(119905)] to determine a new 120576 value Since CUDArsquosarchitecture does not offer static variables the informationfor each thread has to be memorized in the global memoryThe accesses to the global memory and the computationalcosts due to these additional calculations slow down theintegration process with respect to the integration of ODEsperformed according to strictly mass-action based kinetics(as in the cupSODA simulator) Nevertheless in Section 33we show that our parallel implementation largely outper-forms a sequential counterpart of the LSODA algorithm
BioMed Research International 11
0 100 200 300 400 500 600 700
Fibr
inog
en co
ncen
trat
ion
(M)
Time (s)
30 of fibrinogen
0
1eminus6
2eminus6
3eminus6
4eminus6
5eminus6
6eminus6
7eminus6
8eminus6
9eminus6
Figure 5 Dynamics of fibrinogen in physiological condition The clotting time is defined as the time necessary to convert the 70 of thefibrinogen into fibrin
As in the case of cupSODA [24 25] coagSODA exploitsthe shared memory to improve performances by storing thecurrent state and time of the simulations in these low-latencymemory banks Despite the improvement of performancesensured by this solution it strongly affects the occupancyof GPUrsquos multiprocessors and therefore it represents thelimiting factor for the number of blocks that can be executedsimultaneously as a matter of fact coagSODA is limited to2 blocks per streaming multiprocessor on GPUs based onthe Fermi architecture with a reduced exploitation of theGPU with respect to the theoretical 8 blocks allowed by thisarchitecture
3 Results and Discussion
In this section we discuss the results of the parameter sweepanalysis (PSA) carried out on the BCC model to the aimof investigating either the prolongation or the reduction ofthe clotting time in response to perturbed values of somereaction constants and of the initial concentration of somemolecular species chosen according to their meaning withinthe whole pathway PSA was performed by generating aset of different initial conditions corresponding to differentparameterizations of the model and then automaticallyexecuting the deterministic simulations with coagSODATheuse ofGPU technology is fundamental in this type of analysisespecially for large biological systems as the BCC is becauseit drastically reduces the computational time
The sweep analysis for single parameters (PSA-1D) wasperformed considering a logarithmic sampling of numericalvalues of each parameter under investigation (reaction con-stant or initial molecular concentration) within a specifiedrange with respect to its physiological reference value Thesweep analysis over pairs of parameters (PSA-2D) was per-formed by simultaneously varying the values of two param-eters within a specified range considering a logarithmicsampling on the resulting lattice The logarithmic samplingallows uniformly spanning different orders of magnitude ofthe parameters value using a reduced and fine-grained set
of samples therefore efficiently analyzing the response of thesystem in a broad range of conditions
To determine the response of the BCC to perturbed con-ditions we chose the clotting time (CT) as output of the PSAThe CT is defined as the time necessary to convert the 70of the fibrinogen into fibrin (see Figure 5) conventionallyassumed to correspond to the time required to form the clot[47] and it is generally used in laboratory tests formonitoringthe therapywith anticoagulant drugs According to themodeldefined in [30] the reference value of CT is around 300seconds in physiological conditions We investigate herethe response of the BCC by evaluating the CT in variousconditions corresponding to different values of the reactionconstants varying over six orders of magnitude with respectto their physiological values (ie three below and three abovethe reference values if not otherwise specified) as well asto different values of the initial molecular concentrationsand varying over twelve orders of magnitude with respect totheir physiological values (ie six below and six above thereference values if not otherwise specified)
The total number of parallel simulations executed tocarry out these analyses was 100 for PSA-1D over reactionconstants 200 for PSA-1D over initial concentrations and1600 for PSA-2D
Finally we present the comparison of the performanceof the CPU and GPU to run an increasing number ofsimulations of the BCC model to prove the efficiency ofcoagSODA
31 PSA-1D of the BCC Model
311 Reaction 11990344 Thefirst PSA was performed to determine
the value of the kinetic constant for the autoactivation offactor XII (reaction 119903
44in Table 1) which corresponds to an
upstream process in the intrinsic pathway This analysis wasmotivated by two considerations Firstly by using the fullparameterization given in [30] the action of the intrinsicpathway turns out to be fundamental for the BCC in vivowhich is in contrast to experimental observations which
12 BioMed Research International
324
326
328
330
332
334
336
338
340
Clot
ting
time (
s)
1eminus
10
1eminus
9
1eminus
8
1eminus
7
1eminus
6
1eminus
5
1eminus
4
1eminus
3
k44 (sminus1)
Figure 6 Clotting time according to PSA-1D over constant 11989644of reaction 119903
44 XII rarr XIIaThe reference value of 119896
44used in [30] is 5sdot10minus4sminus1
the sweep range is [5 sdot 10minus10 5 sdot 10minus4]
280
300
320
340
360
380
400
1e4 1e5 1e6 1e7 1e8 1e9 1e10 1e11
Clot
ting
time (
s)
k27 (Mminus1sminus1)
Figure 7 Clotting time according to PSA-1D over constant 11989627of reaction 119903
27 IIa + V rarr IIa + Va The reference value is 2 sdot 107Mminus1 sminus1 the
sweep range is [2 sdot 104 2 sdot 1010]
indicate that the extrinsic pathway is the main responsibleof clot formation Secondly in [30] all constants values havea reference to experimental measurements except for theconstant of this reaction (which corresponds to entry 29 inTable 1 in [30])
Figure 6 shows the results of this PSA-1D where 11989644was
varied over six orders of magnitude considering the valuegiven in [30] as the upper limit of the sweep interval We canobserve that by decreasing the value of 119896
44with respect to
the value considered in [30] the CT increases with respect toits reference value however for values of this constant lowerthan 100 sdot 10minus6 sminus1 the CT remains unaltered and this canbe intuitively explained by the fact that in this situation thefibrinogen is mainly activated by the extrinsic pathway
Consequently we assigned the value 700 sdot 10minus6 sminus1 to11989644 achieving a CT that is comparable to the experimental
observations of the BCC in vivo This new value was used inall PSA discussed in what follows
312 Reactions 11990327
and 11990358 In the next PSA we investigated
the effect of the perturbation of the kinetics of two pivotal
reactions of the BCC model Reaction 11990327 which describes
the catalytic activation of factor V by factor IIa was chosenbecause it represents the main positive feedback within thecommon pathway reaction 119903
58 which is involved in the
activation of factor XI by binding to factor IIa was chosenbecause it represents the main positive feedback in theintrinsic pathway (Figure 1) Moreover preliminary PSA overall reaction constants of the BCC model given in Table 1evidenced that these two reactions are among the mostrelevant steps of the coagulation network
The PSA-1D over 11989627(Figure 7) shows that the CT is very
sensitive to the perturbation of the rate of this reaction whenthe reference value of its constant (ie 119896
27= 2 sdot 10
7Mminus1 sminus1)is either increased or decreased in particular when 119896
27is
very low a plateau in CT is reached since the strength ofthe positive feedback exerted by factor IIa is largely reduceda condition where the contribution of the amplification ofthe hemostatic stimulus (due by the common pathway) tothe formation of the clot is basically not effective On theother side the PSA-1D over 119896
58(Figure 8) shows that while
a decrease of the constant with respect to its reference value
BioMed Research International 13
310
320
330
340
350
360
370
1e5 1e6 1e7 1e8 1e9 1e10 1e11
Clot
ting
time (
s)
k58 (Mminus1sminus1)
Figure 8 Clotting time according to PSA-1D over constant 11989658of reaction 119903
58 IIa + XI rarr IIa-XI The reference value is 1 sdot 108Mminus1 sminus1 the
sweep range is [1 sdot 105 2 sdot 1011]
220
240
260
280
300
320
340
360
380
400
Clot
ting
time (
s)
Factor VIII (M)
Seve
re h
aem
ophi
lia
Mild
hae
mop
hilia
Prot
hrom
botic
stat
e
1eminus
16
1eminus
14
1eminus
12
1eminus
10
1eminus
8
1eminus
6
1eminus
4
1eminus
2
Figure 9 Clotting time at different initial concentrations of factor VIII The reference value is 700 sdot 10minus10M (dashed black line) the sweeprange is [700 sdot 10minus16 700 sdot 10minus4] The blue area indicates a condition in which factor VIII concentration is less than 1 of its physiologicalconcentration corresponding to a situation of severe haemophilia the red area indicates a condition in which factor VIII concentration isbetween 1 and 30 of its physiological concentration corresponding to a situation of mild haemophilia the green area indicates a conditionin which factor VIII concentration is greater than 130 of its physiological concentration corresponding to a situation of hypercoagulability
(ie 11989658= 1 sdot 10
8Mminus1 sminus1) does not have any substantialeffect an increase of its value leads to a progressive increasein the CT This increase is due to the fact that factor IIa issequestered in the formation of a complex with factor XI andhence it is no longer available as a free component in bloodto participate in other reactions especially those reactionsof the extrinsic pathway which principally lead to the clotformation in vivo This behavior highlights that the intrinsicpathway has a secondary role in blood coagulation in vivocompared with the extrinsic pathway as also evidenced byvarious experimental observations [48]
313 Factors VIII IX and II The next set of PSA-1D wasrealized by varying the initial concentrations of factors VIIIIX and II These factors were selected since both an excess
and a deficiency of their concentrations lead to diseasesrelated to blood clotting
The PSA-1D over factor VIII (Figure 9) shows thatincreasing the initial concentration of this factor resultsin decreasing the CT suggesting the possible presence ofhypercoagulable states in these perturbed conditions As amatter of fact high levels of factor VIII cause an increasedrisk of deep vein thrombosis and pulmonary embolism[49] On the other hand individuals with less than 1of the average concentration of factor VIII show a severehaemophilia A characterized by higher CT (Figure 9) andrequire infusions of plasma containing the deficient factorotherwise frequent spontaneous bleeding would occur [50]When the concentration of this factor is between 5 and 30of the average concentration individuals still risk bleeding incase of trauma [50]
14 BioMed Research International
250
300
350
400
450
500
550
600
650
700
Clot
ting
time (
s)
Factor IX (M)
1eminus
14
1eminus
12
1eminus
10
1eminus
8
1eminus
6
1eminus
4
1eminus
2
1e0
Figure 10 Clotting time at different initial concentrations of factor IXThe reference value is 900sdot10minus8M the sweep range is [900sdot10minus14 900sdot10minus2]
0
100
200
300
400
500
600
700
Clot
ting
time (
s)
Factor II (M)
1eminus
12
1eminus
10
1eminus
8
1eminus
6
1eminus
4
1eminus
2
1e0
1e2
Figure 11 Clotting time at different initial concentrations of thrombinThe reference value is 140sdot10minus6M the sweep range is [140sdot10minus12 140sdot100]
The PSA-1D over factor IX (Figure 10) shows only aslight decrease of the CT as the initial concentration offIX increases this is in contrast to recent studies thatdemonstrated how the excess of factor IX leads to anincreased risk of deep vein thrombosis [51] This result canbe explained by considering that (i) a high concentrationof factor IX is not sufficient to bring about coagulationproblems though when the concentration of other factorsis above the average value (yet not at pathological levels)prothrombotic states can be observed (ii) in this model weconsider average values as initial concentrations of factorshowever individuals are characterized by different (balanced)combinations of procoagulant and anticoagulant factor levelsthat altogether contribute to define a unique coagulationphenotype that reflects the developmental environmentalgenetic nutritional and pharmacological influences of eachindividual [52] On the contrary the lack of factor IX causes
haemophilia B characterized by higher CT with respect tothe reference value (Figure 10)
Furthermore by comparing the PSA-1D of factors VIIIand IX (Figures 9 and 10) it is clear that haemophilia A ismore serious than haemophilia B since the CT achieved inconditions of factor VIII deficiency is higher than the CTobtained in the case of factor IX deficiency
In both PSA-1D over factors VIII and IX we observedafter the initial decrease of the CT an unexpected increaseof the CT as the factor concentration increasesThis counter-intuitive behavior arises at very high concentrations of thesefactors (with respect to the average physiological levels) andto the best of our knowledge it was never observed in vivoNonetheless it would be interesting to verify by means of adhoc laboratory experiments if the model correctly describesthe behavior of the BCC even in these conditions or on thecontrary it is not predictive in these extreme situations
BioMed Research International 15
The PSA-1D over factor II (Figure 11) shows a dramaticdecrease of the CT as the initial concentration of this factorincreases (with respect to the average physiological level)This behavior resembles the effects of hypercoagulability (orthrombophilia) a disease caused by mutation G20210A inthe prothrombin gene [53] that causes an increase of theprothrombin level (factor II) in the blood flow resulting inan excessive formation of the active form of this factor thusheightening venous thrombosis risks [54] Hypercoagulabil-ity is usually treated with warfarin therapy or with otheranticoagulants with a similar effect These drugs decrease thecapacity of coagulation factors to become active preventingthe formation of unwanted thrombi
On the other hand when the initial concentration offactor II is low we achieved the effects of prothrombindeficiency a rare autosomal recessive disease that causes atendency to severe bleeding [29 55] As shown in Figure 11 aconcentration equal to 10of the physiological value of factorII (ie 14 sdot 10minus6 M) leads to clotting effects similar to severehaemophilia A
32 PSA-2D of the BCCModel We present here the results ofthe PSA-2D on the BCCmodel where couples of parameterswere varied to analyze the possible effects arising from thecombined perturbation of their values
321 Reactions 11990327
and 11990358 Figure 12 shows the effect of
the simultaneous variation of constants 11989627
and 11989658
(overthe same sweep ranges considered in the two PSA-1DSection 31) This result remarks that reaction 119903
27 involved
in the common pathway has a stronger influence on theBCC and that there is a synergic interplay between thesetwo reactions In particular when the value of 119896
27is low and
the value of 11989658
is high the CT is higher than the valuesachieved when only a single constant is changed because inthis condition both the intrinsic and the common pathwaysare simultaneously inhibited
322 Factor VIII Factor IX and Tissue Factor In the last twoPSA-2Dwe varied the initial concentration of factor VIII andTissue Factor and the initial concentration of factor IX andTissue Factor respectively
The initial concentrations of factors VIII and IX were var-ied over four orders of magnitude using their physiologicalvalues as upper limit for the sweep ranges the concentrationof Tissue Factor was varied over four orders of magnitudetwo above and two below its reference value (see Table 2)The rationale behind this choice is to observe how the BCCmodel in conditions corresponding to different states ofhaemophilia (obtained by decreasing the concentrations offactors VIII and IX) behaves with different initial concentra-tions of the Tissue Factor which is the upstream factor of theextrinsic pathway that is the most important element of theBCC
The results of these PSA-2D show that with respect tothe condition of haemophilia B in the case of haemophiliaA the amount of Tissue Factor (below its reference value)has a negligible influence on the CT as indicated by the
presence of a plateau in Figure 13 on the contrary concerninghaemophilia B a deficiency of Tissue Factor leads to anincrease of the CT especially when factor IX is present in lowconcentrations (Figure 14)
The different results achieved in the two PSA-2D are dueto the presence in the BCC model of a direct interactionbetweenTissue Factor and factor IX bymeans of theTF-fVIIacomplex (see reactions 119903
13 119903
15in Table 1) Indeed the lack
of Tissue Factor directly affects the concentration of activefactor IX which results in a strong alteration of the CT withrespect to physiological conditions
33 CPU versus GPU Performance Comparison In orderto show the relevant speedup achieved by coagSODA wepresent here the comparison of the computational effortrequired by GPU and CPU for the simulation of the BCCmodel The performances of our GPU simulator were com-pared with those obtained using the LSODA algorithmimplemented in the software COPASI [56] executing onthe CPU the same set of simulations that were run on theGPU COPASI is single-threaded and does not exploit thephysical and logical cores of the CPU therefore it representsa good benchmark as a single-node CPU-bound simulator ofbiological systems
In all simulations executed on both GPU and CPU westored 100 samples uniformly distributed in the time intervalconsidered for each simulation that is [0 700] seconds of thedynamics of all chemical species involved in the BCCmodelThe settings for the LSODA algorithm were the followingrelative error 1 sdot 10minus7 absolute error 1 sdot 10minus13 and maximumnumber of internal steps set to 20000
331 Benchmark Details The GPU used for the simulationsis a Nvidia Tesla K20c equipped with 2496 cores organized in13 streamingmultiprocessors GPU clock 706MHz and 5GBofDDR5RAM In all tests coagSODAwas compiled and exe-cuted with version 55 of CUDA libraries Even though thisGPU has compute capability 35 and is based on the GK110Kepler architecture currently coagSODA does not exploitany of the new features (eg Dynamic Parallelism Hyper-QRemote DMA) with the exception of the new ISA encodingwhich allows threads to exploit an increased number ofregisters (255 instead of 63) reducing register spilling intoglobal memory and increasing performances [57] Moreoveras described in Section 22 the Kepler architecture offers thepossibility to reconfigure the 64KB on-chip cache balancingbetween L1 cache and shared memory Since coagSODAexploits the shared memory to reduce the latencies duringthe access to the concentration values of the BCC model weopted for the following configuration 48KB for the sharedmemory 16 KB for the L1 cache
The performance of the GPU was compared against apersonal computer equippedwith a dual-core CPU Intel Corei5 frequency 23 GHz 4GB of DDR3 RAM running theoperating systemMacOSXLion 1075The software used as acoagSODA-equivalent single-threaded CPU implementationis COPASI version 48 (build 35)
16 BioMed Research International
280
300
320
340
360
380
400
420
440
1e41e6
1e91e11 1e5
1e7
1e9
1e11
280 320 360 400 440
Clot
ting
time (
s)
k 58(M
minus1 sminus1 )
k27 (Mminus1sminus1)
Figure 12 Clotting time according to PSA-2D over reaction constants 11989627and 119896
58 The sweep ranges are [2 sdot 104 2 sdot 1010] and [1 sdot 105 2 sdot 1011]
respectively
100
200
300
400
Tissue factor (M) Factor VIII
(M)
Clot
ting
time (
s)
100
150
200
250
300
350
400
450
1eminus131eminus11
1eminus91eminus14
1eminus12
1eminus10
Figure 13 Clotting time at different initial concentrations of Tissue Factor and factor VIII The sweep ranges are [500 sdot 10minus14 500 sdot 10minus10]and [700 sdot 10minus14 500 sdot 10minus10] respectively
A direct comparison of the performances between thesetwo different architectures and implementations is not aneasy task since CPUs are optimized for single-thread exe-cution and exploit a large number of optimizations (eghigher clock frequency instruction caching pipelining outof order execution and branch prediction) whereas GPUsare optimized for graphics processing and parallel exe-cution of identical code relying only on multilevel datacaching For this reason in this paper we propose anempirical comparison of the performances based on therunning times of identical batches of simulations of the BCCmodel
332 Benchmark Results In Figure 15 we show the compar-ison of the running times required to run several batchesof simulations executed to carry out the PSA-1D over thereaction constant 119896
27 The choice of comparing the perfor-
mances of the GPU and CPU implementations by executingbatches of simulations that are related to a PSA instead ofrunning 119899 independent but identical simulations (ie allcharacterized by the same parameterization of the model)is due to the fact that these results represent a more realisticscenario in the computational analysis of biological systemswhereby it is useful to investigate large search spaces ofparameters corresponding to different perturbed conditionsof themodel [26]Moreover for the evaluation of the running
time the execution of a batch of 119899 identical deterministicsimulations would be futile The figure clearly shows thatcoagSODA always performs better than the CPU counter-part In particular while the CPU performance increaseslinearly with the number of simulations the running timesare strongly reduced on the GPU in this case the overallrunning time roughly corresponds to the time required forthe execution of the slowest simulations This is due to thefact that different parameterizations may require differenttime steps for LSODA to converge and the execution ofa block terminates as long as all the threads it containsterminate In turn the execution of a kernel terminates whenall blocks terminate for this reason a single simulationwhose dynamics requires more steps than the others mayaffect the overall running time
In Table 3 we report the running times of all batchesof simulations along with the speedup achieved on theGPU In particular these results highlight that the advantageof exploiting the GPUs for the simulation of the BCCmodel becomes more evident as the number of simula-tions increases with a 181times speedup when a PSA with10000 different parameterizations is executed Therefore theGPU accelerated analysis of the BCC model with coag-SODA represents a novel relevant computational means toinvestigate the behavior of this complex biological systemunder nonphysiological conditions and could be exploited
BioMed Research International 17
0
200
400
600
Tissue factor (M) Factor IX (M
)
Clot
ting
time (
s)
0
100
200
300
400
500
600
700
1eminus12
1eminus10
1eminus8
1eminus131eminus11
1eminus9
Figure 14 Clotting time at different initial concentrations of Tissue Factor and factor IXThe sweep ranges are [500 sdot 10minus14 500 sdot 10minus10] and[900 sdot 10
minus12 900 sdot 10
minus8] respectively
5 10 50 100 500 1000 5000 10000
Runn
ing
time (
s)
Number of simulations
CPUGPU
1e2
1e3
1e4
1e5
1e6
1e7
Figure 15 Comparison of the computational time required to execute an increasing number of simulations of the BCCmodel on CPU (IntelCore i5 23 GHz) and GPU (Nvidia Tesla K20c GPU clock 706MHz) The computational time is expressed in seconds The time valuesrelated to 1000 5000 and 10000 simulations on the CPU were estimated by regression (see also Table 3)
Table 3 CPU versus GPU performance comparison
Number ofsimulations CPU time (sec) GPU time (sec) Speedup
5 3808 1278 298times10 10604 2329 456times50 51704 8058 642times100 116528 10977 1061times500 536058 17394 3081times1000 1073582lowast 19982 5373times5000 5363507lowast 30960 1732times10000 10724975lowast 58951 1819timeslowastValues estimated by regression
to efficiently determine the response of the BCC to differenttherapeutical drugs
4 Conclusions
Thanks to their high-performance computing capabilitiesand the very low costs GPUs nowadays represent a suitable
technology for the development and the application of par-allel computational methods for in silico analysis of complexbiological systems However the implementation of efficientcomputational tools able to fully exploit the large potentialityof GPUs is still challenging since good programming skillsare required to implement GPU-based algorithmic methodsand to handle specific features as an optimal usage ofmemoryor the communication bandwidth between GPU and CPUMoreover algorithms cannot be directly ported on the GPUbecause of the limited programming capabilities allowed byGPUkernels as amatter of fact they need to be redesigned tomeet the requirements of the underlying SIMD architecture
In this work we presented coagSODA a GPU-poweredsimulator specifically developed to carry out fast parallelsimulations of the BCC model coagSODA was designed tooffer a black-box solution usable by any final user in aneasy way It relies on cupSODA [24] a numerical integratorfor ODEs that we previously implemented for the GPUarchitecture based on the LSODA algorithm and capableof automatically translating a reaction-based model into aset of coupled ODEs In addition to mass-action kinet-ics coagSODA implements specific functions to compute
18 BioMed Research International
the kinetics of Hill function based reactions such as thoseinvolved in the platelets activity of the BCC model coag-SODA exploits the massive parallel capabilities of modernGPUs and our results demonstrated that it can achieve arelevant reduction of the computational time required toexecutemany concurrent and independent simulations of theBCC dynamics The mutual independence of the simulationsallows fully exploiting the underlying SIMD architecturemoreover coagSODA benefits from an additional speedupthanks to our choice of storing the state of the systeminto the low-latency shared memory (a solution that wasalready implemented in cupSODA) Since the BCC modelis large (96 reactions 71 chemical species) a large amountof shared memory was assigned to each thread stronglyreducing the theoretical occupancy of the GPU that is theratio of active warps with respect to the maximum numberof warps supported by each streaming multiprocessor of theGPU However the results of the analysis presented in thiswork performed on the BCC model show that coagSODAachieves a relevant boost with respect to a reference CPUimplementation For instance in the case of 10000 simula-tions we achieved a noticeable 181times speedup Interestinglythe performances of coagSODA are better than COPASI evenfor small batches of simulations These results indicate thatfor biological models consisting in many reactions and manyspecies our GPU implementation of LSODA becomes moreprofitable than the CPU counterpart as the number of con-current simulations increases making it suitable especiallywhen performing demanding computational analysis suchas for example parameter sweep parameter estimation orsensitivity analysis
As a test case in this paper we presented several parame-ter sweep analyses over reaction constants and initial concen-trations of factors involved in the BCC model Other com-putational analyses on mathematical models of this pathwaywere previously presented For instance in [58] a parameterestimation of the reaction constants of different modelsinvolving the activation of factor X factor V prothrombinand the inactivation of factors was performed The aim ofthis analysis was to discriminate between different modelsto unravel the mechanisms on the basis of the BCC Similaranalyses were executed on a model describing thrombingeneration in plasma since the reaction mechanism thereaction constants and initial concentrations were unknown[59] In [34] a sensitivity analysis of a model consisting of44 species over 34 chemical species was presented reactionconstants were varied between 10 and 1000 of theirreference value to the aim of identifying the most influentialfactors of the BCC
All these analyses were performed bymeans of sequentialsimulations of the models under investigation thereforein general only small batches of simulations could be run(for instance in [34] only 836 simulations were executed toexecute the sensitivity analysis of the model) On the otherhand in this paper we presented the results of different PSAefficiently executed bymeans of coagSODA overall providinguseful information regarding unknownparameters and inter-esting insights into the functioning of the BCC A thoroughsensitivity analysis of the whole BCC model consisting in
around 5 sdot 105 simulations is currently in progress on GPUsby our group exploiting the great computational efficiency ofcoagSODA
The results of our computational analyses should benow validated by means of specifically designed laboratoryexperiments In particular we identified a plausible valuefor the constant of the reaction describing the autoactivationof factor XII (since no reference values can be found inliterature) moreover the results of the PSA over factor IXsuggest that a deficiency of this factor is not enough to causesevere bleeding disorders as haemophilia B but an alterationof other factors seems to be necessary for the occurrence ofsuch condition (eg the lack of Tissue Factor as suggestedby the PSA-2D over Tissue Factor and factor IX) Finallythe PSA over factors VIII and IX showed in a situationcharacterized by very high concentrations of these factorsa counterintuitive behavior in which the clotting time isincreased with respect to the value obtained in physiologicalconditions Indeed it would be interesting to design ad hoclaboratory experiments to verify if the BCCmodel is actuallypredictive in such extreme situations
The coagSODA software and the SBML version of theBCC model used in this work are available from the authorsupon request
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This work is partially supported by SYSBIO Centre forSystems Biology
References
[1] B B Aldridge JM BurkeDA Lauffenburger andP K SorgerldquoPhysicochemicalmodelling of cell signalling pathwaysrdquoNatureCell Biology vol 8 no 11 pp 1195ndash1203 2006
[2] I-C Chou and E O Voit ldquoRecent developments in parameterestimation and structure identification of biochemical andgenomic systemsrdquoMathematical Biosciences vol 219 no 2 pp57ndash83 2009
[3] D Besozzi P Cazzaniga D Pescini G Mauri S Colomboand EMartegani ldquoThe role of feedback control mechanisms onthe establishment of oscillatory regimes in the RascAMPPKApathway in S cerevisiaerdquo EURASIP Journal on Bioinformaticsand Systems Biology vol 2012 no 10 2012
[4] A Saltelli M Ratto T Andres et al Global Sensitivity AnalysisThe Primer Wiley-Interscience 2008
[5] A Raue C Kreutz T Maiwald et al ldquoStructural and practicalidentifiability analysis of partially observed dynamical modelsby exploiting the profile likelihoodrdquo Bioinformatics vol 25 no15 pp 1923ndash1929 2009
[6] A Drager M Kronfeld M J Ziller et al ldquoModeling metabolicnetworks in C glutamicum a comparison of rate laws incombination with various parameter optimization strategiesrdquoBMC Systems Biology vol 3 article 5 2009
BioMed Research International 19
[7] D Besozzi P Cazzaniga G Mauri D Pescini and L Van-neschi ldquoA comparison of genetic algorithms and particle swarmoptimization for parameter estimation in stochastic biochem-ical systemsrdquo in Proceedings of the 7th European ConferenceEvolutionary Computation Machine Learning and Data Miningin Bioinformatics (EvoBIO rsquo09) C Pizzuti M Ritchie and MGiacobini Eds vol 5483 of Lecture Notes in Computer Sciencepp 116ndash127 Springer 2009
[8] M S Nobile D Besozzi P Cazzaniga GMauri andD PescinildquocupSODA aGPU-basedmulti-swarmPSOmethod for param-eter estimation in stochastic biological systems exploitingdiscrete-time target seriesrdquo in Proceedings of the 10th EuropeanConference Evolutionary Computation Machine Learning andData Mining in Bioinformatics (EvoBIO rsquo12) M Giacobini LVanneschi and W Bush Eds vol 7246 of Lecture Notes inComputer Science pp 74ndash85 Springer
[9] M S Nobile D Besozzi P Cazzaniga D Pescini and GMaurildquoReverse engineering of kinetic reaction networks by meansof Cartesian Genetic Programming and Particle Swarm Opti-mizationrdquo in Proceedings of the IEEE Congress on EvolutionaryComputation (CEC rsquo13) pp 1594ndash1601 June 2013
[10] M S Nobile ldquoEvolutionary inference of biochemical inter-action networks accelerated on Graphics Processing Unitsrdquoin Proceedings of the 11th International Conference on HighPerformance Computingamp Simulation 2013 (HPCS rsquo13) pp 668ndash670 Helsinki Finland 2013
[11] K Kentzoglanakis and M Poole ldquoA swarm intelligence frame-work for reconstructing gene networks searching for bio-logically plausible architecturesrdquo IEEEACM Transactions onComputational Biology and Bioinformatics vol 9 no 2 pp 358ndash371 2012
[12] J R Koza W Mydlowec G Lanza Y Jessen and M A KeaneldquoAutomatic computational discovery of chemical reaction net-works using genetic programmingrdquo inComputationalDiscoveryof Scientific Knowledge S Dzeroski and L Todorovski Edsvol 4660 of Lecture Notes in Computer Science pp 205ndash227Springer 2007
[13] S Ando E Sakamoto and H Iba ldquoEvolutionary modeling andinference of gene networkrdquo Information Sciences vol 145 no3-4 pp 237ndash259 2002
[14] L Dematte and D Prandi ldquoGPU computing for systemsbiologyrdquo Briefings in Bioinformatics vol 11 no 3 Article IDbbq006 pp 323ndash333 2010
[15] J L Payne N A Sinnott-Armstrong and J HMoore ldquoExploit-ing graphics processing units for computational biology andbioinformaticsrdquo Interdisciplinary Sciences Computational LifeSciences vol 2 no 3 pp 213ndash220 2010
[16] M J Harvey and G De Fabritiis ldquoA survey of computationalmolecular science using graphics processing unitsrdquoWiley Inter-disciplinary Reviews Computational Molecular Science vol 2no 5 pp 734ndash742 2012
[17] Y Zhou J Liepe X Sheng M P H Stumpf and C BarnesldquoGPU accelerated biochemical network simulationrdquo Bioinfor-matics vol 27 no 6 Article ID btr015 pp 874ndash876 2011
[18] M Vigelius A Lane and B Meyer ldquoAccelerating reaction-diffusion simulations with general-purpose graphics processingunitsrdquo Bioinformatics vol 27 no 2 Article ID btq622 pp 288ndash290 2011
[19] D J Wilkinson ldquoStochastic modelling for quantitative descrip-tion of heterogeneous biological systemsrdquo Nature ReviewsGenetics vol 10 no 2 pp 122ndash133 2009
[20] J C Butcher ldquoTrees and numerical methods for ordinarydifferential equationsrdquo Numerical Algorithms vol 53 no 2-3pp 153ndash170 2010
[21] D T Gillespie ldquoStochastic simulation of chemical kineticsrdquoAnnual Review of Physical Chemistry vol 58 pp 35ndash55 2007
[22] L Petzold ldquoAutomatic selection of methods for solving stiffand nonstiff systems of ordinary differential equationsrdquo SIAMJournal of Scientific and Statistical Computing vol 4 no 1 pp136ndash148 1983
[23] Y Cao D T Gillespie and L R Petzold ldquoEfficient step sizeselection for the tau-leaping simulation methodrdquo The Journalof Chemical Physics vol 124 no 4 2006
[24] M S Nobile D Besozzi P Cazzaniga GMauri andD PescinildquocupSODA a CUDA-powered simulator of mass-action kinet-icsrdquo in Proceedings of the 12th International Conference onParallel Computing Technologies (PaCT rsquo13) V Malyshkin Edvol 7979 of Lecture Notes in Computer Science pp 344ndash357
[25] M S Nobile P Cazzaniga D Besozzi and G Mauri ldquoGPU-accelerated simulations of mass-action kinetics models withcupSODArdquoThe Journal of Supercomputing 2014
[26] M S Nobile P Cazzaniga D Besozzi D Pescini and GMauri ldquocuTauLeaping a GPU-powered tau-leaping stochasticsimulator for massive parallel analyses of biological systemsrdquoPLoS ONE vol 9 no 3 2014
[27] K Rogers Blood Physiology and CirculationTheHuman BodyBritannica Educational Publishing 2010
[28] A D Michelson Platelets Elsevier 2nd edition 2007[29] R W Colman J Hirsh V J Marder A W Clowes and J
N George Hemostasis and Thrombosis Basic Principles andClinical Practice Lippincott Williams ampWilkins 2006
[30] M S Chatterjee W S Denney H Jing and S L DiamondldquoSystems biology of coagulation initiation Kinetics of thrombingeneration in resting and activated human bloodrdquo PLoS Com-putational Biology vol 6 no 9 Article ID e1000950 2010
[31] M Bellini D Besozzi P Cazzaniga GMauri andM S NobileldquoSimulation and analysis of the blood coagulation cascadeaccelerated on GPUrdquo in Proceedings of the 22nd EuromicroInternational Conference on Parallel Distributed and Network-Based Processing (PDP) pp 590ndash593 IEEE Computer SocietyConference Publishing Services 2014
[32] T Wajima G K Isbister and S B Duffull ldquoA comprehensivemodel for the humoral coagulation network in humansrdquo Clini-cal Pharmacology and Therapeutics vol 86 no 3 pp 290ndash2982009
[33] K Lo W S Denney and S L Diamond ldquoStochastic modelingof blood coagulation initiationrdquo Pathophysiology of Haemostasisand Thrombosis vol 34 no 2-3 pp 80ndash90 2006
[34] C M Danforth T Orfeo K G Mann K E Brummel-Ziedins and S J Everse ldquoThe impact of uncertainty in a bloodcoagulation modelrdquo Mathematical Medicine and Biology vol26 no 4 Article ID dqp011 pp 323ndash336 2009
[35] S D Lewis P P Shields and J A Shafer ldquoCharacterizationof the kinetic pathway for liberation of fibrinopeptides duringassembly of fibrinrdquoThe Journal of Biological Chemistry vol 260no 18 pp 10192ndash10199 1985
[36] M F Hockin K C Jones S J Everse and K G Mann ldquoAmodel for the stoichiometric regulation of blood coagulationrdquoThe Journal of Biological Chemistry vol 277 no 21 pp 18322ndash18333 2002
[37] K Brummel-Ziedins C YVossen F R Rosendaal KUmezakiand K G Mann ldquoThe plasma hemostatic proteome thrombin
20 BioMed Research International
generation in healthy individualsrdquo Journal of Thrombosis andHaemostasis vol 3 no 7 pp 1472ndash1481 2005
[38] K E Brummel-Ziedins C Y Vossen S Butenas K G Mannand F R Rosendaal ldquoThrombin generation profiles in deepvenous thrombosisrdquo Journal of Thrombosis and Haemostasisvol 3 no 11 pp 2497ndash2505 2005
[39] S D Bungay P A Gentry and R D Gentry ldquoA mathematicalmodel of lipid-mediated thrombin generationrdquo MathematicalMedicine and Biology vol 20 no 1 pp 105ndash129 2003
[40] S Butenas T Orfeo M T Gissel K E Brummel and K GMann ldquoThe significance of circulating factor IXa in bloodrdquoThe Journal of Biological Chemistry vol 279 no 22 pp 22875ndash22882 2004
[41] D Voet and J G Voet Biochemistry Wiley 2011[42] A C Guyton and J E Hall Textbook of Medical Physiology
Saunders 2010[43] M F M James and A M Roche ldquoDose-response relationship
between plasma ionized calcium concentration and thrombe-lastographyrdquo Journal of Cardiothoracic and Vascular Anesthesiavol 18 no 5 pp 581ndash586 2004
[44] D L Nelson and M M Cox Lehninger Principles of Biochem-istry W H Freeman Company 2004
[45] R M Farber ldquoTopical perspective on massive threading andparallelismrdquo Journal of Molecular Graphics and Modelling vol30 pp 82ndash89 2011
[46] OWolkenhauerMUllahWKolch andK-HCho ldquoModelingand simulation of intracellular dynamics choosing an appropri-ate frameworkrdquo IEEETransactions onNanobioscience vol 3 no3 pp 200ndash207 2004
[47] B H Estridge A P Reynolds and N J Walters Basic MedicalLaboratory Techniques Cengage Learning 4th edition 2000
[48] T Renne A H Schmaier K F Nickel M Blomback and CMaas ldquoIn vivo roles of factor XIIrdquo Blood vol 120 no 22 pp4296ndash4303 2012
[49] P V Jenkins O Rawley O P Smith and J S ODonnellldquoElevated factor VIII levels and risk of venous thrombosisrdquoBritish Journal of Haematology vol 157 no 6 pp 653ndash663 2012
[50] C E Walsh and K M Batt ldquoHemophilia clinical gene therapybrief reviewrdquo Translational Research vol 161 no 4 pp 307ndash3122013
[51] A Van Hylckama Vlieg I K Van Der Linden R M Bertinaand F R Rosendaal ldquoHigh levels of factor IX increase the risk ofvenous thrombosisrdquo Blood vol 95 no 12 pp 3678ndash3682 2000
[52] K Brummel-Ziedins ldquoModels for thrombin generation and riskof diseaserdquo Journal ofThrombosis and Haemostasis vol 11 no 1pp 212ndash223 2013
[53] U Seligsohn and A Lubetsky ldquoGenetic susceptibility to venousthrombosisrdquoTheNew England Journal of Medicine vol 344 no16 pp 1222ndash1231 2001
[54] I Martinelli P Bucciarelli and P M Mannucci ldquoThromboticrisk factors basic pathophysiologyrdquo Critical Care Medicine vol38 no 2 pp S3ndashS9 2010
[55] K Brummel-Ziedins A Undas T Orfeo et al ldquoThrombingeneration in acute coronary syndrome and stable coronaryartery disease dependence on plasma factor compositionrdquoJournal ofThrombosis andHaemostasis vol 6 no 1 pp 104ndash1102008
[56] S Hoops R Gauges C Lee et al ldquoCOPASImdasha complexpathway simulatorrdquo Bioinformatics vol 22 no 24 pp 3067ndash3074 2006
[57] Nvidia NVIDIArsquos Next Generation CUDA Compute Architec-ture Kepler GK110 2012
[58] W Stortelder W Hemker and H Hemker ldquoMathematicalmodelling in blood coagulation simulation and parameterestimationrdquoModelling Analysis and Simulation vol 20 pp 1ndash111997
[59] S M C Orfao G Jank and K Mottaghy ldquoA new mathematicalapproach to modelling thrombin generationrdquo InternationalJournal of Artificial Organs vol 29 no 7 pp 703ndash708 2006
[60] Nvidia Nvidia CUDA C Programming Guide v50 2012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Anatomy Research International
PeptidesInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
International Journal of
Volume 2014
Zoology
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Molecular Biology International
GenomicsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
BioinformaticsAdvances in
Marine BiologyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Signal TransductionJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
BioMed Research International
Evolutionary BiologyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Biochemistry Research International
ArchaeaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Genetics Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in
Virolog y
Hindawi Publishing Corporationhttpwwwhindawicom
Nucleic AcidsJournal of
Volume 2014
Stem CellsInternational
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Enzyme Research
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Microbiology
6 BioMed Research International
fXIITF-fVII
TFfVIIa
fVII
TF-fVIIa
Fbg
FPA
Fbn1
FPB
Fbn2
Fbn1
(Fbn1)2
(Fbn2)2 FPB
fXa
fIIa
TF-fVIIa-fXa
fIX
fIXa
fX
fmIIa
fII
PKal
KalifXI
fXIa
fIX
fIXa
fVIII
fVIIIa
fIXa-fVIIIa
fVIIIa1 -L
fVIIIa2
fVIIIa2
fIXa
fV
fVa
fXa-fVa
fXa-TFPI
TFPI
TF-fVIIa-fXa-TFPI
fXa-AT
AT
AT
fXIa-AT
C1inh
fXIIa-C1inh
A1AT
A2AP
fXIa-A2APfXIa-A1AT
TF-fVIIa
fXIIa Kal
fVIIIa1 -L
Figure 1 Graphical representation of the blood coagulation cascade model considered in this work Legend Blue box coagulation factor redbox inhibitor and related complexes Black arrow complex formation green arrow catalytic activation violet arrow activation red arrowinhibition The reaction is reversible if the arrow tail consists in a dot
BioMed Research International 7
(iv) the activation of prekallikrein by factor XIIa(reactions 119903
48 11990349 11990350) and by kallikrein (reac-
tion 11990354)
(v) the activation of factor XI by factor XIIa (reac-tions 119903
61 11990362 11990363) and by factor XIa (reaction
11990364)
(vi) the activation of factor IX by factor XIa (reac-tions 119903
69 11990370 11990371)
(vii) the dissociation of free factor VIIIa (reactions11990323and 11990324)
(viii) the dissociation of factor VIIIa in complex withother factors (reactions 119903
25and 11990326)
(3) The third module corresponds to the common path-way which consists in
(i) the activation of factor II by factor Xa (modeledby reaction 119903
16) and by the complex between
factor Xa and factor Va through the formationof the intermediate meizothrombin (reactions11990330 119903
33)
(ii) the activation of factor VII by factor Xa (reac-tion 1199036) and by factor IIa (reaction 119903
7)
(iii) the activation of factor VIII by factor Xa (reac-tions 119903
75 11990376 11990377) and by factor IIa (reaction 119903
17)
(iv) the formation of a complex between factor Xaand factor Va (reactions 119903
28and 11990329)
(v) the activation of fibrinogen by factor IIa (reac-tions 119903
84 119903
96)
(vi) the activation of factor V by factor IIa (reaction11990327)
(vii) the activation of factor XI by factor IIa (reac-tions 119903
58 11990359 11990360)
(4) The fourth module describes the inhibition of coagu-lation carried out through the main inhibitors of theBCC (antithrombin Tissue Factor Pathway InhibitorC1-inihibitor 1205721-antitrypsin and 1205722-antiplasmin)These are tight binding inhibitors belonging to theserpin superfamily which form irreversible com-plexes This module consists in
(i) the inhibition of factor Xa by Tissue FactorPathway Inhibitor (modeled by reactions 119903
34and
11990335)
(ii) the inhibition of the complex between TissueFactor factor VIIa and factor Xa by TissueFactor Pathway Inhibitor (reactions 119903
36 11990337 11990338)
(iii) the inhibition of factor IIa by antithrombin(reactions 119903
40and 11990342)
(iv) the inhibition of factor Xa by antithrombin(reaction 119903
39)
(v) the inhibition of factor IXa by antithrombin(reaction 119903
41)
(vi) the inhibition of factor XIa by antithrombin(reaction 119903
65) and by C1-inhibitor (reaction 119903
66)
Table 2 Initial concentrations of molecular species in the bloodcoagulation cascade model (values taken from [30])
Species Symbol Concentration (M)1205721-Antitrypsin A1AT 450 sdot 10
minus5
1205722-Antiplasmin A2AP 100 sdot 10minus6
Antithrombin III ATIII 340 sdot 10minus6
C1-inhibitor C1inh 250 sdot 10minus6
Fibrinogen Fbg 900 sdot 10minus6
Factor II fII 140 sdot 10minus6
Factor V fV 200 sdot 10minus8
Factor VII fVII 100 sdot 10minus8
Active factor VII fVIIa 100 sdot 10minus10
Factor VIII fVIII 700 sdot 10minus10
Factor IX fIX 900 sdot 10minus8
Factor X fX 16 sdot 10minus7
Factor XI fXI 310 sdot 10minus8
Factor XII fXII 340 sdot 10minus7
Prekallikrein Pkal 450 sdot 10minus7
Tissue Factor TF 500 sdot 10minus12
Tissue Factor Pathway Inhibitor TFPI 250 sdot 10minus9
(vii) the inhibition of factor XIIa by C1-inhibitor(reaction 119903
56) by antithrombin (reaction 119903
57)
by 1205721-antitrypsin (reaction 11990367) and by 1205722-
antiplasmin (reaction 11990368)
(viii) the inhibition of the complex between TissueFactor and factor VIIa by antithrombin (reac-tion 11990343)
(ix) the inhibition of kallikrein (reaction 11990355)
The values of the initial concentrations of the molecularspecies occurring in the BCC model are given in Table 2According to [30] the concentrations of complexes and activefactors were set to 0 except for the active factor VII which isphysiologically present in the blood circulation even in theabsence of damage in a concentration that is approximatelyequal to 1 of the corresponding inactive factor [42]
The system of ordinary differential equations (ODEs)needed to carry out the simulations and the parametersweep analysis presented in Section 3 was derived fromthe reactions given in Table 1 according to the mass-actionlaw with the exception of 14 reactions belonging to the set119878120576= 11990319 11990321 11990323 11990329 11990331 11990335 11990346 11990349 11990352 11990362 11990370 11990373 11990376 11990382
The mass-action law is the fundamental empirical law thatgoverns biochemical reaction rates which states that in adilute solution the rate of an elementary reaction is propor-tional to the product of the concentration of its reactantsraised to the power of the corresponding stoichiometriccoefficient [44] The reactions in 119878
120576are instead influenced by
a specifically defined variable 120576 depending on aHill functionfit against experimental data which quantify the plateletactivation status which is used to model the physiologicallevels of thrombin concentration as a function of plateletactivation as thoroughly described in [30] More preciselythe value of 120576 influences the formation of some complexes
8 BioMed Research International
0 0 100 200 300 400 500 600 700
Fact
or II
a con
cent
ratio
n (M
)
Time (s)
1eminus7
2eminus7
3eminus7
4eminus7
5eminus7
6eminus7
7eminus7
8eminus7
9eminus7
Figure 2 Dynamics of thrombin (factor IIa) in physiological condition
occurring on the plateletsrsquo surface by modifying the activityof reactions of the form
119860 + 119861
1198961
999445999468
1198962
119860119861 (1)
Namely 120576 intervenes with the dissociation constant of thesereactions so that the corresponding standard ODEs arechanged to yield new equations of the form
119889 [119860119861]
119889119905
= 1198961sdot [119860] sdot [119861] minus
1198962sdot [119860] sdot [119861]
120576
(2)
The value of 120576 in (2) depends on the following HillfunctionH which quantifies the state of platelets activationaccording to the thrombin concentration (here denoted as[fIIa]) that is the factor catalyzing the formation of the fibrinclot
H ([fIIalowast (119905)]) =[fIIalowast (119905)]16123
[fIIalowast (119905)]16123 + (24279 sdot 10minus9)16123
(3)
where [fIIalowast(119905)] = max1199051015840isin[0119905][fIIa(1199051015840)] for a chosen time
interval [0 119905] of simulationThe value [fIIalowast(119905)] represents the maximum transient
thrombin concentration and is needed to simulate the factthat in physiological conditions the thrombin concentrationstarts decreasing after rising to a peak (Figure 2) So doing[fIIalowast(119905)] never decreases once it reaches its maximummagnitude [fIIalowast(119905)] is equivalent to [fIIa(119905)] until the con-centration of factor IIa reaches the peak while thereafter itremains constant at that value which is the maximum in theconsidered time interval [0 119905] FunctionH allows simulatingthe physiological condition whereby platelets remain activealso when the thrombin concentration decreases
For a given concentration of factor IIa the maximumplatelets activation state 120576max is defined as
120576max = 120576max0
+ (1 minus 120576max0
) sdotH ([fIIalowast (119905)]) (4)
where 120576max0
defines the basal activation state of the plateletsat simulation time 119905 = 0 The value 120576max
0
is initially set to 001assuming a basal 1 binding strength of coagulation factorsto the resting plateletsrsquo surface When the full activationof platelets is reached 120576max is equal to 1 and the complexdissociation constants are minimized (see [30] for moredetails)
22 CUDA Architecture and the coagSODA Simulator Intro-duced by Nvidia in 2006 Compute Unified Device Architec-ture (CUDA) is a parallel computing platform and program-ming model that provides programmers with a frameworkto exploit GPUs in general-purpose computational tasks(GPGPU computing) GPGPU computing is a low-cost andenergy-wise alternative to the traditional high-performancecomputing infrastructures (eg clusters of machines) whichgives access to the tera-scale computing on common work-stations of mid-range price However due to the innovativearchitecture and the intrinsic limitations of GPUs a directporting of sequential code on the GPU is most of the timesunfeasible therefore making it challenging to fully exploit itscomputational power and massive parallelism [45]
CUDA combines the single instruction multiple data(SIMD) architecture with multithreading which automat-ically handles the conditional divergence between threadsThe drawback of such flexibility is that any divergence ofthe execution flow between threads causes a serialization ofthe execution which affects the overall performances UnderCUDArsquos naming conventions the programmer implementsthe kernel that is a CC++ function which is loaded fromthe host (the CPU) to the devices (one or more GPUs)and replicated in many copies named threads Threads canbe organized in three-dimensional structures named blockswhich in turn are contained in three-dimensional grids(a schematic description is given in Figure 3 left side)Whenever the host runs a kernel the GPU creates thecorresponding grid and automatically schedules each blockon one free streaming multiprocessor available on the GPUallowing a transparent scaling of performances on different
BioMed Research International 9
Host
Kernel 1
Kernel 2
Device
Grid 1
Grid 2
Block(0 0)
Block(1 0)
Block(2 0)
Block(0 1)
Block(1 1)
Block(2 1)
Block (1 1)
Thread(0 0)
Thread(1 0)
Thread(2 0)
Thread(3 0)
Thread(4 0)
Thread(0 1)
Thread(1 1)
Thread(2 1)
Thread(3 1)
Thread(4 1)
Thread(0 2)
Thread(1 2)
Thread(2 2)
Thread(3 2)
Thread(4 2)
Global memory
Texture memory
Constant memory
Shared memory Shared memory
Block (0 0) Block (1 0)
Localmemory
Localmemory
Localmemory
Localmemory
Thread(0 0)
Thread(1 0)
Registers Registers Registers Registers
Thread(0 0)
Thread(1 0)
Grid
Figure 3 Schematic description of CUDArsquos architecture in terms of threads and memory hierarchy Left Side Threads organization a singlekernel is launched from the host (the CPU) and is executed in multiple threads on the device (the GPU) Threads can be organized in three-dimensional structures named blocks which can be in turn organized in three-dimensional grids The dimensions of blocks and grids areexplicitly defined by the programmer Right Side Memory hierarchy threads can access data from many different memories with differentscopes Registers and local memories are private for each thread Shared memory lets threads belonging to the same block communicate andhas low access latency All threads can access the global memory which suffers high latencies but it is cached since the introduction of theFermi architecture Texture and constant memory can be read from any thread and are equipped with a cache as well in this work we exploitthe constant memory Figures are taken from Nvidiarsquos CUDA programming guide [60]
devices Threads within a block are executed in groups of 32threads named warps
The GPU is equipped with different kinds of memoryIn this work we exploit the global memory (accessible fromall threads) the shared memory (accessible from threadsof the same block) the local memory (registers and arraysaccessible from owner thread) and the constant memory(cached and not modifiable) A schematic representation ofthis memory hierarchy is shown in Figure 3 right side Toachieve the best performances the sharedmemory should beexploited as much as possible however it is very limited (ie49152 bytes for each multiprocessor since the introductionof the Fermi architecture) and it introduces restrictions onthe blocksrsquo size On the other hand the global memory isvery large (thousands ofMBs) but suffers from high latenciesA solution to this problem was implemented on the Fermiarchitecture where the global memory is equipped with aL2 cache This architecture also introduced the possibility tobalance 64KB of fast on-chip memory between the sharedmemory and L1 cache using two possible configurations48KB for the shared memory and 16KB for L1 cache or16 KB for the shared memory and 48 for L1 cacheThe Keplerarchitecture used in this paper allows a third and perfectly
balanced configuration where shared memory and L1 cacheobtain the same amount of memory (32KB) See Figure 4 fora schematization of the memory architecture
The systematic analysis of models of biological systemsoften consists in the execution of large batches of simulationsOne of the standard analyses that can be executed on suchkind of models regards an intensive search within the param-eters space which requires large numbers of independentsimulations In order to reduce the computational burdenwe previously implemented on the CUDA architecture oneof the most efficient numerical integration algorithms forODEs LSODA that is able to automatically recognize stiffand nonstiff systems and to dynamically select between themost appropriate integration procedure (ie Adams methodin the absence of stiffness and the Backward DifferentiationFormulae otherwise) [22] This GPU-powered tool is calledcupSODA [24 25] and exploits CUDArsquos massive parallelismto execute different and independent simulations in eachthread thus reducing the computational time required by astandard CPU counterpart of LSODA
Besides being very efficient for the simulation of manyindependent simulations cupSODA is also user-friendlyLSODA was originally designed to solve ODEs systems
10 BioMed Research International
Thread
Shared memory L1 cache Read-onlydata cache
L2 cache
DRAM
Figure 4 Schematic description of memory hierarchies in Fermi and Kepler architectures GPUs relying on these architectures are equippedwith a two-level data cache and a read-only data cache Shared memory and L1 cache share the same on-chip 64KB memory banks theamount of memory can be reconfigured by the user according to the specific needs of the application Figure taken from Nvidiarsquos KeplerGK110 whitepaper [57]
written in the canonical form but the user is supposed tospecify the system of ODEs by implementing a custom Cfunction that is passed to the algorithmmoreover in order tospeed up the computation when dealing with stiff systems inLSODA the Jacobianmatrix associatedwith theODEs systemmust be implemented as a custom C function as well On thecontrary cupSODA was conceived as a black-box simulatorthat can be easily used without any programming skillscupSODA consists in a tool to automatically convert a genericmechanistic reaction-based model of a biological system intothe corresponding set of ODEs to comply with the mass-action kinetics [46] and to directly encode the obtainedsystem along with the corresponding Jacobian matrix as Carrays
This fully automaticmethodology can be exploited for thesimulation of models whose chemical kinetics is based on themass-action law The BCC model described in Section 21though includes a set 119878
120576of reactions that do not follow
the mass-action kinetics [30] the platelets activity is notexplicitly modeled by biochemical reactions but it is realizedby modulating the rate of the dissociation of the complexesformed on a plateletrsquos surface by means of the variable120576 which is calculated with a special equation during theintegration steps For this reason cupSODA was used asa starting point for the development of a new tool ableto compute at run-time the specific kinetics of this set ofreactions
This new CUDA-powered simulation tool named coag-SODA is specifically tailored for the simulation of the BCC
model developed in [30] In particular coagSODA realizesthe run-time calculation of the 120576 value required to correctlysimulate the activity of platelets which is determined accord-ing to the equations described in Section 21 The plateletsactivation state 120576 is calculated at each time instant by solvingthe following differential equation
119889120576
119889119905
= 119896 (120576max minus 120576) (5)
where the constant 119896 is inversely proportional to the timescale of platelets activation and is set to 0005 This isconsistent with the fact that platelets do not instantly achievetheir maximum attainable activation state (120576max) but theyreach it on a physiologically relevant timescale [30]
Dealing with the 14 reactions in the set 119878120576that are
influenced by the Hill functionH (see Section 21) the valueof [fIIalowast(119905)]must be stored on the GPU because during eachintegration step coagSODA recalculates (4) which exploitsthe value [fIIalowast(119905)] to determine a new 120576 value Since CUDArsquosarchitecture does not offer static variables the informationfor each thread has to be memorized in the global memoryThe accesses to the global memory and the computationalcosts due to these additional calculations slow down theintegration process with respect to the integration of ODEsperformed according to strictly mass-action based kinetics(as in the cupSODA simulator) Nevertheless in Section 33we show that our parallel implementation largely outper-forms a sequential counterpart of the LSODA algorithm
BioMed Research International 11
0 100 200 300 400 500 600 700
Fibr
inog
en co
ncen
trat
ion
(M)
Time (s)
30 of fibrinogen
0
1eminus6
2eminus6
3eminus6
4eminus6
5eminus6
6eminus6
7eminus6
8eminus6
9eminus6
Figure 5 Dynamics of fibrinogen in physiological condition The clotting time is defined as the time necessary to convert the 70 of thefibrinogen into fibrin
As in the case of cupSODA [24 25] coagSODA exploitsthe shared memory to improve performances by storing thecurrent state and time of the simulations in these low-latencymemory banks Despite the improvement of performancesensured by this solution it strongly affects the occupancyof GPUrsquos multiprocessors and therefore it represents thelimiting factor for the number of blocks that can be executedsimultaneously as a matter of fact coagSODA is limited to2 blocks per streaming multiprocessor on GPUs based onthe Fermi architecture with a reduced exploitation of theGPU with respect to the theoretical 8 blocks allowed by thisarchitecture
3 Results and Discussion
In this section we discuss the results of the parameter sweepanalysis (PSA) carried out on the BCC model to the aimof investigating either the prolongation or the reduction ofthe clotting time in response to perturbed values of somereaction constants and of the initial concentration of somemolecular species chosen according to their meaning withinthe whole pathway PSA was performed by generating aset of different initial conditions corresponding to differentparameterizations of the model and then automaticallyexecuting the deterministic simulations with coagSODATheuse ofGPU technology is fundamental in this type of analysisespecially for large biological systems as the BCC is becauseit drastically reduces the computational time
The sweep analysis for single parameters (PSA-1D) wasperformed considering a logarithmic sampling of numericalvalues of each parameter under investigation (reaction con-stant or initial molecular concentration) within a specifiedrange with respect to its physiological reference value Thesweep analysis over pairs of parameters (PSA-2D) was per-formed by simultaneously varying the values of two param-eters within a specified range considering a logarithmicsampling on the resulting lattice The logarithmic samplingallows uniformly spanning different orders of magnitude ofthe parameters value using a reduced and fine-grained set
of samples therefore efficiently analyzing the response of thesystem in a broad range of conditions
To determine the response of the BCC to perturbed con-ditions we chose the clotting time (CT) as output of the PSAThe CT is defined as the time necessary to convert the 70of the fibrinogen into fibrin (see Figure 5) conventionallyassumed to correspond to the time required to form the clot[47] and it is generally used in laboratory tests formonitoringthe therapywith anticoagulant drugs According to themodeldefined in [30] the reference value of CT is around 300seconds in physiological conditions We investigate herethe response of the BCC by evaluating the CT in variousconditions corresponding to different values of the reactionconstants varying over six orders of magnitude with respectto their physiological values (ie three below and three abovethe reference values if not otherwise specified) as well asto different values of the initial molecular concentrationsand varying over twelve orders of magnitude with respect totheir physiological values (ie six below and six above thereference values if not otherwise specified)
The total number of parallel simulations executed tocarry out these analyses was 100 for PSA-1D over reactionconstants 200 for PSA-1D over initial concentrations and1600 for PSA-2D
Finally we present the comparison of the performanceof the CPU and GPU to run an increasing number ofsimulations of the BCC model to prove the efficiency ofcoagSODA
31 PSA-1D of the BCC Model
311 Reaction 11990344 Thefirst PSA was performed to determine
the value of the kinetic constant for the autoactivation offactor XII (reaction 119903
44in Table 1) which corresponds to an
upstream process in the intrinsic pathway This analysis wasmotivated by two considerations Firstly by using the fullparameterization given in [30] the action of the intrinsicpathway turns out to be fundamental for the BCC in vivowhich is in contrast to experimental observations which
12 BioMed Research International
324
326
328
330
332
334
336
338
340
Clot
ting
time (
s)
1eminus
10
1eminus
9
1eminus
8
1eminus
7
1eminus
6
1eminus
5
1eminus
4
1eminus
3
k44 (sminus1)
Figure 6 Clotting time according to PSA-1D over constant 11989644of reaction 119903
44 XII rarr XIIaThe reference value of 119896
44used in [30] is 5sdot10minus4sminus1
the sweep range is [5 sdot 10minus10 5 sdot 10minus4]
280
300
320
340
360
380
400
1e4 1e5 1e6 1e7 1e8 1e9 1e10 1e11
Clot
ting
time (
s)
k27 (Mminus1sminus1)
Figure 7 Clotting time according to PSA-1D over constant 11989627of reaction 119903
27 IIa + V rarr IIa + Va The reference value is 2 sdot 107Mminus1 sminus1 the
sweep range is [2 sdot 104 2 sdot 1010]
indicate that the extrinsic pathway is the main responsibleof clot formation Secondly in [30] all constants values havea reference to experimental measurements except for theconstant of this reaction (which corresponds to entry 29 inTable 1 in [30])
Figure 6 shows the results of this PSA-1D where 11989644was
varied over six orders of magnitude considering the valuegiven in [30] as the upper limit of the sweep interval We canobserve that by decreasing the value of 119896
44with respect to
the value considered in [30] the CT increases with respect toits reference value however for values of this constant lowerthan 100 sdot 10minus6 sminus1 the CT remains unaltered and this canbe intuitively explained by the fact that in this situation thefibrinogen is mainly activated by the extrinsic pathway
Consequently we assigned the value 700 sdot 10minus6 sminus1 to11989644 achieving a CT that is comparable to the experimental
observations of the BCC in vivo This new value was used inall PSA discussed in what follows
312 Reactions 11990327
and 11990358 In the next PSA we investigated
the effect of the perturbation of the kinetics of two pivotal
reactions of the BCC model Reaction 11990327 which describes
the catalytic activation of factor V by factor IIa was chosenbecause it represents the main positive feedback within thecommon pathway reaction 119903
58 which is involved in the
activation of factor XI by binding to factor IIa was chosenbecause it represents the main positive feedback in theintrinsic pathway (Figure 1) Moreover preliminary PSA overall reaction constants of the BCC model given in Table 1evidenced that these two reactions are among the mostrelevant steps of the coagulation network
The PSA-1D over 11989627(Figure 7) shows that the CT is very
sensitive to the perturbation of the rate of this reaction whenthe reference value of its constant (ie 119896
27= 2 sdot 10
7Mminus1 sminus1)is either increased or decreased in particular when 119896
27is
very low a plateau in CT is reached since the strength ofthe positive feedback exerted by factor IIa is largely reduceda condition where the contribution of the amplification ofthe hemostatic stimulus (due by the common pathway) tothe formation of the clot is basically not effective On theother side the PSA-1D over 119896
58(Figure 8) shows that while
a decrease of the constant with respect to its reference value
BioMed Research International 13
310
320
330
340
350
360
370
1e5 1e6 1e7 1e8 1e9 1e10 1e11
Clot
ting
time (
s)
k58 (Mminus1sminus1)
Figure 8 Clotting time according to PSA-1D over constant 11989658of reaction 119903
58 IIa + XI rarr IIa-XI The reference value is 1 sdot 108Mminus1 sminus1 the
sweep range is [1 sdot 105 2 sdot 1011]
220
240
260
280
300
320
340
360
380
400
Clot
ting
time (
s)
Factor VIII (M)
Seve
re h
aem
ophi
lia
Mild
hae
mop
hilia
Prot
hrom
botic
stat
e
1eminus
16
1eminus
14
1eminus
12
1eminus
10
1eminus
8
1eminus
6
1eminus
4
1eminus
2
Figure 9 Clotting time at different initial concentrations of factor VIII The reference value is 700 sdot 10minus10M (dashed black line) the sweeprange is [700 sdot 10minus16 700 sdot 10minus4] The blue area indicates a condition in which factor VIII concentration is less than 1 of its physiologicalconcentration corresponding to a situation of severe haemophilia the red area indicates a condition in which factor VIII concentration isbetween 1 and 30 of its physiological concentration corresponding to a situation of mild haemophilia the green area indicates a conditionin which factor VIII concentration is greater than 130 of its physiological concentration corresponding to a situation of hypercoagulability
(ie 11989658= 1 sdot 10
8Mminus1 sminus1) does not have any substantialeffect an increase of its value leads to a progressive increasein the CT This increase is due to the fact that factor IIa issequestered in the formation of a complex with factor XI andhence it is no longer available as a free component in bloodto participate in other reactions especially those reactionsof the extrinsic pathway which principally lead to the clotformation in vivo This behavior highlights that the intrinsicpathway has a secondary role in blood coagulation in vivocompared with the extrinsic pathway as also evidenced byvarious experimental observations [48]
313 Factors VIII IX and II The next set of PSA-1D wasrealized by varying the initial concentrations of factors VIIIIX and II These factors were selected since both an excess
and a deficiency of their concentrations lead to diseasesrelated to blood clotting
The PSA-1D over factor VIII (Figure 9) shows thatincreasing the initial concentration of this factor resultsin decreasing the CT suggesting the possible presence ofhypercoagulable states in these perturbed conditions As amatter of fact high levels of factor VIII cause an increasedrisk of deep vein thrombosis and pulmonary embolism[49] On the other hand individuals with less than 1of the average concentration of factor VIII show a severehaemophilia A characterized by higher CT (Figure 9) andrequire infusions of plasma containing the deficient factorotherwise frequent spontaneous bleeding would occur [50]When the concentration of this factor is between 5 and 30of the average concentration individuals still risk bleeding incase of trauma [50]
14 BioMed Research International
250
300
350
400
450
500
550
600
650
700
Clot
ting
time (
s)
Factor IX (M)
1eminus
14
1eminus
12
1eminus
10
1eminus
8
1eminus
6
1eminus
4
1eminus
2
1e0
Figure 10 Clotting time at different initial concentrations of factor IXThe reference value is 900sdot10minus8M the sweep range is [900sdot10minus14 900sdot10minus2]
0
100
200
300
400
500
600
700
Clot
ting
time (
s)
Factor II (M)
1eminus
12
1eminus
10
1eminus
8
1eminus
6
1eminus
4
1eminus
2
1e0
1e2
Figure 11 Clotting time at different initial concentrations of thrombinThe reference value is 140sdot10minus6M the sweep range is [140sdot10minus12 140sdot100]
The PSA-1D over factor IX (Figure 10) shows only aslight decrease of the CT as the initial concentration offIX increases this is in contrast to recent studies thatdemonstrated how the excess of factor IX leads to anincreased risk of deep vein thrombosis [51] This result canbe explained by considering that (i) a high concentrationof factor IX is not sufficient to bring about coagulationproblems though when the concentration of other factorsis above the average value (yet not at pathological levels)prothrombotic states can be observed (ii) in this model weconsider average values as initial concentrations of factorshowever individuals are characterized by different (balanced)combinations of procoagulant and anticoagulant factor levelsthat altogether contribute to define a unique coagulationphenotype that reflects the developmental environmentalgenetic nutritional and pharmacological influences of eachindividual [52] On the contrary the lack of factor IX causes
haemophilia B characterized by higher CT with respect tothe reference value (Figure 10)
Furthermore by comparing the PSA-1D of factors VIIIand IX (Figures 9 and 10) it is clear that haemophilia A ismore serious than haemophilia B since the CT achieved inconditions of factor VIII deficiency is higher than the CTobtained in the case of factor IX deficiency
In both PSA-1D over factors VIII and IX we observedafter the initial decrease of the CT an unexpected increaseof the CT as the factor concentration increasesThis counter-intuitive behavior arises at very high concentrations of thesefactors (with respect to the average physiological levels) andto the best of our knowledge it was never observed in vivoNonetheless it would be interesting to verify by means of adhoc laboratory experiments if the model correctly describesthe behavior of the BCC even in these conditions or on thecontrary it is not predictive in these extreme situations
BioMed Research International 15
The PSA-1D over factor II (Figure 11) shows a dramaticdecrease of the CT as the initial concentration of this factorincreases (with respect to the average physiological level)This behavior resembles the effects of hypercoagulability (orthrombophilia) a disease caused by mutation G20210A inthe prothrombin gene [53] that causes an increase of theprothrombin level (factor II) in the blood flow resulting inan excessive formation of the active form of this factor thusheightening venous thrombosis risks [54] Hypercoagulabil-ity is usually treated with warfarin therapy or with otheranticoagulants with a similar effect These drugs decrease thecapacity of coagulation factors to become active preventingthe formation of unwanted thrombi
On the other hand when the initial concentration offactor II is low we achieved the effects of prothrombindeficiency a rare autosomal recessive disease that causes atendency to severe bleeding [29 55] As shown in Figure 11 aconcentration equal to 10of the physiological value of factorII (ie 14 sdot 10minus6 M) leads to clotting effects similar to severehaemophilia A
32 PSA-2D of the BCCModel We present here the results ofthe PSA-2D on the BCCmodel where couples of parameterswere varied to analyze the possible effects arising from thecombined perturbation of their values
321 Reactions 11990327
and 11990358 Figure 12 shows the effect of
the simultaneous variation of constants 11989627
and 11989658
(overthe same sweep ranges considered in the two PSA-1DSection 31) This result remarks that reaction 119903
27 involved
in the common pathway has a stronger influence on theBCC and that there is a synergic interplay between thesetwo reactions In particular when the value of 119896
27is low and
the value of 11989658
is high the CT is higher than the valuesachieved when only a single constant is changed because inthis condition both the intrinsic and the common pathwaysare simultaneously inhibited
322 Factor VIII Factor IX and Tissue Factor In the last twoPSA-2Dwe varied the initial concentration of factor VIII andTissue Factor and the initial concentration of factor IX andTissue Factor respectively
The initial concentrations of factors VIII and IX were var-ied over four orders of magnitude using their physiologicalvalues as upper limit for the sweep ranges the concentrationof Tissue Factor was varied over four orders of magnitudetwo above and two below its reference value (see Table 2)The rationale behind this choice is to observe how the BCCmodel in conditions corresponding to different states ofhaemophilia (obtained by decreasing the concentrations offactors VIII and IX) behaves with different initial concentra-tions of the Tissue Factor which is the upstream factor of theextrinsic pathway that is the most important element of theBCC
The results of these PSA-2D show that with respect tothe condition of haemophilia B in the case of haemophiliaA the amount of Tissue Factor (below its reference value)has a negligible influence on the CT as indicated by the
presence of a plateau in Figure 13 on the contrary concerninghaemophilia B a deficiency of Tissue Factor leads to anincrease of the CT especially when factor IX is present in lowconcentrations (Figure 14)
The different results achieved in the two PSA-2D are dueto the presence in the BCC model of a direct interactionbetweenTissue Factor and factor IX bymeans of theTF-fVIIacomplex (see reactions 119903
13 119903
15in Table 1) Indeed the lack
of Tissue Factor directly affects the concentration of activefactor IX which results in a strong alteration of the CT withrespect to physiological conditions
33 CPU versus GPU Performance Comparison In orderto show the relevant speedup achieved by coagSODA wepresent here the comparison of the computational effortrequired by GPU and CPU for the simulation of the BCCmodel The performances of our GPU simulator were com-pared with those obtained using the LSODA algorithmimplemented in the software COPASI [56] executing onthe CPU the same set of simulations that were run on theGPU COPASI is single-threaded and does not exploit thephysical and logical cores of the CPU therefore it representsa good benchmark as a single-node CPU-bound simulator ofbiological systems
In all simulations executed on both GPU and CPU westored 100 samples uniformly distributed in the time intervalconsidered for each simulation that is [0 700] seconds of thedynamics of all chemical species involved in the BCCmodelThe settings for the LSODA algorithm were the followingrelative error 1 sdot 10minus7 absolute error 1 sdot 10minus13 and maximumnumber of internal steps set to 20000
331 Benchmark Details The GPU used for the simulationsis a Nvidia Tesla K20c equipped with 2496 cores organized in13 streamingmultiprocessors GPU clock 706MHz and 5GBofDDR5RAM In all tests coagSODAwas compiled and exe-cuted with version 55 of CUDA libraries Even though thisGPU has compute capability 35 and is based on the GK110Kepler architecture currently coagSODA does not exploitany of the new features (eg Dynamic Parallelism Hyper-QRemote DMA) with the exception of the new ISA encodingwhich allows threads to exploit an increased number ofregisters (255 instead of 63) reducing register spilling intoglobal memory and increasing performances [57] Moreoveras described in Section 22 the Kepler architecture offers thepossibility to reconfigure the 64KB on-chip cache balancingbetween L1 cache and shared memory Since coagSODAexploits the shared memory to reduce the latencies duringthe access to the concentration values of the BCC model weopted for the following configuration 48KB for the sharedmemory 16 KB for the L1 cache
The performance of the GPU was compared against apersonal computer equippedwith a dual-core CPU Intel Corei5 frequency 23 GHz 4GB of DDR3 RAM running theoperating systemMacOSXLion 1075The software used as acoagSODA-equivalent single-threaded CPU implementationis COPASI version 48 (build 35)
16 BioMed Research International
280
300
320
340
360
380
400
420
440
1e41e6
1e91e11 1e5
1e7
1e9
1e11
280 320 360 400 440
Clot
ting
time (
s)
k 58(M
minus1 sminus1 )
k27 (Mminus1sminus1)
Figure 12 Clotting time according to PSA-2D over reaction constants 11989627and 119896
58 The sweep ranges are [2 sdot 104 2 sdot 1010] and [1 sdot 105 2 sdot 1011]
respectively
100
200
300
400
Tissue factor (M) Factor VIII
(M)
Clot
ting
time (
s)
100
150
200
250
300
350
400
450
1eminus131eminus11
1eminus91eminus14
1eminus12
1eminus10
Figure 13 Clotting time at different initial concentrations of Tissue Factor and factor VIII The sweep ranges are [500 sdot 10minus14 500 sdot 10minus10]and [700 sdot 10minus14 500 sdot 10minus10] respectively
A direct comparison of the performances between thesetwo different architectures and implementations is not aneasy task since CPUs are optimized for single-thread exe-cution and exploit a large number of optimizations (eghigher clock frequency instruction caching pipelining outof order execution and branch prediction) whereas GPUsare optimized for graphics processing and parallel exe-cution of identical code relying only on multilevel datacaching For this reason in this paper we propose anempirical comparison of the performances based on therunning times of identical batches of simulations of the BCCmodel
332 Benchmark Results In Figure 15 we show the compar-ison of the running times required to run several batchesof simulations executed to carry out the PSA-1D over thereaction constant 119896
27 The choice of comparing the perfor-
mances of the GPU and CPU implementations by executingbatches of simulations that are related to a PSA instead ofrunning 119899 independent but identical simulations (ie allcharacterized by the same parameterization of the model)is due to the fact that these results represent a more realisticscenario in the computational analysis of biological systemswhereby it is useful to investigate large search spaces ofparameters corresponding to different perturbed conditionsof themodel [26]Moreover for the evaluation of the running
time the execution of a batch of 119899 identical deterministicsimulations would be futile The figure clearly shows thatcoagSODA always performs better than the CPU counter-part In particular while the CPU performance increaseslinearly with the number of simulations the running timesare strongly reduced on the GPU in this case the overallrunning time roughly corresponds to the time required forthe execution of the slowest simulations This is due to thefact that different parameterizations may require differenttime steps for LSODA to converge and the execution ofa block terminates as long as all the threads it containsterminate In turn the execution of a kernel terminates whenall blocks terminate for this reason a single simulationwhose dynamics requires more steps than the others mayaffect the overall running time
In Table 3 we report the running times of all batchesof simulations along with the speedup achieved on theGPU In particular these results highlight that the advantageof exploiting the GPUs for the simulation of the BCCmodel becomes more evident as the number of simula-tions increases with a 181times speedup when a PSA with10000 different parameterizations is executed Therefore theGPU accelerated analysis of the BCC model with coag-SODA represents a novel relevant computational means toinvestigate the behavior of this complex biological systemunder nonphysiological conditions and could be exploited
BioMed Research International 17
0
200
400
600
Tissue factor (M) Factor IX (M
)
Clot
ting
time (
s)
0
100
200
300
400
500
600
700
1eminus12
1eminus10
1eminus8
1eminus131eminus11
1eminus9
Figure 14 Clotting time at different initial concentrations of Tissue Factor and factor IXThe sweep ranges are [500 sdot 10minus14 500 sdot 10minus10] and[900 sdot 10
minus12 900 sdot 10
minus8] respectively
5 10 50 100 500 1000 5000 10000
Runn
ing
time (
s)
Number of simulations
CPUGPU
1e2
1e3
1e4
1e5
1e6
1e7
Figure 15 Comparison of the computational time required to execute an increasing number of simulations of the BCCmodel on CPU (IntelCore i5 23 GHz) and GPU (Nvidia Tesla K20c GPU clock 706MHz) The computational time is expressed in seconds The time valuesrelated to 1000 5000 and 10000 simulations on the CPU were estimated by regression (see also Table 3)
Table 3 CPU versus GPU performance comparison
Number ofsimulations CPU time (sec) GPU time (sec) Speedup
5 3808 1278 298times10 10604 2329 456times50 51704 8058 642times100 116528 10977 1061times500 536058 17394 3081times1000 1073582lowast 19982 5373times5000 5363507lowast 30960 1732times10000 10724975lowast 58951 1819timeslowastValues estimated by regression
to efficiently determine the response of the BCC to differenttherapeutical drugs
4 Conclusions
Thanks to their high-performance computing capabilitiesand the very low costs GPUs nowadays represent a suitable
technology for the development and the application of par-allel computational methods for in silico analysis of complexbiological systems However the implementation of efficientcomputational tools able to fully exploit the large potentialityof GPUs is still challenging since good programming skillsare required to implement GPU-based algorithmic methodsand to handle specific features as an optimal usage ofmemoryor the communication bandwidth between GPU and CPUMoreover algorithms cannot be directly ported on the GPUbecause of the limited programming capabilities allowed byGPUkernels as amatter of fact they need to be redesigned tomeet the requirements of the underlying SIMD architecture
In this work we presented coagSODA a GPU-poweredsimulator specifically developed to carry out fast parallelsimulations of the BCC model coagSODA was designed tooffer a black-box solution usable by any final user in aneasy way It relies on cupSODA [24] a numerical integratorfor ODEs that we previously implemented for the GPUarchitecture based on the LSODA algorithm and capableof automatically translating a reaction-based model into aset of coupled ODEs In addition to mass-action kinet-ics coagSODA implements specific functions to compute
18 BioMed Research International
the kinetics of Hill function based reactions such as thoseinvolved in the platelets activity of the BCC model coag-SODA exploits the massive parallel capabilities of modernGPUs and our results demonstrated that it can achieve arelevant reduction of the computational time required toexecutemany concurrent and independent simulations of theBCC dynamics The mutual independence of the simulationsallows fully exploiting the underlying SIMD architecturemoreover coagSODA benefits from an additional speedupthanks to our choice of storing the state of the systeminto the low-latency shared memory (a solution that wasalready implemented in cupSODA) Since the BCC modelis large (96 reactions 71 chemical species) a large amountof shared memory was assigned to each thread stronglyreducing the theoretical occupancy of the GPU that is theratio of active warps with respect to the maximum numberof warps supported by each streaming multiprocessor of theGPU However the results of the analysis presented in thiswork performed on the BCC model show that coagSODAachieves a relevant boost with respect to a reference CPUimplementation For instance in the case of 10000 simula-tions we achieved a noticeable 181times speedup Interestinglythe performances of coagSODA are better than COPASI evenfor small batches of simulations These results indicate thatfor biological models consisting in many reactions and manyspecies our GPU implementation of LSODA becomes moreprofitable than the CPU counterpart as the number of con-current simulations increases making it suitable especiallywhen performing demanding computational analysis suchas for example parameter sweep parameter estimation orsensitivity analysis
As a test case in this paper we presented several parame-ter sweep analyses over reaction constants and initial concen-trations of factors involved in the BCC model Other com-putational analyses on mathematical models of this pathwaywere previously presented For instance in [58] a parameterestimation of the reaction constants of different modelsinvolving the activation of factor X factor V prothrombinand the inactivation of factors was performed The aim ofthis analysis was to discriminate between different modelsto unravel the mechanisms on the basis of the BCC Similaranalyses were executed on a model describing thrombingeneration in plasma since the reaction mechanism thereaction constants and initial concentrations were unknown[59] In [34] a sensitivity analysis of a model consisting of44 species over 34 chemical species was presented reactionconstants were varied between 10 and 1000 of theirreference value to the aim of identifying the most influentialfactors of the BCC
All these analyses were performed bymeans of sequentialsimulations of the models under investigation thereforein general only small batches of simulations could be run(for instance in [34] only 836 simulations were executed toexecute the sensitivity analysis of the model) On the otherhand in this paper we presented the results of different PSAefficiently executed bymeans of coagSODA overall providinguseful information regarding unknownparameters and inter-esting insights into the functioning of the BCC A thoroughsensitivity analysis of the whole BCC model consisting in
around 5 sdot 105 simulations is currently in progress on GPUsby our group exploiting the great computational efficiency ofcoagSODA
The results of our computational analyses should benow validated by means of specifically designed laboratoryexperiments In particular we identified a plausible valuefor the constant of the reaction describing the autoactivationof factor XII (since no reference values can be found inliterature) moreover the results of the PSA over factor IXsuggest that a deficiency of this factor is not enough to causesevere bleeding disorders as haemophilia B but an alterationof other factors seems to be necessary for the occurrence ofsuch condition (eg the lack of Tissue Factor as suggestedby the PSA-2D over Tissue Factor and factor IX) Finallythe PSA over factors VIII and IX showed in a situationcharacterized by very high concentrations of these factorsa counterintuitive behavior in which the clotting time isincreased with respect to the value obtained in physiologicalconditions Indeed it would be interesting to design ad hoclaboratory experiments to verify if the BCCmodel is actuallypredictive in such extreme situations
The coagSODA software and the SBML version of theBCC model used in this work are available from the authorsupon request
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This work is partially supported by SYSBIO Centre forSystems Biology
References
[1] B B Aldridge JM BurkeDA Lauffenburger andP K SorgerldquoPhysicochemicalmodelling of cell signalling pathwaysrdquoNatureCell Biology vol 8 no 11 pp 1195ndash1203 2006
[2] I-C Chou and E O Voit ldquoRecent developments in parameterestimation and structure identification of biochemical andgenomic systemsrdquoMathematical Biosciences vol 219 no 2 pp57ndash83 2009
[3] D Besozzi P Cazzaniga D Pescini G Mauri S Colomboand EMartegani ldquoThe role of feedback control mechanisms onthe establishment of oscillatory regimes in the RascAMPPKApathway in S cerevisiaerdquo EURASIP Journal on Bioinformaticsand Systems Biology vol 2012 no 10 2012
[4] A Saltelli M Ratto T Andres et al Global Sensitivity AnalysisThe Primer Wiley-Interscience 2008
[5] A Raue C Kreutz T Maiwald et al ldquoStructural and practicalidentifiability analysis of partially observed dynamical modelsby exploiting the profile likelihoodrdquo Bioinformatics vol 25 no15 pp 1923ndash1929 2009
[6] A Drager M Kronfeld M J Ziller et al ldquoModeling metabolicnetworks in C glutamicum a comparison of rate laws incombination with various parameter optimization strategiesrdquoBMC Systems Biology vol 3 article 5 2009
BioMed Research International 19
[7] D Besozzi P Cazzaniga G Mauri D Pescini and L Van-neschi ldquoA comparison of genetic algorithms and particle swarmoptimization for parameter estimation in stochastic biochem-ical systemsrdquo in Proceedings of the 7th European ConferenceEvolutionary Computation Machine Learning and Data Miningin Bioinformatics (EvoBIO rsquo09) C Pizzuti M Ritchie and MGiacobini Eds vol 5483 of Lecture Notes in Computer Sciencepp 116ndash127 Springer 2009
[8] M S Nobile D Besozzi P Cazzaniga GMauri andD PescinildquocupSODA aGPU-basedmulti-swarmPSOmethod for param-eter estimation in stochastic biological systems exploitingdiscrete-time target seriesrdquo in Proceedings of the 10th EuropeanConference Evolutionary Computation Machine Learning andData Mining in Bioinformatics (EvoBIO rsquo12) M Giacobini LVanneschi and W Bush Eds vol 7246 of Lecture Notes inComputer Science pp 74ndash85 Springer
[9] M S Nobile D Besozzi P Cazzaniga D Pescini and GMaurildquoReverse engineering of kinetic reaction networks by meansof Cartesian Genetic Programming and Particle Swarm Opti-mizationrdquo in Proceedings of the IEEE Congress on EvolutionaryComputation (CEC rsquo13) pp 1594ndash1601 June 2013
[10] M S Nobile ldquoEvolutionary inference of biochemical inter-action networks accelerated on Graphics Processing Unitsrdquoin Proceedings of the 11th International Conference on HighPerformance Computingamp Simulation 2013 (HPCS rsquo13) pp 668ndash670 Helsinki Finland 2013
[11] K Kentzoglanakis and M Poole ldquoA swarm intelligence frame-work for reconstructing gene networks searching for bio-logically plausible architecturesrdquo IEEEACM Transactions onComputational Biology and Bioinformatics vol 9 no 2 pp 358ndash371 2012
[12] J R Koza W Mydlowec G Lanza Y Jessen and M A KeaneldquoAutomatic computational discovery of chemical reaction net-works using genetic programmingrdquo inComputationalDiscoveryof Scientific Knowledge S Dzeroski and L Todorovski Edsvol 4660 of Lecture Notes in Computer Science pp 205ndash227Springer 2007
[13] S Ando E Sakamoto and H Iba ldquoEvolutionary modeling andinference of gene networkrdquo Information Sciences vol 145 no3-4 pp 237ndash259 2002
[14] L Dematte and D Prandi ldquoGPU computing for systemsbiologyrdquo Briefings in Bioinformatics vol 11 no 3 Article IDbbq006 pp 323ndash333 2010
[15] J L Payne N A Sinnott-Armstrong and J HMoore ldquoExploit-ing graphics processing units for computational biology andbioinformaticsrdquo Interdisciplinary Sciences Computational LifeSciences vol 2 no 3 pp 213ndash220 2010
[16] M J Harvey and G De Fabritiis ldquoA survey of computationalmolecular science using graphics processing unitsrdquoWiley Inter-disciplinary Reviews Computational Molecular Science vol 2no 5 pp 734ndash742 2012
[17] Y Zhou J Liepe X Sheng M P H Stumpf and C BarnesldquoGPU accelerated biochemical network simulationrdquo Bioinfor-matics vol 27 no 6 Article ID btr015 pp 874ndash876 2011
[18] M Vigelius A Lane and B Meyer ldquoAccelerating reaction-diffusion simulations with general-purpose graphics processingunitsrdquo Bioinformatics vol 27 no 2 Article ID btq622 pp 288ndash290 2011
[19] D J Wilkinson ldquoStochastic modelling for quantitative descrip-tion of heterogeneous biological systemsrdquo Nature ReviewsGenetics vol 10 no 2 pp 122ndash133 2009
[20] J C Butcher ldquoTrees and numerical methods for ordinarydifferential equationsrdquo Numerical Algorithms vol 53 no 2-3pp 153ndash170 2010
[21] D T Gillespie ldquoStochastic simulation of chemical kineticsrdquoAnnual Review of Physical Chemistry vol 58 pp 35ndash55 2007
[22] L Petzold ldquoAutomatic selection of methods for solving stiffand nonstiff systems of ordinary differential equationsrdquo SIAMJournal of Scientific and Statistical Computing vol 4 no 1 pp136ndash148 1983
[23] Y Cao D T Gillespie and L R Petzold ldquoEfficient step sizeselection for the tau-leaping simulation methodrdquo The Journalof Chemical Physics vol 124 no 4 2006
[24] M S Nobile D Besozzi P Cazzaniga GMauri andD PescinildquocupSODA a CUDA-powered simulator of mass-action kinet-icsrdquo in Proceedings of the 12th International Conference onParallel Computing Technologies (PaCT rsquo13) V Malyshkin Edvol 7979 of Lecture Notes in Computer Science pp 344ndash357
[25] M S Nobile P Cazzaniga D Besozzi and G Mauri ldquoGPU-accelerated simulations of mass-action kinetics models withcupSODArdquoThe Journal of Supercomputing 2014
[26] M S Nobile P Cazzaniga D Besozzi D Pescini and GMauri ldquocuTauLeaping a GPU-powered tau-leaping stochasticsimulator for massive parallel analyses of biological systemsrdquoPLoS ONE vol 9 no 3 2014
[27] K Rogers Blood Physiology and CirculationTheHuman BodyBritannica Educational Publishing 2010
[28] A D Michelson Platelets Elsevier 2nd edition 2007[29] R W Colman J Hirsh V J Marder A W Clowes and J
N George Hemostasis and Thrombosis Basic Principles andClinical Practice Lippincott Williams ampWilkins 2006
[30] M S Chatterjee W S Denney H Jing and S L DiamondldquoSystems biology of coagulation initiation Kinetics of thrombingeneration in resting and activated human bloodrdquo PLoS Com-putational Biology vol 6 no 9 Article ID e1000950 2010
[31] M Bellini D Besozzi P Cazzaniga GMauri andM S NobileldquoSimulation and analysis of the blood coagulation cascadeaccelerated on GPUrdquo in Proceedings of the 22nd EuromicroInternational Conference on Parallel Distributed and Network-Based Processing (PDP) pp 590ndash593 IEEE Computer SocietyConference Publishing Services 2014
[32] T Wajima G K Isbister and S B Duffull ldquoA comprehensivemodel for the humoral coagulation network in humansrdquo Clini-cal Pharmacology and Therapeutics vol 86 no 3 pp 290ndash2982009
[33] K Lo W S Denney and S L Diamond ldquoStochastic modelingof blood coagulation initiationrdquo Pathophysiology of Haemostasisand Thrombosis vol 34 no 2-3 pp 80ndash90 2006
[34] C M Danforth T Orfeo K G Mann K E Brummel-Ziedins and S J Everse ldquoThe impact of uncertainty in a bloodcoagulation modelrdquo Mathematical Medicine and Biology vol26 no 4 Article ID dqp011 pp 323ndash336 2009
[35] S D Lewis P P Shields and J A Shafer ldquoCharacterizationof the kinetic pathway for liberation of fibrinopeptides duringassembly of fibrinrdquoThe Journal of Biological Chemistry vol 260no 18 pp 10192ndash10199 1985
[36] M F Hockin K C Jones S J Everse and K G Mann ldquoAmodel for the stoichiometric regulation of blood coagulationrdquoThe Journal of Biological Chemistry vol 277 no 21 pp 18322ndash18333 2002
[37] K Brummel-Ziedins C YVossen F R Rosendaal KUmezakiand K G Mann ldquoThe plasma hemostatic proteome thrombin
20 BioMed Research International
generation in healthy individualsrdquo Journal of Thrombosis andHaemostasis vol 3 no 7 pp 1472ndash1481 2005
[38] K E Brummel-Ziedins C Y Vossen S Butenas K G Mannand F R Rosendaal ldquoThrombin generation profiles in deepvenous thrombosisrdquo Journal of Thrombosis and Haemostasisvol 3 no 11 pp 2497ndash2505 2005
[39] S D Bungay P A Gentry and R D Gentry ldquoA mathematicalmodel of lipid-mediated thrombin generationrdquo MathematicalMedicine and Biology vol 20 no 1 pp 105ndash129 2003
[40] S Butenas T Orfeo M T Gissel K E Brummel and K GMann ldquoThe significance of circulating factor IXa in bloodrdquoThe Journal of Biological Chemistry vol 279 no 22 pp 22875ndash22882 2004
[41] D Voet and J G Voet Biochemistry Wiley 2011[42] A C Guyton and J E Hall Textbook of Medical Physiology
Saunders 2010[43] M F M James and A M Roche ldquoDose-response relationship
between plasma ionized calcium concentration and thrombe-lastographyrdquo Journal of Cardiothoracic and Vascular Anesthesiavol 18 no 5 pp 581ndash586 2004
[44] D L Nelson and M M Cox Lehninger Principles of Biochem-istry W H Freeman Company 2004
[45] R M Farber ldquoTopical perspective on massive threading andparallelismrdquo Journal of Molecular Graphics and Modelling vol30 pp 82ndash89 2011
[46] OWolkenhauerMUllahWKolch andK-HCho ldquoModelingand simulation of intracellular dynamics choosing an appropri-ate frameworkrdquo IEEETransactions onNanobioscience vol 3 no3 pp 200ndash207 2004
[47] B H Estridge A P Reynolds and N J Walters Basic MedicalLaboratory Techniques Cengage Learning 4th edition 2000
[48] T Renne A H Schmaier K F Nickel M Blomback and CMaas ldquoIn vivo roles of factor XIIrdquo Blood vol 120 no 22 pp4296ndash4303 2012
[49] P V Jenkins O Rawley O P Smith and J S ODonnellldquoElevated factor VIII levels and risk of venous thrombosisrdquoBritish Journal of Haematology vol 157 no 6 pp 653ndash663 2012
[50] C E Walsh and K M Batt ldquoHemophilia clinical gene therapybrief reviewrdquo Translational Research vol 161 no 4 pp 307ndash3122013
[51] A Van Hylckama Vlieg I K Van Der Linden R M Bertinaand F R Rosendaal ldquoHigh levels of factor IX increase the risk ofvenous thrombosisrdquo Blood vol 95 no 12 pp 3678ndash3682 2000
[52] K Brummel-Ziedins ldquoModels for thrombin generation and riskof diseaserdquo Journal ofThrombosis and Haemostasis vol 11 no 1pp 212ndash223 2013
[53] U Seligsohn and A Lubetsky ldquoGenetic susceptibility to venousthrombosisrdquoTheNew England Journal of Medicine vol 344 no16 pp 1222ndash1231 2001
[54] I Martinelli P Bucciarelli and P M Mannucci ldquoThromboticrisk factors basic pathophysiologyrdquo Critical Care Medicine vol38 no 2 pp S3ndashS9 2010
[55] K Brummel-Ziedins A Undas T Orfeo et al ldquoThrombingeneration in acute coronary syndrome and stable coronaryartery disease dependence on plasma factor compositionrdquoJournal ofThrombosis andHaemostasis vol 6 no 1 pp 104ndash1102008
[56] S Hoops R Gauges C Lee et al ldquoCOPASImdasha complexpathway simulatorrdquo Bioinformatics vol 22 no 24 pp 3067ndash3074 2006
[57] Nvidia NVIDIArsquos Next Generation CUDA Compute Architec-ture Kepler GK110 2012
[58] W Stortelder W Hemker and H Hemker ldquoMathematicalmodelling in blood coagulation simulation and parameterestimationrdquoModelling Analysis and Simulation vol 20 pp 1ndash111997
[59] S M C Orfao G Jank and K Mottaghy ldquoA new mathematicalapproach to modelling thrombin generationrdquo InternationalJournal of Artificial Organs vol 29 no 7 pp 703ndash708 2006
[60] Nvidia Nvidia CUDA C Programming Guide v50 2012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Anatomy Research International
PeptidesInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
International Journal of
Volume 2014
Zoology
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Molecular Biology International
GenomicsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
BioinformaticsAdvances in
Marine BiologyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Signal TransductionJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
BioMed Research International
Evolutionary BiologyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Biochemistry Research International
ArchaeaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Genetics Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in
Virolog y
Hindawi Publishing Corporationhttpwwwhindawicom
Nucleic AcidsJournal of
Volume 2014
Stem CellsInternational
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Enzyme Research
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Microbiology
BioMed Research International 7
(iv) the activation of prekallikrein by factor XIIa(reactions 119903
48 11990349 11990350) and by kallikrein (reac-
tion 11990354)
(v) the activation of factor XI by factor XIIa (reac-tions 119903
61 11990362 11990363) and by factor XIa (reaction
11990364)
(vi) the activation of factor IX by factor XIa (reac-tions 119903
69 11990370 11990371)
(vii) the dissociation of free factor VIIIa (reactions11990323and 11990324)
(viii) the dissociation of factor VIIIa in complex withother factors (reactions 119903
25and 11990326)
(3) The third module corresponds to the common path-way which consists in
(i) the activation of factor II by factor Xa (modeledby reaction 119903
16) and by the complex between
factor Xa and factor Va through the formationof the intermediate meizothrombin (reactions11990330 119903
33)
(ii) the activation of factor VII by factor Xa (reac-tion 1199036) and by factor IIa (reaction 119903
7)
(iii) the activation of factor VIII by factor Xa (reac-tions 119903
75 11990376 11990377) and by factor IIa (reaction 119903
17)
(iv) the formation of a complex between factor Xaand factor Va (reactions 119903
28and 11990329)
(v) the activation of fibrinogen by factor IIa (reac-tions 119903
84 119903
96)
(vi) the activation of factor V by factor IIa (reaction11990327)
(vii) the activation of factor XI by factor IIa (reac-tions 119903
58 11990359 11990360)
(4) The fourth module describes the inhibition of coagu-lation carried out through the main inhibitors of theBCC (antithrombin Tissue Factor Pathway InhibitorC1-inihibitor 1205721-antitrypsin and 1205722-antiplasmin)These are tight binding inhibitors belonging to theserpin superfamily which form irreversible com-plexes This module consists in
(i) the inhibition of factor Xa by Tissue FactorPathway Inhibitor (modeled by reactions 119903
34and
11990335)
(ii) the inhibition of the complex between TissueFactor factor VIIa and factor Xa by TissueFactor Pathway Inhibitor (reactions 119903
36 11990337 11990338)
(iii) the inhibition of factor IIa by antithrombin(reactions 119903
40and 11990342)
(iv) the inhibition of factor Xa by antithrombin(reaction 119903
39)
(v) the inhibition of factor IXa by antithrombin(reaction 119903
41)
(vi) the inhibition of factor XIa by antithrombin(reaction 119903
65) and by C1-inhibitor (reaction 119903
66)
Table 2 Initial concentrations of molecular species in the bloodcoagulation cascade model (values taken from [30])
Species Symbol Concentration (M)1205721-Antitrypsin A1AT 450 sdot 10
minus5
1205722-Antiplasmin A2AP 100 sdot 10minus6
Antithrombin III ATIII 340 sdot 10minus6
C1-inhibitor C1inh 250 sdot 10minus6
Fibrinogen Fbg 900 sdot 10minus6
Factor II fII 140 sdot 10minus6
Factor V fV 200 sdot 10minus8
Factor VII fVII 100 sdot 10minus8
Active factor VII fVIIa 100 sdot 10minus10
Factor VIII fVIII 700 sdot 10minus10
Factor IX fIX 900 sdot 10minus8
Factor X fX 16 sdot 10minus7
Factor XI fXI 310 sdot 10minus8
Factor XII fXII 340 sdot 10minus7
Prekallikrein Pkal 450 sdot 10minus7
Tissue Factor TF 500 sdot 10minus12
Tissue Factor Pathway Inhibitor TFPI 250 sdot 10minus9
(vii) the inhibition of factor XIIa by C1-inhibitor(reaction 119903
56) by antithrombin (reaction 119903
57)
by 1205721-antitrypsin (reaction 11990367) and by 1205722-
antiplasmin (reaction 11990368)
(viii) the inhibition of the complex between TissueFactor and factor VIIa by antithrombin (reac-tion 11990343)
(ix) the inhibition of kallikrein (reaction 11990355)
The values of the initial concentrations of the molecularspecies occurring in the BCC model are given in Table 2According to [30] the concentrations of complexes and activefactors were set to 0 except for the active factor VII which isphysiologically present in the blood circulation even in theabsence of damage in a concentration that is approximatelyequal to 1 of the corresponding inactive factor [42]
The system of ordinary differential equations (ODEs)needed to carry out the simulations and the parametersweep analysis presented in Section 3 was derived fromthe reactions given in Table 1 according to the mass-actionlaw with the exception of 14 reactions belonging to the set119878120576= 11990319 11990321 11990323 11990329 11990331 11990335 11990346 11990349 11990352 11990362 11990370 11990373 11990376 11990382
The mass-action law is the fundamental empirical law thatgoverns biochemical reaction rates which states that in adilute solution the rate of an elementary reaction is propor-tional to the product of the concentration of its reactantsraised to the power of the corresponding stoichiometriccoefficient [44] The reactions in 119878
120576are instead influenced by
a specifically defined variable 120576 depending on aHill functionfit against experimental data which quantify the plateletactivation status which is used to model the physiologicallevels of thrombin concentration as a function of plateletactivation as thoroughly described in [30] More preciselythe value of 120576 influences the formation of some complexes
8 BioMed Research International
0 0 100 200 300 400 500 600 700
Fact
or II
a con
cent
ratio
n (M
)
Time (s)
1eminus7
2eminus7
3eminus7
4eminus7
5eminus7
6eminus7
7eminus7
8eminus7
9eminus7
Figure 2 Dynamics of thrombin (factor IIa) in physiological condition
occurring on the plateletsrsquo surface by modifying the activityof reactions of the form
119860 + 119861
1198961
999445999468
1198962
119860119861 (1)
Namely 120576 intervenes with the dissociation constant of thesereactions so that the corresponding standard ODEs arechanged to yield new equations of the form
119889 [119860119861]
119889119905
= 1198961sdot [119860] sdot [119861] minus
1198962sdot [119860] sdot [119861]
120576
(2)
The value of 120576 in (2) depends on the following HillfunctionH which quantifies the state of platelets activationaccording to the thrombin concentration (here denoted as[fIIa]) that is the factor catalyzing the formation of the fibrinclot
H ([fIIalowast (119905)]) =[fIIalowast (119905)]16123
[fIIalowast (119905)]16123 + (24279 sdot 10minus9)16123
(3)
where [fIIalowast(119905)] = max1199051015840isin[0119905][fIIa(1199051015840)] for a chosen time
interval [0 119905] of simulationThe value [fIIalowast(119905)] represents the maximum transient
thrombin concentration and is needed to simulate the factthat in physiological conditions the thrombin concentrationstarts decreasing after rising to a peak (Figure 2) So doing[fIIalowast(119905)] never decreases once it reaches its maximummagnitude [fIIalowast(119905)] is equivalent to [fIIa(119905)] until the con-centration of factor IIa reaches the peak while thereafter itremains constant at that value which is the maximum in theconsidered time interval [0 119905] FunctionH allows simulatingthe physiological condition whereby platelets remain activealso when the thrombin concentration decreases
For a given concentration of factor IIa the maximumplatelets activation state 120576max is defined as
120576max = 120576max0
+ (1 minus 120576max0
) sdotH ([fIIalowast (119905)]) (4)
where 120576max0
defines the basal activation state of the plateletsat simulation time 119905 = 0 The value 120576max
0
is initially set to 001assuming a basal 1 binding strength of coagulation factorsto the resting plateletsrsquo surface When the full activationof platelets is reached 120576max is equal to 1 and the complexdissociation constants are minimized (see [30] for moredetails)
22 CUDA Architecture and the coagSODA Simulator Intro-duced by Nvidia in 2006 Compute Unified Device Architec-ture (CUDA) is a parallel computing platform and program-ming model that provides programmers with a frameworkto exploit GPUs in general-purpose computational tasks(GPGPU computing) GPGPU computing is a low-cost andenergy-wise alternative to the traditional high-performancecomputing infrastructures (eg clusters of machines) whichgives access to the tera-scale computing on common work-stations of mid-range price However due to the innovativearchitecture and the intrinsic limitations of GPUs a directporting of sequential code on the GPU is most of the timesunfeasible therefore making it challenging to fully exploit itscomputational power and massive parallelism [45]
CUDA combines the single instruction multiple data(SIMD) architecture with multithreading which automat-ically handles the conditional divergence between threadsThe drawback of such flexibility is that any divergence ofthe execution flow between threads causes a serialization ofthe execution which affects the overall performances UnderCUDArsquos naming conventions the programmer implementsthe kernel that is a CC++ function which is loaded fromthe host (the CPU) to the devices (one or more GPUs)and replicated in many copies named threads Threads canbe organized in three-dimensional structures named blockswhich in turn are contained in three-dimensional grids(a schematic description is given in Figure 3 left side)Whenever the host runs a kernel the GPU creates thecorresponding grid and automatically schedules each blockon one free streaming multiprocessor available on the GPUallowing a transparent scaling of performances on different
BioMed Research International 9
Host
Kernel 1
Kernel 2
Device
Grid 1
Grid 2
Block(0 0)
Block(1 0)
Block(2 0)
Block(0 1)
Block(1 1)
Block(2 1)
Block (1 1)
Thread(0 0)
Thread(1 0)
Thread(2 0)
Thread(3 0)
Thread(4 0)
Thread(0 1)
Thread(1 1)
Thread(2 1)
Thread(3 1)
Thread(4 1)
Thread(0 2)
Thread(1 2)
Thread(2 2)
Thread(3 2)
Thread(4 2)
Global memory
Texture memory
Constant memory
Shared memory Shared memory
Block (0 0) Block (1 0)
Localmemory
Localmemory
Localmemory
Localmemory
Thread(0 0)
Thread(1 0)
Registers Registers Registers Registers
Thread(0 0)
Thread(1 0)
Grid
Figure 3 Schematic description of CUDArsquos architecture in terms of threads and memory hierarchy Left Side Threads organization a singlekernel is launched from the host (the CPU) and is executed in multiple threads on the device (the GPU) Threads can be organized in three-dimensional structures named blocks which can be in turn organized in three-dimensional grids The dimensions of blocks and grids areexplicitly defined by the programmer Right Side Memory hierarchy threads can access data from many different memories with differentscopes Registers and local memories are private for each thread Shared memory lets threads belonging to the same block communicate andhas low access latency All threads can access the global memory which suffers high latencies but it is cached since the introduction of theFermi architecture Texture and constant memory can be read from any thread and are equipped with a cache as well in this work we exploitthe constant memory Figures are taken from Nvidiarsquos CUDA programming guide [60]
devices Threads within a block are executed in groups of 32threads named warps
The GPU is equipped with different kinds of memoryIn this work we exploit the global memory (accessible fromall threads) the shared memory (accessible from threadsof the same block) the local memory (registers and arraysaccessible from owner thread) and the constant memory(cached and not modifiable) A schematic representation ofthis memory hierarchy is shown in Figure 3 right side Toachieve the best performances the sharedmemory should beexploited as much as possible however it is very limited (ie49152 bytes for each multiprocessor since the introductionof the Fermi architecture) and it introduces restrictions onthe blocksrsquo size On the other hand the global memory isvery large (thousands ofMBs) but suffers from high latenciesA solution to this problem was implemented on the Fermiarchitecture where the global memory is equipped with aL2 cache This architecture also introduced the possibility tobalance 64KB of fast on-chip memory between the sharedmemory and L1 cache using two possible configurations48KB for the shared memory and 16KB for L1 cache or16 KB for the shared memory and 48 for L1 cacheThe Keplerarchitecture used in this paper allows a third and perfectly
balanced configuration where shared memory and L1 cacheobtain the same amount of memory (32KB) See Figure 4 fora schematization of the memory architecture
The systematic analysis of models of biological systemsoften consists in the execution of large batches of simulationsOne of the standard analyses that can be executed on suchkind of models regards an intensive search within the param-eters space which requires large numbers of independentsimulations In order to reduce the computational burdenwe previously implemented on the CUDA architecture oneof the most efficient numerical integration algorithms forODEs LSODA that is able to automatically recognize stiffand nonstiff systems and to dynamically select between themost appropriate integration procedure (ie Adams methodin the absence of stiffness and the Backward DifferentiationFormulae otherwise) [22] This GPU-powered tool is calledcupSODA [24 25] and exploits CUDArsquos massive parallelismto execute different and independent simulations in eachthread thus reducing the computational time required by astandard CPU counterpart of LSODA
Besides being very efficient for the simulation of manyindependent simulations cupSODA is also user-friendlyLSODA was originally designed to solve ODEs systems
10 BioMed Research International
Thread
Shared memory L1 cache Read-onlydata cache
L2 cache
DRAM
Figure 4 Schematic description of memory hierarchies in Fermi and Kepler architectures GPUs relying on these architectures are equippedwith a two-level data cache and a read-only data cache Shared memory and L1 cache share the same on-chip 64KB memory banks theamount of memory can be reconfigured by the user according to the specific needs of the application Figure taken from Nvidiarsquos KeplerGK110 whitepaper [57]
written in the canonical form but the user is supposed tospecify the system of ODEs by implementing a custom Cfunction that is passed to the algorithmmoreover in order tospeed up the computation when dealing with stiff systems inLSODA the Jacobianmatrix associatedwith theODEs systemmust be implemented as a custom C function as well On thecontrary cupSODA was conceived as a black-box simulatorthat can be easily used without any programming skillscupSODA consists in a tool to automatically convert a genericmechanistic reaction-based model of a biological system intothe corresponding set of ODEs to comply with the mass-action kinetics [46] and to directly encode the obtainedsystem along with the corresponding Jacobian matrix as Carrays
This fully automaticmethodology can be exploited for thesimulation of models whose chemical kinetics is based on themass-action law The BCC model described in Section 21though includes a set 119878
120576of reactions that do not follow
the mass-action kinetics [30] the platelets activity is notexplicitly modeled by biochemical reactions but it is realizedby modulating the rate of the dissociation of the complexesformed on a plateletrsquos surface by means of the variable120576 which is calculated with a special equation during theintegration steps For this reason cupSODA was used asa starting point for the development of a new tool ableto compute at run-time the specific kinetics of this set ofreactions
This new CUDA-powered simulation tool named coag-SODA is specifically tailored for the simulation of the BCC
model developed in [30] In particular coagSODA realizesthe run-time calculation of the 120576 value required to correctlysimulate the activity of platelets which is determined accord-ing to the equations described in Section 21 The plateletsactivation state 120576 is calculated at each time instant by solvingthe following differential equation
119889120576
119889119905
= 119896 (120576max minus 120576) (5)
where the constant 119896 is inversely proportional to the timescale of platelets activation and is set to 0005 This isconsistent with the fact that platelets do not instantly achievetheir maximum attainable activation state (120576max) but theyreach it on a physiologically relevant timescale [30]
Dealing with the 14 reactions in the set 119878120576that are
influenced by the Hill functionH (see Section 21) the valueof [fIIalowast(119905)]must be stored on the GPU because during eachintegration step coagSODA recalculates (4) which exploitsthe value [fIIalowast(119905)] to determine a new 120576 value Since CUDArsquosarchitecture does not offer static variables the informationfor each thread has to be memorized in the global memoryThe accesses to the global memory and the computationalcosts due to these additional calculations slow down theintegration process with respect to the integration of ODEsperformed according to strictly mass-action based kinetics(as in the cupSODA simulator) Nevertheless in Section 33we show that our parallel implementation largely outper-forms a sequential counterpart of the LSODA algorithm
BioMed Research International 11
0 100 200 300 400 500 600 700
Fibr
inog
en co
ncen
trat
ion
(M)
Time (s)
30 of fibrinogen
0
1eminus6
2eminus6
3eminus6
4eminus6
5eminus6
6eminus6
7eminus6
8eminus6
9eminus6
Figure 5 Dynamics of fibrinogen in physiological condition The clotting time is defined as the time necessary to convert the 70 of thefibrinogen into fibrin
As in the case of cupSODA [24 25] coagSODA exploitsthe shared memory to improve performances by storing thecurrent state and time of the simulations in these low-latencymemory banks Despite the improvement of performancesensured by this solution it strongly affects the occupancyof GPUrsquos multiprocessors and therefore it represents thelimiting factor for the number of blocks that can be executedsimultaneously as a matter of fact coagSODA is limited to2 blocks per streaming multiprocessor on GPUs based onthe Fermi architecture with a reduced exploitation of theGPU with respect to the theoretical 8 blocks allowed by thisarchitecture
3 Results and Discussion
In this section we discuss the results of the parameter sweepanalysis (PSA) carried out on the BCC model to the aimof investigating either the prolongation or the reduction ofthe clotting time in response to perturbed values of somereaction constants and of the initial concentration of somemolecular species chosen according to their meaning withinthe whole pathway PSA was performed by generating aset of different initial conditions corresponding to differentparameterizations of the model and then automaticallyexecuting the deterministic simulations with coagSODATheuse ofGPU technology is fundamental in this type of analysisespecially for large biological systems as the BCC is becauseit drastically reduces the computational time
The sweep analysis for single parameters (PSA-1D) wasperformed considering a logarithmic sampling of numericalvalues of each parameter under investigation (reaction con-stant or initial molecular concentration) within a specifiedrange with respect to its physiological reference value Thesweep analysis over pairs of parameters (PSA-2D) was per-formed by simultaneously varying the values of two param-eters within a specified range considering a logarithmicsampling on the resulting lattice The logarithmic samplingallows uniformly spanning different orders of magnitude ofthe parameters value using a reduced and fine-grained set
of samples therefore efficiently analyzing the response of thesystem in a broad range of conditions
To determine the response of the BCC to perturbed con-ditions we chose the clotting time (CT) as output of the PSAThe CT is defined as the time necessary to convert the 70of the fibrinogen into fibrin (see Figure 5) conventionallyassumed to correspond to the time required to form the clot[47] and it is generally used in laboratory tests formonitoringthe therapywith anticoagulant drugs According to themodeldefined in [30] the reference value of CT is around 300seconds in physiological conditions We investigate herethe response of the BCC by evaluating the CT in variousconditions corresponding to different values of the reactionconstants varying over six orders of magnitude with respectto their physiological values (ie three below and three abovethe reference values if not otherwise specified) as well asto different values of the initial molecular concentrationsand varying over twelve orders of magnitude with respect totheir physiological values (ie six below and six above thereference values if not otherwise specified)
The total number of parallel simulations executed tocarry out these analyses was 100 for PSA-1D over reactionconstants 200 for PSA-1D over initial concentrations and1600 for PSA-2D
Finally we present the comparison of the performanceof the CPU and GPU to run an increasing number ofsimulations of the BCC model to prove the efficiency ofcoagSODA
31 PSA-1D of the BCC Model
311 Reaction 11990344 Thefirst PSA was performed to determine
the value of the kinetic constant for the autoactivation offactor XII (reaction 119903
44in Table 1) which corresponds to an
upstream process in the intrinsic pathway This analysis wasmotivated by two considerations Firstly by using the fullparameterization given in [30] the action of the intrinsicpathway turns out to be fundamental for the BCC in vivowhich is in contrast to experimental observations which
12 BioMed Research International
324
326
328
330
332
334
336
338
340
Clot
ting
time (
s)
1eminus
10
1eminus
9
1eminus
8
1eminus
7
1eminus
6
1eminus
5
1eminus
4
1eminus
3
k44 (sminus1)
Figure 6 Clotting time according to PSA-1D over constant 11989644of reaction 119903
44 XII rarr XIIaThe reference value of 119896
44used in [30] is 5sdot10minus4sminus1
the sweep range is [5 sdot 10minus10 5 sdot 10minus4]
280
300
320
340
360
380
400
1e4 1e5 1e6 1e7 1e8 1e9 1e10 1e11
Clot
ting
time (
s)
k27 (Mminus1sminus1)
Figure 7 Clotting time according to PSA-1D over constant 11989627of reaction 119903
27 IIa + V rarr IIa + Va The reference value is 2 sdot 107Mminus1 sminus1 the
sweep range is [2 sdot 104 2 sdot 1010]
indicate that the extrinsic pathway is the main responsibleof clot formation Secondly in [30] all constants values havea reference to experimental measurements except for theconstant of this reaction (which corresponds to entry 29 inTable 1 in [30])
Figure 6 shows the results of this PSA-1D where 11989644was
varied over six orders of magnitude considering the valuegiven in [30] as the upper limit of the sweep interval We canobserve that by decreasing the value of 119896
44with respect to
the value considered in [30] the CT increases with respect toits reference value however for values of this constant lowerthan 100 sdot 10minus6 sminus1 the CT remains unaltered and this canbe intuitively explained by the fact that in this situation thefibrinogen is mainly activated by the extrinsic pathway
Consequently we assigned the value 700 sdot 10minus6 sminus1 to11989644 achieving a CT that is comparable to the experimental
observations of the BCC in vivo This new value was used inall PSA discussed in what follows
312 Reactions 11990327
and 11990358 In the next PSA we investigated
the effect of the perturbation of the kinetics of two pivotal
reactions of the BCC model Reaction 11990327 which describes
the catalytic activation of factor V by factor IIa was chosenbecause it represents the main positive feedback within thecommon pathway reaction 119903
58 which is involved in the
activation of factor XI by binding to factor IIa was chosenbecause it represents the main positive feedback in theintrinsic pathway (Figure 1) Moreover preliminary PSA overall reaction constants of the BCC model given in Table 1evidenced that these two reactions are among the mostrelevant steps of the coagulation network
The PSA-1D over 11989627(Figure 7) shows that the CT is very
sensitive to the perturbation of the rate of this reaction whenthe reference value of its constant (ie 119896
27= 2 sdot 10
7Mminus1 sminus1)is either increased or decreased in particular when 119896
27is
very low a plateau in CT is reached since the strength ofthe positive feedback exerted by factor IIa is largely reduceda condition where the contribution of the amplification ofthe hemostatic stimulus (due by the common pathway) tothe formation of the clot is basically not effective On theother side the PSA-1D over 119896
58(Figure 8) shows that while
a decrease of the constant with respect to its reference value
BioMed Research International 13
310
320
330
340
350
360
370
1e5 1e6 1e7 1e8 1e9 1e10 1e11
Clot
ting
time (
s)
k58 (Mminus1sminus1)
Figure 8 Clotting time according to PSA-1D over constant 11989658of reaction 119903
58 IIa + XI rarr IIa-XI The reference value is 1 sdot 108Mminus1 sminus1 the
sweep range is [1 sdot 105 2 sdot 1011]
220
240
260
280
300
320
340
360
380
400
Clot
ting
time (
s)
Factor VIII (M)
Seve
re h
aem
ophi
lia
Mild
hae
mop
hilia
Prot
hrom
botic
stat
e
1eminus
16
1eminus
14
1eminus
12
1eminus
10
1eminus
8
1eminus
6
1eminus
4
1eminus
2
Figure 9 Clotting time at different initial concentrations of factor VIII The reference value is 700 sdot 10minus10M (dashed black line) the sweeprange is [700 sdot 10minus16 700 sdot 10minus4] The blue area indicates a condition in which factor VIII concentration is less than 1 of its physiologicalconcentration corresponding to a situation of severe haemophilia the red area indicates a condition in which factor VIII concentration isbetween 1 and 30 of its physiological concentration corresponding to a situation of mild haemophilia the green area indicates a conditionin which factor VIII concentration is greater than 130 of its physiological concentration corresponding to a situation of hypercoagulability
(ie 11989658= 1 sdot 10
8Mminus1 sminus1) does not have any substantialeffect an increase of its value leads to a progressive increasein the CT This increase is due to the fact that factor IIa issequestered in the formation of a complex with factor XI andhence it is no longer available as a free component in bloodto participate in other reactions especially those reactionsof the extrinsic pathway which principally lead to the clotformation in vivo This behavior highlights that the intrinsicpathway has a secondary role in blood coagulation in vivocompared with the extrinsic pathway as also evidenced byvarious experimental observations [48]
313 Factors VIII IX and II The next set of PSA-1D wasrealized by varying the initial concentrations of factors VIIIIX and II These factors were selected since both an excess
and a deficiency of their concentrations lead to diseasesrelated to blood clotting
The PSA-1D over factor VIII (Figure 9) shows thatincreasing the initial concentration of this factor resultsin decreasing the CT suggesting the possible presence ofhypercoagulable states in these perturbed conditions As amatter of fact high levels of factor VIII cause an increasedrisk of deep vein thrombosis and pulmonary embolism[49] On the other hand individuals with less than 1of the average concentration of factor VIII show a severehaemophilia A characterized by higher CT (Figure 9) andrequire infusions of plasma containing the deficient factorotherwise frequent spontaneous bleeding would occur [50]When the concentration of this factor is between 5 and 30of the average concentration individuals still risk bleeding incase of trauma [50]
14 BioMed Research International
250
300
350
400
450
500
550
600
650
700
Clot
ting
time (
s)
Factor IX (M)
1eminus
14
1eminus
12
1eminus
10
1eminus
8
1eminus
6
1eminus
4
1eminus
2
1e0
Figure 10 Clotting time at different initial concentrations of factor IXThe reference value is 900sdot10minus8M the sweep range is [900sdot10minus14 900sdot10minus2]
0
100
200
300
400
500
600
700
Clot
ting
time (
s)
Factor II (M)
1eminus
12
1eminus
10
1eminus
8
1eminus
6
1eminus
4
1eminus
2
1e0
1e2
Figure 11 Clotting time at different initial concentrations of thrombinThe reference value is 140sdot10minus6M the sweep range is [140sdot10minus12 140sdot100]
The PSA-1D over factor IX (Figure 10) shows only aslight decrease of the CT as the initial concentration offIX increases this is in contrast to recent studies thatdemonstrated how the excess of factor IX leads to anincreased risk of deep vein thrombosis [51] This result canbe explained by considering that (i) a high concentrationof factor IX is not sufficient to bring about coagulationproblems though when the concentration of other factorsis above the average value (yet not at pathological levels)prothrombotic states can be observed (ii) in this model weconsider average values as initial concentrations of factorshowever individuals are characterized by different (balanced)combinations of procoagulant and anticoagulant factor levelsthat altogether contribute to define a unique coagulationphenotype that reflects the developmental environmentalgenetic nutritional and pharmacological influences of eachindividual [52] On the contrary the lack of factor IX causes
haemophilia B characterized by higher CT with respect tothe reference value (Figure 10)
Furthermore by comparing the PSA-1D of factors VIIIand IX (Figures 9 and 10) it is clear that haemophilia A ismore serious than haemophilia B since the CT achieved inconditions of factor VIII deficiency is higher than the CTobtained in the case of factor IX deficiency
In both PSA-1D over factors VIII and IX we observedafter the initial decrease of the CT an unexpected increaseof the CT as the factor concentration increasesThis counter-intuitive behavior arises at very high concentrations of thesefactors (with respect to the average physiological levels) andto the best of our knowledge it was never observed in vivoNonetheless it would be interesting to verify by means of adhoc laboratory experiments if the model correctly describesthe behavior of the BCC even in these conditions or on thecontrary it is not predictive in these extreme situations
BioMed Research International 15
The PSA-1D over factor II (Figure 11) shows a dramaticdecrease of the CT as the initial concentration of this factorincreases (with respect to the average physiological level)This behavior resembles the effects of hypercoagulability (orthrombophilia) a disease caused by mutation G20210A inthe prothrombin gene [53] that causes an increase of theprothrombin level (factor II) in the blood flow resulting inan excessive formation of the active form of this factor thusheightening venous thrombosis risks [54] Hypercoagulabil-ity is usually treated with warfarin therapy or with otheranticoagulants with a similar effect These drugs decrease thecapacity of coagulation factors to become active preventingthe formation of unwanted thrombi
On the other hand when the initial concentration offactor II is low we achieved the effects of prothrombindeficiency a rare autosomal recessive disease that causes atendency to severe bleeding [29 55] As shown in Figure 11 aconcentration equal to 10of the physiological value of factorII (ie 14 sdot 10minus6 M) leads to clotting effects similar to severehaemophilia A
32 PSA-2D of the BCCModel We present here the results ofthe PSA-2D on the BCCmodel where couples of parameterswere varied to analyze the possible effects arising from thecombined perturbation of their values
321 Reactions 11990327
and 11990358 Figure 12 shows the effect of
the simultaneous variation of constants 11989627
and 11989658
(overthe same sweep ranges considered in the two PSA-1DSection 31) This result remarks that reaction 119903
27 involved
in the common pathway has a stronger influence on theBCC and that there is a synergic interplay between thesetwo reactions In particular when the value of 119896
27is low and
the value of 11989658
is high the CT is higher than the valuesachieved when only a single constant is changed because inthis condition both the intrinsic and the common pathwaysare simultaneously inhibited
322 Factor VIII Factor IX and Tissue Factor In the last twoPSA-2Dwe varied the initial concentration of factor VIII andTissue Factor and the initial concentration of factor IX andTissue Factor respectively
The initial concentrations of factors VIII and IX were var-ied over four orders of magnitude using their physiologicalvalues as upper limit for the sweep ranges the concentrationof Tissue Factor was varied over four orders of magnitudetwo above and two below its reference value (see Table 2)The rationale behind this choice is to observe how the BCCmodel in conditions corresponding to different states ofhaemophilia (obtained by decreasing the concentrations offactors VIII and IX) behaves with different initial concentra-tions of the Tissue Factor which is the upstream factor of theextrinsic pathway that is the most important element of theBCC
The results of these PSA-2D show that with respect tothe condition of haemophilia B in the case of haemophiliaA the amount of Tissue Factor (below its reference value)has a negligible influence on the CT as indicated by the
presence of a plateau in Figure 13 on the contrary concerninghaemophilia B a deficiency of Tissue Factor leads to anincrease of the CT especially when factor IX is present in lowconcentrations (Figure 14)
The different results achieved in the two PSA-2D are dueto the presence in the BCC model of a direct interactionbetweenTissue Factor and factor IX bymeans of theTF-fVIIacomplex (see reactions 119903
13 119903
15in Table 1) Indeed the lack
of Tissue Factor directly affects the concentration of activefactor IX which results in a strong alteration of the CT withrespect to physiological conditions
33 CPU versus GPU Performance Comparison In orderto show the relevant speedup achieved by coagSODA wepresent here the comparison of the computational effortrequired by GPU and CPU for the simulation of the BCCmodel The performances of our GPU simulator were com-pared with those obtained using the LSODA algorithmimplemented in the software COPASI [56] executing onthe CPU the same set of simulations that were run on theGPU COPASI is single-threaded and does not exploit thephysical and logical cores of the CPU therefore it representsa good benchmark as a single-node CPU-bound simulator ofbiological systems
In all simulations executed on both GPU and CPU westored 100 samples uniformly distributed in the time intervalconsidered for each simulation that is [0 700] seconds of thedynamics of all chemical species involved in the BCCmodelThe settings for the LSODA algorithm were the followingrelative error 1 sdot 10minus7 absolute error 1 sdot 10minus13 and maximumnumber of internal steps set to 20000
331 Benchmark Details The GPU used for the simulationsis a Nvidia Tesla K20c equipped with 2496 cores organized in13 streamingmultiprocessors GPU clock 706MHz and 5GBofDDR5RAM In all tests coagSODAwas compiled and exe-cuted with version 55 of CUDA libraries Even though thisGPU has compute capability 35 and is based on the GK110Kepler architecture currently coagSODA does not exploitany of the new features (eg Dynamic Parallelism Hyper-QRemote DMA) with the exception of the new ISA encodingwhich allows threads to exploit an increased number ofregisters (255 instead of 63) reducing register spilling intoglobal memory and increasing performances [57] Moreoveras described in Section 22 the Kepler architecture offers thepossibility to reconfigure the 64KB on-chip cache balancingbetween L1 cache and shared memory Since coagSODAexploits the shared memory to reduce the latencies duringthe access to the concentration values of the BCC model weopted for the following configuration 48KB for the sharedmemory 16 KB for the L1 cache
The performance of the GPU was compared against apersonal computer equippedwith a dual-core CPU Intel Corei5 frequency 23 GHz 4GB of DDR3 RAM running theoperating systemMacOSXLion 1075The software used as acoagSODA-equivalent single-threaded CPU implementationis COPASI version 48 (build 35)
16 BioMed Research International
280
300
320
340
360
380
400
420
440
1e41e6
1e91e11 1e5
1e7
1e9
1e11
280 320 360 400 440
Clot
ting
time (
s)
k 58(M
minus1 sminus1 )
k27 (Mminus1sminus1)
Figure 12 Clotting time according to PSA-2D over reaction constants 11989627and 119896
58 The sweep ranges are [2 sdot 104 2 sdot 1010] and [1 sdot 105 2 sdot 1011]
respectively
100
200
300
400
Tissue factor (M) Factor VIII
(M)
Clot
ting
time (
s)
100
150
200
250
300
350
400
450
1eminus131eminus11
1eminus91eminus14
1eminus12
1eminus10
Figure 13 Clotting time at different initial concentrations of Tissue Factor and factor VIII The sweep ranges are [500 sdot 10minus14 500 sdot 10minus10]and [700 sdot 10minus14 500 sdot 10minus10] respectively
A direct comparison of the performances between thesetwo different architectures and implementations is not aneasy task since CPUs are optimized for single-thread exe-cution and exploit a large number of optimizations (eghigher clock frequency instruction caching pipelining outof order execution and branch prediction) whereas GPUsare optimized for graphics processing and parallel exe-cution of identical code relying only on multilevel datacaching For this reason in this paper we propose anempirical comparison of the performances based on therunning times of identical batches of simulations of the BCCmodel
332 Benchmark Results In Figure 15 we show the compar-ison of the running times required to run several batchesof simulations executed to carry out the PSA-1D over thereaction constant 119896
27 The choice of comparing the perfor-
mances of the GPU and CPU implementations by executingbatches of simulations that are related to a PSA instead ofrunning 119899 independent but identical simulations (ie allcharacterized by the same parameterization of the model)is due to the fact that these results represent a more realisticscenario in the computational analysis of biological systemswhereby it is useful to investigate large search spaces ofparameters corresponding to different perturbed conditionsof themodel [26]Moreover for the evaluation of the running
time the execution of a batch of 119899 identical deterministicsimulations would be futile The figure clearly shows thatcoagSODA always performs better than the CPU counter-part In particular while the CPU performance increaseslinearly with the number of simulations the running timesare strongly reduced on the GPU in this case the overallrunning time roughly corresponds to the time required forthe execution of the slowest simulations This is due to thefact that different parameterizations may require differenttime steps for LSODA to converge and the execution ofa block terminates as long as all the threads it containsterminate In turn the execution of a kernel terminates whenall blocks terminate for this reason a single simulationwhose dynamics requires more steps than the others mayaffect the overall running time
In Table 3 we report the running times of all batchesof simulations along with the speedup achieved on theGPU In particular these results highlight that the advantageof exploiting the GPUs for the simulation of the BCCmodel becomes more evident as the number of simula-tions increases with a 181times speedup when a PSA with10000 different parameterizations is executed Therefore theGPU accelerated analysis of the BCC model with coag-SODA represents a novel relevant computational means toinvestigate the behavior of this complex biological systemunder nonphysiological conditions and could be exploited
BioMed Research International 17
0
200
400
600
Tissue factor (M) Factor IX (M
)
Clot
ting
time (
s)
0
100
200
300
400
500
600
700
1eminus12
1eminus10
1eminus8
1eminus131eminus11
1eminus9
Figure 14 Clotting time at different initial concentrations of Tissue Factor and factor IXThe sweep ranges are [500 sdot 10minus14 500 sdot 10minus10] and[900 sdot 10
minus12 900 sdot 10
minus8] respectively
5 10 50 100 500 1000 5000 10000
Runn
ing
time (
s)
Number of simulations
CPUGPU
1e2
1e3
1e4
1e5
1e6
1e7
Figure 15 Comparison of the computational time required to execute an increasing number of simulations of the BCCmodel on CPU (IntelCore i5 23 GHz) and GPU (Nvidia Tesla K20c GPU clock 706MHz) The computational time is expressed in seconds The time valuesrelated to 1000 5000 and 10000 simulations on the CPU were estimated by regression (see also Table 3)
Table 3 CPU versus GPU performance comparison
Number ofsimulations CPU time (sec) GPU time (sec) Speedup
5 3808 1278 298times10 10604 2329 456times50 51704 8058 642times100 116528 10977 1061times500 536058 17394 3081times1000 1073582lowast 19982 5373times5000 5363507lowast 30960 1732times10000 10724975lowast 58951 1819timeslowastValues estimated by regression
to efficiently determine the response of the BCC to differenttherapeutical drugs
4 Conclusions
Thanks to their high-performance computing capabilitiesand the very low costs GPUs nowadays represent a suitable
technology for the development and the application of par-allel computational methods for in silico analysis of complexbiological systems However the implementation of efficientcomputational tools able to fully exploit the large potentialityof GPUs is still challenging since good programming skillsare required to implement GPU-based algorithmic methodsand to handle specific features as an optimal usage ofmemoryor the communication bandwidth between GPU and CPUMoreover algorithms cannot be directly ported on the GPUbecause of the limited programming capabilities allowed byGPUkernels as amatter of fact they need to be redesigned tomeet the requirements of the underlying SIMD architecture
In this work we presented coagSODA a GPU-poweredsimulator specifically developed to carry out fast parallelsimulations of the BCC model coagSODA was designed tooffer a black-box solution usable by any final user in aneasy way It relies on cupSODA [24] a numerical integratorfor ODEs that we previously implemented for the GPUarchitecture based on the LSODA algorithm and capableof automatically translating a reaction-based model into aset of coupled ODEs In addition to mass-action kinet-ics coagSODA implements specific functions to compute
18 BioMed Research International
the kinetics of Hill function based reactions such as thoseinvolved in the platelets activity of the BCC model coag-SODA exploits the massive parallel capabilities of modernGPUs and our results demonstrated that it can achieve arelevant reduction of the computational time required toexecutemany concurrent and independent simulations of theBCC dynamics The mutual independence of the simulationsallows fully exploiting the underlying SIMD architecturemoreover coagSODA benefits from an additional speedupthanks to our choice of storing the state of the systeminto the low-latency shared memory (a solution that wasalready implemented in cupSODA) Since the BCC modelis large (96 reactions 71 chemical species) a large amountof shared memory was assigned to each thread stronglyreducing the theoretical occupancy of the GPU that is theratio of active warps with respect to the maximum numberof warps supported by each streaming multiprocessor of theGPU However the results of the analysis presented in thiswork performed on the BCC model show that coagSODAachieves a relevant boost with respect to a reference CPUimplementation For instance in the case of 10000 simula-tions we achieved a noticeable 181times speedup Interestinglythe performances of coagSODA are better than COPASI evenfor small batches of simulations These results indicate thatfor biological models consisting in many reactions and manyspecies our GPU implementation of LSODA becomes moreprofitable than the CPU counterpart as the number of con-current simulations increases making it suitable especiallywhen performing demanding computational analysis suchas for example parameter sweep parameter estimation orsensitivity analysis
As a test case in this paper we presented several parame-ter sweep analyses over reaction constants and initial concen-trations of factors involved in the BCC model Other com-putational analyses on mathematical models of this pathwaywere previously presented For instance in [58] a parameterestimation of the reaction constants of different modelsinvolving the activation of factor X factor V prothrombinand the inactivation of factors was performed The aim ofthis analysis was to discriminate between different modelsto unravel the mechanisms on the basis of the BCC Similaranalyses were executed on a model describing thrombingeneration in plasma since the reaction mechanism thereaction constants and initial concentrations were unknown[59] In [34] a sensitivity analysis of a model consisting of44 species over 34 chemical species was presented reactionconstants were varied between 10 and 1000 of theirreference value to the aim of identifying the most influentialfactors of the BCC
All these analyses were performed bymeans of sequentialsimulations of the models under investigation thereforein general only small batches of simulations could be run(for instance in [34] only 836 simulations were executed toexecute the sensitivity analysis of the model) On the otherhand in this paper we presented the results of different PSAefficiently executed bymeans of coagSODA overall providinguseful information regarding unknownparameters and inter-esting insights into the functioning of the BCC A thoroughsensitivity analysis of the whole BCC model consisting in
around 5 sdot 105 simulations is currently in progress on GPUsby our group exploiting the great computational efficiency ofcoagSODA
The results of our computational analyses should benow validated by means of specifically designed laboratoryexperiments In particular we identified a plausible valuefor the constant of the reaction describing the autoactivationof factor XII (since no reference values can be found inliterature) moreover the results of the PSA over factor IXsuggest that a deficiency of this factor is not enough to causesevere bleeding disorders as haemophilia B but an alterationof other factors seems to be necessary for the occurrence ofsuch condition (eg the lack of Tissue Factor as suggestedby the PSA-2D over Tissue Factor and factor IX) Finallythe PSA over factors VIII and IX showed in a situationcharacterized by very high concentrations of these factorsa counterintuitive behavior in which the clotting time isincreased with respect to the value obtained in physiologicalconditions Indeed it would be interesting to design ad hoclaboratory experiments to verify if the BCCmodel is actuallypredictive in such extreme situations
The coagSODA software and the SBML version of theBCC model used in this work are available from the authorsupon request
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This work is partially supported by SYSBIO Centre forSystems Biology
References
[1] B B Aldridge JM BurkeDA Lauffenburger andP K SorgerldquoPhysicochemicalmodelling of cell signalling pathwaysrdquoNatureCell Biology vol 8 no 11 pp 1195ndash1203 2006
[2] I-C Chou and E O Voit ldquoRecent developments in parameterestimation and structure identification of biochemical andgenomic systemsrdquoMathematical Biosciences vol 219 no 2 pp57ndash83 2009
[3] D Besozzi P Cazzaniga D Pescini G Mauri S Colomboand EMartegani ldquoThe role of feedback control mechanisms onthe establishment of oscillatory regimes in the RascAMPPKApathway in S cerevisiaerdquo EURASIP Journal on Bioinformaticsand Systems Biology vol 2012 no 10 2012
[4] A Saltelli M Ratto T Andres et al Global Sensitivity AnalysisThe Primer Wiley-Interscience 2008
[5] A Raue C Kreutz T Maiwald et al ldquoStructural and practicalidentifiability analysis of partially observed dynamical modelsby exploiting the profile likelihoodrdquo Bioinformatics vol 25 no15 pp 1923ndash1929 2009
[6] A Drager M Kronfeld M J Ziller et al ldquoModeling metabolicnetworks in C glutamicum a comparison of rate laws incombination with various parameter optimization strategiesrdquoBMC Systems Biology vol 3 article 5 2009
BioMed Research International 19
[7] D Besozzi P Cazzaniga G Mauri D Pescini and L Van-neschi ldquoA comparison of genetic algorithms and particle swarmoptimization for parameter estimation in stochastic biochem-ical systemsrdquo in Proceedings of the 7th European ConferenceEvolutionary Computation Machine Learning and Data Miningin Bioinformatics (EvoBIO rsquo09) C Pizzuti M Ritchie and MGiacobini Eds vol 5483 of Lecture Notes in Computer Sciencepp 116ndash127 Springer 2009
[8] M S Nobile D Besozzi P Cazzaniga GMauri andD PescinildquocupSODA aGPU-basedmulti-swarmPSOmethod for param-eter estimation in stochastic biological systems exploitingdiscrete-time target seriesrdquo in Proceedings of the 10th EuropeanConference Evolutionary Computation Machine Learning andData Mining in Bioinformatics (EvoBIO rsquo12) M Giacobini LVanneschi and W Bush Eds vol 7246 of Lecture Notes inComputer Science pp 74ndash85 Springer
[9] M S Nobile D Besozzi P Cazzaniga D Pescini and GMaurildquoReverse engineering of kinetic reaction networks by meansof Cartesian Genetic Programming and Particle Swarm Opti-mizationrdquo in Proceedings of the IEEE Congress on EvolutionaryComputation (CEC rsquo13) pp 1594ndash1601 June 2013
[10] M S Nobile ldquoEvolutionary inference of biochemical inter-action networks accelerated on Graphics Processing Unitsrdquoin Proceedings of the 11th International Conference on HighPerformance Computingamp Simulation 2013 (HPCS rsquo13) pp 668ndash670 Helsinki Finland 2013
[11] K Kentzoglanakis and M Poole ldquoA swarm intelligence frame-work for reconstructing gene networks searching for bio-logically plausible architecturesrdquo IEEEACM Transactions onComputational Biology and Bioinformatics vol 9 no 2 pp 358ndash371 2012
[12] J R Koza W Mydlowec G Lanza Y Jessen and M A KeaneldquoAutomatic computational discovery of chemical reaction net-works using genetic programmingrdquo inComputationalDiscoveryof Scientific Knowledge S Dzeroski and L Todorovski Edsvol 4660 of Lecture Notes in Computer Science pp 205ndash227Springer 2007
[13] S Ando E Sakamoto and H Iba ldquoEvolutionary modeling andinference of gene networkrdquo Information Sciences vol 145 no3-4 pp 237ndash259 2002
[14] L Dematte and D Prandi ldquoGPU computing for systemsbiologyrdquo Briefings in Bioinformatics vol 11 no 3 Article IDbbq006 pp 323ndash333 2010
[15] J L Payne N A Sinnott-Armstrong and J HMoore ldquoExploit-ing graphics processing units for computational biology andbioinformaticsrdquo Interdisciplinary Sciences Computational LifeSciences vol 2 no 3 pp 213ndash220 2010
[16] M J Harvey and G De Fabritiis ldquoA survey of computationalmolecular science using graphics processing unitsrdquoWiley Inter-disciplinary Reviews Computational Molecular Science vol 2no 5 pp 734ndash742 2012
[17] Y Zhou J Liepe X Sheng M P H Stumpf and C BarnesldquoGPU accelerated biochemical network simulationrdquo Bioinfor-matics vol 27 no 6 Article ID btr015 pp 874ndash876 2011
[18] M Vigelius A Lane and B Meyer ldquoAccelerating reaction-diffusion simulations with general-purpose graphics processingunitsrdquo Bioinformatics vol 27 no 2 Article ID btq622 pp 288ndash290 2011
[19] D J Wilkinson ldquoStochastic modelling for quantitative descrip-tion of heterogeneous biological systemsrdquo Nature ReviewsGenetics vol 10 no 2 pp 122ndash133 2009
[20] J C Butcher ldquoTrees and numerical methods for ordinarydifferential equationsrdquo Numerical Algorithms vol 53 no 2-3pp 153ndash170 2010
[21] D T Gillespie ldquoStochastic simulation of chemical kineticsrdquoAnnual Review of Physical Chemistry vol 58 pp 35ndash55 2007
[22] L Petzold ldquoAutomatic selection of methods for solving stiffand nonstiff systems of ordinary differential equationsrdquo SIAMJournal of Scientific and Statistical Computing vol 4 no 1 pp136ndash148 1983
[23] Y Cao D T Gillespie and L R Petzold ldquoEfficient step sizeselection for the tau-leaping simulation methodrdquo The Journalof Chemical Physics vol 124 no 4 2006
[24] M S Nobile D Besozzi P Cazzaniga GMauri andD PescinildquocupSODA a CUDA-powered simulator of mass-action kinet-icsrdquo in Proceedings of the 12th International Conference onParallel Computing Technologies (PaCT rsquo13) V Malyshkin Edvol 7979 of Lecture Notes in Computer Science pp 344ndash357
[25] M S Nobile P Cazzaniga D Besozzi and G Mauri ldquoGPU-accelerated simulations of mass-action kinetics models withcupSODArdquoThe Journal of Supercomputing 2014
[26] M S Nobile P Cazzaniga D Besozzi D Pescini and GMauri ldquocuTauLeaping a GPU-powered tau-leaping stochasticsimulator for massive parallel analyses of biological systemsrdquoPLoS ONE vol 9 no 3 2014
[27] K Rogers Blood Physiology and CirculationTheHuman BodyBritannica Educational Publishing 2010
[28] A D Michelson Platelets Elsevier 2nd edition 2007[29] R W Colman J Hirsh V J Marder A W Clowes and J
N George Hemostasis and Thrombosis Basic Principles andClinical Practice Lippincott Williams ampWilkins 2006
[30] M S Chatterjee W S Denney H Jing and S L DiamondldquoSystems biology of coagulation initiation Kinetics of thrombingeneration in resting and activated human bloodrdquo PLoS Com-putational Biology vol 6 no 9 Article ID e1000950 2010
[31] M Bellini D Besozzi P Cazzaniga GMauri andM S NobileldquoSimulation and analysis of the blood coagulation cascadeaccelerated on GPUrdquo in Proceedings of the 22nd EuromicroInternational Conference on Parallel Distributed and Network-Based Processing (PDP) pp 590ndash593 IEEE Computer SocietyConference Publishing Services 2014
[32] T Wajima G K Isbister and S B Duffull ldquoA comprehensivemodel for the humoral coagulation network in humansrdquo Clini-cal Pharmacology and Therapeutics vol 86 no 3 pp 290ndash2982009
[33] K Lo W S Denney and S L Diamond ldquoStochastic modelingof blood coagulation initiationrdquo Pathophysiology of Haemostasisand Thrombosis vol 34 no 2-3 pp 80ndash90 2006
[34] C M Danforth T Orfeo K G Mann K E Brummel-Ziedins and S J Everse ldquoThe impact of uncertainty in a bloodcoagulation modelrdquo Mathematical Medicine and Biology vol26 no 4 Article ID dqp011 pp 323ndash336 2009
[35] S D Lewis P P Shields and J A Shafer ldquoCharacterizationof the kinetic pathway for liberation of fibrinopeptides duringassembly of fibrinrdquoThe Journal of Biological Chemistry vol 260no 18 pp 10192ndash10199 1985
[36] M F Hockin K C Jones S J Everse and K G Mann ldquoAmodel for the stoichiometric regulation of blood coagulationrdquoThe Journal of Biological Chemistry vol 277 no 21 pp 18322ndash18333 2002
[37] K Brummel-Ziedins C YVossen F R Rosendaal KUmezakiand K G Mann ldquoThe plasma hemostatic proteome thrombin
20 BioMed Research International
generation in healthy individualsrdquo Journal of Thrombosis andHaemostasis vol 3 no 7 pp 1472ndash1481 2005
[38] K E Brummel-Ziedins C Y Vossen S Butenas K G Mannand F R Rosendaal ldquoThrombin generation profiles in deepvenous thrombosisrdquo Journal of Thrombosis and Haemostasisvol 3 no 11 pp 2497ndash2505 2005
[39] S D Bungay P A Gentry and R D Gentry ldquoA mathematicalmodel of lipid-mediated thrombin generationrdquo MathematicalMedicine and Biology vol 20 no 1 pp 105ndash129 2003
[40] S Butenas T Orfeo M T Gissel K E Brummel and K GMann ldquoThe significance of circulating factor IXa in bloodrdquoThe Journal of Biological Chemistry vol 279 no 22 pp 22875ndash22882 2004
[41] D Voet and J G Voet Biochemistry Wiley 2011[42] A C Guyton and J E Hall Textbook of Medical Physiology
Saunders 2010[43] M F M James and A M Roche ldquoDose-response relationship
between plasma ionized calcium concentration and thrombe-lastographyrdquo Journal of Cardiothoracic and Vascular Anesthesiavol 18 no 5 pp 581ndash586 2004
[44] D L Nelson and M M Cox Lehninger Principles of Biochem-istry W H Freeman Company 2004
[45] R M Farber ldquoTopical perspective on massive threading andparallelismrdquo Journal of Molecular Graphics and Modelling vol30 pp 82ndash89 2011
[46] OWolkenhauerMUllahWKolch andK-HCho ldquoModelingand simulation of intracellular dynamics choosing an appropri-ate frameworkrdquo IEEETransactions onNanobioscience vol 3 no3 pp 200ndash207 2004
[47] B H Estridge A P Reynolds and N J Walters Basic MedicalLaboratory Techniques Cengage Learning 4th edition 2000
[48] T Renne A H Schmaier K F Nickel M Blomback and CMaas ldquoIn vivo roles of factor XIIrdquo Blood vol 120 no 22 pp4296ndash4303 2012
[49] P V Jenkins O Rawley O P Smith and J S ODonnellldquoElevated factor VIII levels and risk of venous thrombosisrdquoBritish Journal of Haematology vol 157 no 6 pp 653ndash663 2012
[50] C E Walsh and K M Batt ldquoHemophilia clinical gene therapybrief reviewrdquo Translational Research vol 161 no 4 pp 307ndash3122013
[51] A Van Hylckama Vlieg I K Van Der Linden R M Bertinaand F R Rosendaal ldquoHigh levels of factor IX increase the risk ofvenous thrombosisrdquo Blood vol 95 no 12 pp 3678ndash3682 2000
[52] K Brummel-Ziedins ldquoModels for thrombin generation and riskof diseaserdquo Journal ofThrombosis and Haemostasis vol 11 no 1pp 212ndash223 2013
[53] U Seligsohn and A Lubetsky ldquoGenetic susceptibility to venousthrombosisrdquoTheNew England Journal of Medicine vol 344 no16 pp 1222ndash1231 2001
[54] I Martinelli P Bucciarelli and P M Mannucci ldquoThromboticrisk factors basic pathophysiologyrdquo Critical Care Medicine vol38 no 2 pp S3ndashS9 2010
[55] K Brummel-Ziedins A Undas T Orfeo et al ldquoThrombingeneration in acute coronary syndrome and stable coronaryartery disease dependence on plasma factor compositionrdquoJournal ofThrombosis andHaemostasis vol 6 no 1 pp 104ndash1102008
[56] S Hoops R Gauges C Lee et al ldquoCOPASImdasha complexpathway simulatorrdquo Bioinformatics vol 22 no 24 pp 3067ndash3074 2006
[57] Nvidia NVIDIArsquos Next Generation CUDA Compute Architec-ture Kepler GK110 2012
[58] W Stortelder W Hemker and H Hemker ldquoMathematicalmodelling in blood coagulation simulation and parameterestimationrdquoModelling Analysis and Simulation vol 20 pp 1ndash111997
[59] S M C Orfao G Jank and K Mottaghy ldquoA new mathematicalapproach to modelling thrombin generationrdquo InternationalJournal of Artificial Organs vol 29 no 7 pp 703ndash708 2006
[60] Nvidia Nvidia CUDA C Programming Guide v50 2012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Anatomy Research International
PeptidesInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
International Journal of
Volume 2014
Zoology
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Molecular Biology International
GenomicsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
BioinformaticsAdvances in
Marine BiologyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Signal TransductionJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
BioMed Research International
Evolutionary BiologyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Biochemistry Research International
ArchaeaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Genetics Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in
Virolog y
Hindawi Publishing Corporationhttpwwwhindawicom
Nucleic AcidsJournal of
Volume 2014
Stem CellsInternational
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Enzyme Research
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Microbiology
8 BioMed Research International
0 0 100 200 300 400 500 600 700
Fact
or II
a con
cent
ratio
n (M
)
Time (s)
1eminus7
2eminus7
3eminus7
4eminus7
5eminus7
6eminus7
7eminus7
8eminus7
9eminus7
Figure 2 Dynamics of thrombin (factor IIa) in physiological condition
occurring on the plateletsrsquo surface by modifying the activityof reactions of the form
119860 + 119861
1198961
999445999468
1198962
119860119861 (1)
Namely 120576 intervenes with the dissociation constant of thesereactions so that the corresponding standard ODEs arechanged to yield new equations of the form
119889 [119860119861]
119889119905
= 1198961sdot [119860] sdot [119861] minus
1198962sdot [119860] sdot [119861]
120576
(2)
The value of 120576 in (2) depends on the following HillfunctionH which quantifies the state of platelets activationaccording to the thrombin concentration (here denoted as[fIIa]) that is the factor catalyzing the formation of the fibrinclot
H ([fIIalowast (119905)]) =[fIIalowast (119905)]16123
[fIIalowast (119905)]16123 + (24279 sdot 10minus9)16123
(3)
where [fIIalowast(119905)] = max1199051015840isin[0119905][fIIa(1199051015840)] for a chosen time
interval [0 119905] of simulationThe value [fIIalowast(119905)] represents the maximum transient
thrombin concentration and is needed to simulate the factthat in physiological conditions the thrombin concentrationstarts decreasing after rising to a peak (Figure 2) So doing[fIIalowast(119905)] never decreases once it reaches its maximummagnitude [fIIalowast(119905)] is equivalent to [fIIa(119905)] until the con-centration of factor IIa reaches the peak while thereafter itremains constant at that value which is the maximum in theconsidered time interval [0 119905] FunctionH allows simulatingthe physiological condition whereby platelets remain activealso when the thrombin concentration decreases
For a given concentration of factor IIa the maximumplatelets activation state 120576max is defined as
120576max = 120576max0
+ (1 minus 120576max0
) sdotH ([fIIalowast (119905)]) (4)
where 120576max0
defines the basal activation state of the plateletsat simulation time 119905 = 0 The value 120576max
0
is initially set to 001assuming a basal 1 binding strength of coagulation factorsto the resting plateletsrsquo surface When the full activationof platelets is reached 120576max is equal to 1 and the complexdissociation constants are minimized (see [30] for moredetails)
22 CUDA Architecture and the coagSODA Simulator Intro-duced by Nvidia in 2006 Compute Unified Device Architec-ture (CUDA) is a parallel computing platform and program-ming model that provides programmers with a frameworkto exploit GPUs in general-purpose computational tasks(GPGPU computing) GPGPU computing is a low-cost andenergy-wise alternative to the traditional high-performancecomputing infrastructures (eg clusters of machines) whichgives access to the tera-scale computing on common work-stations of mid-range price However due to the innovativearchitecture and the intrinsic limitations of GPUs a directporting of sequential code on the GPU is most of the timesunfeasible therefore making it challenging to fully exploit itscomputational power and massive parallelism [45]
CUDA combines the single instruction multiple data(SIMD) architecture with multithreading which automat-ically handles the conditional divergence between threadsThe drawback of such flexibility is that any divergence ofthe execution flow between threads causes a serialization ofthe execution which affects the overall performances UnderCUDArsquos naming conventions the programmer implementsthe kernel that is a CC++ function which is loaded fromthe host (the CPU) to the devices (one or more GPUs)and replicated in many copies named threads Threads canbe organized in three-dimensional structures named blockswhich in turn are contained in three-dimensional grids(a schematic description is given in Figure 3 left side)Whenever the host runs a kernel the GPU creates thecorresponding grid and automatically schedules each blockon one free streaming multiprocessor available on the GPUallowing a transparent scaling of performances on different
BioMed Research International 9
Host
Kernel 1
Kernel 2
Device
Grid 1
Grid 2
Block(0 0)
Block(1 0)
Block(2 0)
Block(0 1)
Block(1 1)
Block(2 1)
Block (1 1)
Thread(0 0)
Thread(1 0)
Thread(2 0)
Thread(3 0)
Thread(4 0)
Thread(0 1)
Thread(1 1)
Thread(2 1)
Thread(3 1)
Thread(4 1)
Thread(0 2)
Thread(1 2)
Thread(2 2)
Thread(3 2)
Thread(4 2)
Global memory
Texture memory
Constant memory
Shared memory Shared memory
Block (0 0) Block (1 0)
Localmemory
Localmemory
Localmemory
Localmemory
Thread(0 0)
Thread(1 0)
Registers Registers Registers Registers
Thread(0 0)
Thread(1 0)
Grid
Figure 3 Schematic description of CUDArsquos architecture in terms of threads and memory hierarchy Left Side Threads organization a singlekernel is launched from the host (the CPU) and is executed in multiple threads on the device (the GPU) Threads can be organized in three-dimensional structures named blocks which can be in turn organized in three-dimensional grids The dimensions of blocks and grids areexplicitly defined by the programmer Right Side Memory hierarchy threads can access data from many different memories with differentscopes Registers and local memories are private for each thread Shared memory lets threads belonging to the same block communicate andhas low access latency All threads can access the global memory which suffers high latencies but it is cached since the introduction of theFermi architecture Texture and constant memory can be read from any thread and are equipped with a cache as well in this work we exploitthe constant memory Figures are taken from Nvidiarsquos CUDA programming guide [60]
devices Threads within a block are executed in groups of 32threads named warps
The GPU is equipped with different kinds of memoryIn this work we exploit the global memory (accessible fromall threads) the shared memory (accessible from threadsof the same block) the local memory (registers and arraysaccessible from owner thread) and the constant memory(cached and not modifiable) A schematic representation ofthis memory hierarchy is shown in Figure 3 right side Toachieve the best performances the sharedmemory should beexploited as much as possible however it is very limited (ie49152 bytes for each multiprocessor since the introductionof the Fermi architecture) and it introduces restrictions onthe blocksrsquo size On the other hand the global memory isvery large (thousands ofMBs) but suffers from high latenciesA solution to this problem was implemented on the Fermiarchitecture where the global memory is equipped with aL2 cache This architecture also introduced the possibility tobalance 64KB of fast on-chip memory between the sharedmemory and L1 cache using two possible configurations48KB for the shared memory and 16KB for L1 cache or16 KB for the shared memory and 48 for L1 cacheThe Keplerarchitecture used in this paper allows a third and perfectly
balanced configuration where shared memory and L1 cacheobtain the same amount of memory (32KB) See Figure 4 fora schematization of the memory architecture
The systematic analysis of models of biological systemsoften consists in the execution of large batches of simulationsOne of the standard analyses that can be executed on suchkind of models regards an intensive search within the param-eters space which requires large numbers of independentsimulations In order to reduce the computational burdenwe previously implemented on the CUDA architecture oneof the most efficient numerical integration algorithms forODEs LSODA that is able to automatically recognize stiffand nonstiff systems and to dynamically select between themost appropriate integration procedure (ie Adams methodin the absence of stiffness and the Backward DifferentiationFormulae otherwise) [22] This GPU-powered tool is calledcupSODA [24 25] and exploits CUDArsquos massive parallelismto execute different and independent simulations in eachthread thus reducing the computational time required by astandard CPU counterpart of LSODA
Besides being very efficient for the simulation of manyindependent simulations cupSODA is also user-friendlyLSODA was originally designed to solve ODEs systems
10 BioMed Research International
Thread
Shared memory L1 cache Read-onlydata cache
L2 cache
DRAM
Figure 4 Schematic description of memory hierarchies in Fermi and Kepler architectures GPUs relying on these architectures are equippedwith a two-level data cache and a read-only data cache Shared memory and L1 cache share the same on-chip 64KB memory banks theamount of memory can be reconfigured by the user according to the specific needs of the application Figure taken from Nvidiarsquos KeplerGK110 whitepaper [57]
written in the canonical form but the user is supposed tospecify the system of ODEs by implementing a custom Cfunction that is passed to the algorithmmoreover in order tospeed up the computation when dealing with stiff systems inLSODA the Jacobianmatrix associatedwith theODEs systemmust be implemented as a custom C function as well On thecontrary cupSODA was conceived as a black-box simulatorthat can be easily used without any programming skillscupSODA consists in a tool to automatically convert a genericmechanistic reaction-based model of a biological system intothe corresponding set of ODEs to comply with the mass-action kinetics [46] and to directly encode the obtainedsystem along with the corresponding Jacobian matrix as Carrays
This fully automaticmethodology can be exploited for thesimulation of models whose chemical kinetics is based on themass-action law The BCC model described in Section 21though includes a set 119878
120576of reactions that do not follow
the mass-action kinetics [30] the platelets activity is notexplicitly modeled by biochemical reactions but it is realizedby modulating the rate of the dissociation of the complexesformed on a plateletrsquos surface by means of the variable120576 which is calculated with a special equation during theintegration steps For this reason cupSODA was used asa starting point for the development of a new tool ableto compute at run-time the specific kinetics of this set ofreactions
This new CUDA-powered simulation tool named coag-SODA is specifically tailored for the simulation of the BCC
model developed in [30] In particular coagSODA realizesthe run-time calculation of the 120576 value required to correctlysimulate the activity of platelets which is determined accord-ing to the equations described in Section 21 The plateletsactivation state 120576 is calculated at each time instant by solvingthe following differential equation
119889120576
119889119905
= 119896 (120576max minus 120576) (5)
where the constant 119896 is inversely proportional to the timescale of platelets activation and is set to 0005 This isconsistent with the fact that platelets do not instantly achievetheir maximum attainable activation state (120576max) but theyreach it on a physiologically relevant timescale [30]
Dealing with the 14 reactions in the set 119878120576that are
influenced by the Hill functionH (see Section 21) the valueof [fIIalowast(119905)]must be stored on the GPU because during eachintegration step coagSODA recalculates (4) which exploitsthe value [fIIalowast(119905)] to determine a new 120576 value Since CUDArsquosarchitecture does not offer static variables the informationfor each thread has to be memorized in the global memoryThe accesses to the global memory and the computationalcosts due to these additional calculations slow down theintegration process with respect to the integration of ODEsperformed according to strictly mass-action based kinetics(as in the cupSODA simulator) Nevertheless in Section 33we show that our parallel implementation largely outper-forms a sequential counterpart of the LSODA algorithm
BioMed Research International 11
0 100 200 300 400 500 600 700
Fibr
inog
en co
ncen
trat
ion
(M)
Time (s)
30 of fibrinogen
0
1eminus6
2eminus6
3eminus6
4eminus6
5eminus6
6eminus6
7eminus6
8eminus6
9eminus6
Figure 5 Dynamics of fibrinogen in physiological condition The clotting time is defined as the time necessary to convert the 70 of thefibrinogen into fibrin
As in the case of cupSODA [24 25] coagSODA exploitsthe shared memory to improve performances by storing thecurrent state and time of the simulations in these low-latencymemory banks Despite the improvement of performancesensured by this solution it strongly affects the occupancyof GPUrsquos multiprocessors and therefore it represents thelimiting factor for the number of blocks that can be executedsimultaneously as a matter of fact coagSODA is limited to2 blocks per streaming multiprocessor on GPUs based onthe Fermi architecture with a reduced exploitation of theGPU with respect to the theoretical 8 blocks allowed by thisarchitecture
3 Results and Discussion
In this section we discuss the results of the parameter sweepanalysis (PSA) carried out on the BCC model to the aimof investigating either the prolongation or the reduction ofthe clotting time in response to perturbed values of somereaction constants and of the initial concentration of somemolecular species chosen according to their meaning withinthe whole pathway PSA was performed by generating aset of different initial conditions corresponding to differentparameterizations of the model and then automaticallyexecuting the deterministic simulations with coagSODATheuse ofGPU technology is fundamental in this type of analysisespecially for large biological systems as the BCC is becauseit drastically reduces the computational time
The sweep analysis for single parameters (PSA-1D) wasperformed considering a logarithmic sampling of numericalvalues of each parameter under investigation (reaction con-stant or initial molecular concentration) within a specifiedrange with respect to its physiological reference value Thesweep analysis over pairs of parameters (PSA-2D) was per-formed by simultaneously varying the values of two param-eters within a specified range considering a logarithmicsampling on the resulting lattice The logarithmic samplingallows uniformly spanning different orders of magnitude ofthe parameters value using a reduced and fine-grained set
of samples therefore efficiently analyzing the response of thesystem in a broad range of conditions
To determine the response of the BCC to perturbed con-ditions we chose the clotting time (CT) as output of the PSAThe CT is defined as the time necessary to convert the 70of the fibrinogen into fibrin (see Figure 5) conventionallyassumed to correspond to the time required to form the clot[47] and it is generally used in laboratory tests formonitoringthe therapywith anticoagulant drugs According to themodeldefined in [30] the reference value of CT is around 300seconds in physiological conditions We investigate herethe response of the BCC by evaluating the CT in variousconditions corresponding to different values of the reactionconstants varying over six orders of magnitude with respectto their physiological values (ie three below and three abovethe reference values if not otherwise specified) as well asto different values of the initial molecular concentrationsand varying over twelve orders of magnitude with respect totheir physiological values (ie six below and six above thereference values if not otherwise specified)
The total number of parallel simulations executed tocarry out these analyses was 100 for PSA-1D over reactionconstants 200 for PSA-1D over initial concentrations and1600 for PSA-2D
Finally we present the comparison of the performanceof the CPU and GPU to run an increasing number ofsimulations of the BCC model to prove the efficiency ofcoagSODA
31 PSA-1D of the BCC Model
311 Reaction 11990344 Thefirst PSA was performed to determine
the value of the kinetic constant for the autoactivation offactor XII (reaction 119903
44in Table 1) which corresponds to an
upstream process in the intrinsic pathway This analysis wasmotivated by two considerations Firstly by using the fullparameterization given in [30] the action of the intrinsicpathway turns out to be fundamental for the BCC in vivowhich is in contrast to experimental observations which
12 BioMed Research International
324
326
328
330
332
334
336
338
340
Clot
ting
time (
s)
1eminus
10
1eminus
9
1eminus
8
1eminus
7
1eminus
6
1eminus
5
1eminus
4
1eminus
3
k44 (sminus1)
Figure 6 Clotting time according to PSA-1D over constant 11989644of reaction 119903
44 XII rarr XIIaThe reference value of 119896
44used in [30] is 5sdot10minus4sminus1
the sweep range is [5 sdot 10minus10 5 sdot 10minus4]
280
300
320
340
360
380
400
1e4 1e5 1e6 1e7 1e8 1e9 1e10 1e11
Clot
ting
time (
s)
k27 (Mminus1sminus1)
Figure 7 Clotting time according to PSA-1D over constant 11989627of reaction 119903
27 IIa + V rarr IIa + Va The reference value is 2 sdot 107Mminus1 sminus1 the
sweep range is [2 sdot 104 2 sdot 1010]
indicate that the extrinsic pathway is the main responsibleof clot formation Secondly in [30] all constants values havea reference to experimental measurements except for theconstant of this reaction (which corresponds to entry 29 inTable 1 in [30])
Figure 6 shows the results of this PSA-1D where 11989644was
varied over six orders of magnitude considering the valuegiven in [30] as the upper limit of the sweep interval We canobserve that by decreasing the value of 119896
44with respect to
the value considered in [30] the CT increases with respect toits reference value however for values of this constant lowerthan 100 sdot 10minus6 sminus1 the CT remains unaltered and this canbe intuitively explained by the fact that in this situation thefibrinogen is mainly activated by the extrinsic pathway
Consequently we assigned the value 700 sdot 10minus6 sminus1 to11989644 achieving a CT that is comparable to the experimental
observations of the BCC in vivo This new value was used inall PSA discussed in what follows
312 Reactions 11990327
and 11990358 In the next PSA we investigated
the effect of the perturbation of the kinetics of two pivotal
reactions of the BCC model Reaction 11990327 which describes
the catalytic activation of factor V by factor IIa was chosenbecause it represents the main positive feedback within thecommon pathway reaction 119903
58 which is involved in the
activation of factor XI by binding to factor IIa was chosenbecause it represents the main positive feedback in theintrinsic pathway (Figure 1) Moreover preliminary PSA overall reaction constants of the BCC model given in Table 1evidenced that these two reactions are among the mostrelevant steps of the coagulation network
The PSA-1D over 11989627(Figure 7) shows that the CT is very
sensitive to the perturbation of the rate of this reaction whenthe reference value of its constant (ie 119896
27= 2 sdot 10
7Mminus1 sminus1)is either increased or decreased in particular when 119896
27is
very low a plateau in CT is reached since the strength ofthe positive feedback exerted by factor IIa is largely reduceda condition where the contribution of the amplification ofthe hemostatic stimulus (due by the common pathway) tothe formation of the clot is basically not effective On theother side the PSA-1D over 119896
58(Figure 8) shows that while
a decrease of the constant with respect to its reference value
BioMed Research International 13
310
320
330
340
350
360
370
1e5 1e6 1e7 1e8 1e9 1e10 1e11
Clot
ting
time (
s)
k58 (Mminus1sminus1)
Figure 8 Clotting time according to PSA-1D over constant 11989658of reaction 119903
58 IIa + XI rarr IIa-XI The reference value is 1 sdot 108Mminus1 sminus1 the
sweep range is [1 sdot 105 2 sdot 1011]
220
240
260
280
300
320
340
360
380
400
Clot
ting
time (
s)
Factor VIII (M)
Seve
re h
aem
ophi
lia
Mild
hae
mop
hilia
Prot
hrom
botic
stat
e
1eminus
16
1eminus
14
1eminus
12
1eminus
10
1eminus
8
1eminus
6
1eminus
4
1eminus
2
Figure 9 Clotting time at different initial concentrations of factor VIII The reference value is 700 sdot 10minus10M (dashed black line) the sweeprange is [700 sdot 10minus16 700 sdot 10minus4] The blue area indicates a condition in which factor VIII concentration is less than 1 of its physiologicalconcentration corresponding to a situation of severe haemophilia the red area indicates a condition in which factor VIII concentration isbetween 1 and 30 of its physiological concentration corresponding to a situation of mild haemophilia the green area indicates a conditionin which factor VIII concentration is greater than 130 of its physiological concentration corresponding to a situation of hypercoagulability
(ie 11989658= 1 sdot 10
8Mminus1 sminus1) does not have any substantialeffect an increase of its value leads to a progressive increasein the CT This increase is due to the fact that factor IIa issequestered in the formation of a complex with factor XI andhence it is no longer available as a free component in bloodto participate in other reactions especially those reactionsof the extrinsic pathway which principally lead to the clotformation in vivo This behavior highlights that the intrinsicpathway has a secondary role in blood coagulation in vivocompared with the extrinsic pathway as also evidenced byvarious experimental observations [48]
313 Factors VIII IX and II The next set of PSA-1D wasrealized by varying the initial concentrations of factors VIIIIX and II These factors were selected since both an excess
and a deficiency of their concentrations lead to diseasesrelated to blood clotting
The PSA-1D over factor VIII (Figure 9) shows thatincreasing the initial concentration of this factor resultsin decreasing the CT suggesting the possible presence ofhypercoagulable states in these perturbed conditions As amatter of fact high levels of factor VIII cause an increasedrisk of deep vein thrombosis and pulmonary embolism[49] On the other hand individuals with less than 1of the average concentration of factor VIII show a severehaemophilia A characterized by higher CT (Figure 9) andrequire infusions of plasma containing the deficient factorotherwise frequent spontaneous bleeding would occur [50]When the concentration of this factor is between 5 and 30of the average concentration individuals still risk bleeding incase of trauma [50]
14 BioMed Research International
250
300
350
400
450
500
550
600
650
700
Clot
ting
time (
s)
Factor IX (M)
1eminus
14
1eminus
12
1eminus
10
1eminus
8
1eminus
6
1eminus
4
1eminus
2
1e0
Figure 10 Clotting time at different initial concentrations of factor IXThe reference value is 900sdot10minus8M the sweep range is [900sdot10minus14 900sdot10minus2]
0
100
200
300
400
500
600
700
Clot
ting
time (
s)
Factor II (M)
1eminus
12
1eminus
10
1eminus
8
1eminus
6
1eminus
4
1eminus
2
1e0
1e2
Figure 11 Clotting time at different initial concentrations of thrombinThe reference value is 140sdot10minus6M the sweep range is [140sdot10minus12 140sdot100]
The PSA-1D over factor IX (Figure 10) shows only aslight decrease of the CT as the initial concentration offIX increases this is in contrast to recent studies thatdemonstrated how the excess of factor IX leads to anincreased risk of deep vein thrombosis [51] This result canbe explained by considering that (i) a high concentrationof factor IX is not sufficient to bring about coagulationproblems though when the concentration of other factorsis above the average value (yet not at pathological levels)prothrombotic states can be observed (ii) in this model weconsider average values as initial concentrations of factorshowever individuals are characterized by different (balanced)combinations of procoagulant and anticoagulant factor levelsthat altogether contribute to define a unique coagulationphenotype that reflects the developmental environmentalgenetic nutritional and pharmacological influences of eachindividual [52] On the contrary the lack of factor IX causes
haemophilia B characterized by higher CT with respect tothe reference value (Figure 10)
Furthermore by comparing the PSA-1D of factors VIIIand IX (Figures 9 and 10) it is clear that haemophilia A ismore serious than haemophilia B since the CT achieved inconditions of factor VIII deficiency is higher than the CTobtained in the case of factor IX deficiency
In both PSA-1D over factors VIII and IX we observedafter the initial decrease of the CT an unexpected increaseof the CT as the factor concentration increasesThis counter-intuitive behavior arises at very high concentrations of thesefactors (with respect to the average physiological levels) andto the best of our knowledge it was never observed in vivoNonetheless it would be interesting to verify by means of adhoc laboratory experiments if the model correctly describesthe behavior of the BCC even in these conditions or on thecontrary it is not predictive in these extreme situations
BioMed Research International 15
The PSA-1D over factor II (Figure 11) shows a dramaticdecrease of the CT as the initial concentration of this factorincreases (with respect to the average physiological level)This behavior resembles the effects of hypercoagulability (orthrombophilia) a disease caused by mutation G20210A inthe prothrombin gene [53] that causes an increase of theprothrombin level (factor II) in the blood flow resulting inan excessive formation of the active form of this factor thusheightening venous thrombosis risks [54] Hypercoagulabil-ity is usually treated with warfarin therapy or with otheranticoagulants with a similar effect These drugs decrease thecapacity of coagulation factors to become active preventingthe formation of unwanted thrombi
On the other hand when the initial concentration offactor II is low we achieved the effects of prothrombindeficiency a rare autosomal recessive disease that causes atendency to severe bleeding [29 55] As shown in Figure 11 aconcentration equal to 10of the physiological value of factorII (ie 14 sdot 10minus6 M) leads to clotting effects similar to severehaemophilia A
32 PSA-2D of the BCCModel We present here the results ofthe PSA-2D on the BCCmodel where couples of parameterswere varied to analyze the possible effects arising from thecombined perturbation of their values
321 Reactions 11990327
and 11990358 Figure 12 shows the effect of
the simultaneous variation of constants 11989627
and 11989658
(overthe same sweep ranges considered in the two PSA-1DSection 31) This result remarks that reaction 119903
27 involved
in the common pathway has a stronger influence on theBCC and that there is a synergic interplay between thesetwo reactions In particular when the value of 119896
27is low and
the value of 11989658
is high the CT is higher than the valuesachieved when only a single constant is changed because inthis condition both the intrinsic and the common pathwaysare simultaneously inhibited
322 Factor VIII Factor IX and Tissue Factor In the last twoPSA-2Dwe varied the initial concentration of factor VIII andTissue Factor and the initial concentration of factor IX andTissue Factor respectively
The initial concentrations of factors VIII and IX were var-ied over four orders of magnitude using their physiologicalvalues as upper limit for the sweep ranges the concentrationof Tissue Factor was varied over four orders of magnitudetwo above and two below its reference value (see Table 2)The rationale behind this choice is to observe how the BCCmodel in conditions corresponding to different states ofhaemophilia (obtained by decreasing the concentrations offactors VIII and IX) behaves with different initial concentra-tions of the Tissue Factor which is the upstream factor of theextrinsic pathway that is the most important element of theBCC
The results of these PSA-2D show that with respect tothe condition of haemophilia B in the case of haemophiliaA the amount of Tissue Factor (below its reference value)has a negligible influence on the CT as indicated by the
presence of a plateau in Figure 13 on the contrary concerninghaemophilia B a deficiency of Tissue Factor leads to anincrease of the CT especially when factor IX is present in lowconcentrations (Figure 14)
The different results achieved in the two PSA-2D are dueto the presence in the BCC model of a direct interactionbetweenTissue Factor and factor IX bymeans of theTF-fVIIacomplex (see reactions 119903
13 119903
15in Table 1) Indeed the lack
of Tissue Factor directly affects the concentration of activefactor IX which results in a strong alteration of the CT withrespect to physiological conditions
33 CPU versus GPU Performance Comparison In orderto show the relevant speedup achieved by coagSODA wepresent here the comparison of the computational effortrequired by GPU and CPU for the simulation of the BCCmodel The performances of our GPU simulator were com-pared with those obtained using the LSODA algorithmimplemented in the software COPASI [56] executing onthe CPU the same set of simulations that were run on theGPU COPASI is single-threaded and does not exploit thephysical and logical cores of the CPU therefore it representsa good benchmark as a single-node CPU-bound simulator ofbiological systems
In all simulations executed on both GPU and CPU westored 100 samples uniformly distributed in the time intervalconsidered for each simulation that is [0 700] seconds of thedynamics of all chemical species involved in the BCCmodelThe settings for the LSODA algorithm were the followingrelative error 1 sdot 10minus7 absolute error 1 sdot 10minus13 and maximumnumber of internal steps set to 20000
331 Benchmark Details The GPU used for the simulationsis a Nvidia Tesla K20c equipped with 2496 cores organized in13 streamingmultiprocessors GPU clock 706MHz and 5GBofDDR5RAM In all tests coagSODAwas compiled and exe-cuted with version 55 of CUDA libraries Even though thisGPU has compute capability 35 and is based on the GK110Kepler architecture currently coagSODA does not exploitany of the new features (eg Dynamic Parallelism Hyper-QRemote DMA) with the exception of the new ISA encodingwhich allows threads to exploit an increased number ofregisters (255 instead of 63) reducing register spilling intoglobal memory and increasing performances [57] Moreoveras described in Section 22 the Kepler architecture offers thepossibility to reconfigure the 64KB on-chip cache balancingbetween L1 cache and shared memory Since coagSODAexploits the shared memory to reduce the latencies duringthe access to the concentration values of the BCC model weopted for the following configuration 48KB for the sharedmemory 16 KB for the L1 cache
The performance of the GPU was compared against apersonal computer equippedwith a dual-core CPU Intel Corei5 frequency 23 GHz 4GB of DDR3 RAM running theoperating systemMacOSXLion 1075The software used as acoagSODA-equivalent single-threaded CPU implementationis COPASI version 48 (build 35)
16 BioMed Research International
280
300
320
340
360
380
400
420
440
1e41e6
1e91e11 1e5
1e7
1e9
1e11
280 320 360 400 440
Clot
ting
time (
s)
k 58(M
minus1 sminus1 )
k27 (Mminus1sminus1)
Figure 12 Clotting time according to PSA-2D over reaction constants 11989627and 119896
58 The sweep ranges are [2 sdot 104 2 sdot 1010] and [1 sdot 105 2 sdot 1011]
respectively
100
200
300
400
Tissue factor (M) Factor VIII
(M)
Clot
ting
time (
s)
100
150
200
250
300
350
400
450
1eminus131eminus11
1eminus91eminus14
1eminus12
1eminus10
Figure 13 Clotting time at different initial concentrations of Tissue Factor and factor VIII The sweep ranges are [500 sdot 10minus14 500 sdot 10minus10]and [700 sdot 10minus14 500 sdot 10minus10] respectively
A direct comparison of the performances between thesetwo different architectures and implementations is not aneasy task since CPUs are optimized for single-thread exe-cution and exploit a large number of optimizations (eghigher clock frequency instruction caching pipelining outof order execution and branch prediction) whereas GPUsare optimized for graphics processing and parallel exe-cution of identical code relying only on multilevel datacaching For this reason in this paper we propose anempirical comparison of the performances based on therunning times of identical batches of simulations of the BCCmodel
332 Benchmark Results In Figure 15 we show the compar-ison of the running times required to run several batchesof simulations executed to carry out the PSA-1D over thereaction constant 119896
27 The choice of comparing the perfor-
mances of the GPU and CPU implementations by executingbatches of simulations that are related to a PSA instead ofrunning 119899 independent but identical simulations (ie allcharacterized by the same parameterization of the model)is due to the fact that these results represent a more realisticscenario in the computational analysis of biological systemswhereby it is useful to investigate large search spaces ofparameters corresponding to different perturbed conditionsof themodel [26]Moreover for the evaluation of the running
time the execution of a batch of 119899 identical deterministicsimulations would be futile The figure clearly shows thatcoagSODA always performs better than the CPU counter-part In particular while the CPU performance increaseslinearly with the number of simulations the running timesare strongly reduced on the GPU in this case the overallrunning time roughly corresponds to the time required forthe execution of the slowest simulations This is due to thefact that different parameterizations may require differenttime steps for LSODA to converge and the execution ofa block terminates as long as all the threads it containsterminate In turn the execution of a kernel terminates whenall blocks terminate for this reason a single simulationwhose dynamics requires more steps than the others mayaffect the overall running time
In Table 3 we report the running times of all batchesof simulations along with the speedup achieved on theGPU In particular these results highlight that the advantageof exploiting the GPUs for the simulation of the BCCmodel becomes more evident as the number of simula-tions increases with a 181times speedup when a PSA with10000 different parameterizations is executed Therefore theGPU accelerated analysis of the BCC model with coag-SODA represents a novel relevant computational means toinvestigate the behavior of this complex biological systemunder nonphysiological conditions and could be exploited
BioMed Research International 17
0
200
400
600
Tissue factor (M) Factor IX (M
)
Clot
ting
time (
s)
0
100
200
300
400
500
600
700
1eminus12
1eminus10
1eminus8
1eminus131eminus11
1eminus9
Figure 14 Clotting time at different initial concentrations of Tissue Factor and factor IXThe sweep ranges are [500 sdot 10minus14 500 sdot 10minus10] and[900 sdot 10
minus12 900 sdot 10
minus8] respectively
5 10 50 100 500 1000 5000 10000
Runn
ing
time (
s)
Number of simulations
CPUGPU
1e2
1e3
1e4
1e5
1e6
1e7
Figure 15 Comparison of the computational time required to execute an increasing number of simulations of the BCCmodel on CPU (IntelCore i5 23 GHz) and GPU (Nvidia Tesla K20c GPU clock 706MHz) The computational time is expressed in seconds The time valuesrelated to 1000 5000 and 10000 simulations on the CPU were estimated by regression (see also Table 3)
Table 3 CPU versus GPU performance comparison
Number ofsimulations CPU time (sec) GPU time (sec) Speedup
5 3808 1278 298times10 10604 2329 456times50 51704 8058 642times100 116528 10977 1061times500 536058 17394 3081times1000 1073582lowast 19982 5373times5000 5363507lowast 30960 1732times10000 10724975lowast 58951 1819timeslowastValues estimated by regression
to efficiently determine the response of the BCC to differenttherapeutical drugs
4 Conclusions
Thanks to their high-performance computing capabilitiesand the very low costs GPUs nowadays represent a suitable
technology for the development and the application of par-allel computational methods for in silico analysis of complexbiological systems However the implementation of efficientcomputational tools able to fully exploit the large potentialityof GPUs is still challenging since good programming skillsare required to implement GPU-based algorithmic methodsand to handle specific features as an optimal usage ofmemoryor the communication bandwidth between GPU and CPUMoreover algorithms cannot be directly ported on the GPUbecause of the limited programming capabilities allowed byGPUkernels as amatter of fact they need to be redesigned tomeet the requirements of the underlying SIMD architecture
In this work we presented coagSODA a GPU-poweredsimulator specifically developed to carry out fast parallelsimulations of the BCC model coagSODA was designed tooffer a black-box solution usable by any final user in aneasy way It relies on cupSODA [24] a numerical integratorfor ODEs that we previously implemented for the GPUarchitecture based on the LSODA algorithm and capableof automatically translating a reaction-based model into aset of coupled ODEs In addition to mass-action kinet-ics coagSODA implements specific functions to compute
18 BioMed Research International
the kinetics of Hill function based reactions such as thoseinvolved in the platelets activity of the BCC model coag-SODA exploits the massive parallel capabilities of modernGPUs and our results demonstrated that it can achieve arelevant reduction of the computational time required toexecutemany concurrent and independent simulations of theBCC dynamics The mutual independence of the simulationsallows fully exploiting the underlying SIMD architecturemoreover coagSODA benefits from an additional speedupthanks to our choice of storing the state of the systeminto the low-latency shared memory (a solution that wasalready implemented in cupSODA) Since the BCC modelis large (96 reactions 71 chemical species) a large amountof shared memory was assigned to each thread stronglyreducing the theoretical occupancy of the GPU that is theratio of active warps with respect to the maximum numberof warps supported by each streaming multiprocessor of theGPU However the results of the analysis presented in thiswork performed on the BCC model show that coagSODAachieves a relevant boost with respect to a reference CPUimplementation For instance in the case of 10000 simula-tions we achieved a noticeable 181times speedup Interestinglythe performances of coagSODA are better than COPASI evenfor small batches of simulations These results indicate thatfor biological models consisting in many reactions and manyspecies our GPU implementation of LSODA becomes moreprofitable than the CPU counterpart as the number of con-current simulations increases making it suitable especiallywhen performing demanding computational analysis suchas for example parameter sweep parameter estimation orsensitivity analysis
As a test case in this paper we presented several parame-ter sweep analyses over reaction constants and initial concen-trations of factors involved in the BCC model Other com-putational analyses on mathematical models of this pathwaywere previously presented For instance in [58] a parameterestimation of the reaction constants of different modelsinvolving the activation of factor X factor V prothrombinand the inactivation of factors was performed The aim ofthis analysis was to discriminate between different modelsto unravel the mechanisms on the basis of the BCC Similaranalyses were executed on a model describing thrombingeneration in plasma since the reaction mechanism thereaction constants and initial concentrations were unknown[59] In [34] a sensitivity analysis of a model consisting of44 species over 34 chemical species was presented reactionconstants were varied between 10 and 1000 of theirreference value to the aim of identifying the most influentialfactors of the BCC
All these analyses were performed bymeans of sequentialsimulations of the models under investigation thereforein general only small batches of simulations could be run(for instance in [34] only 836 simulations were executed toexecute the sensitivity analysis of the model) On the otherhand in this paper we presented the results of different PSAefficiently executed bymeans of coagSODA overall providinguseful information regarding unknownparameters and inter-esting insights into the functioning of the BCC A thoroughsensitivity analysis of the whole BCC model consisting in
around 5 sdot 105 simulations is currently in progress on GPUsby our group exploiting the great computational efficiency ofcoagSODA
The results of our computational analyses should benow validated by means of specifically designed laboratoryexperiments In particular we identified a plausible valuefor the constant of the reaction describing the autoactivationof factor XII (since no reference values can be found inliterature) moreover the results of the PSA over factor IXsuggest that a deficiency of this factor is not enough to causesevere bleeding disorders as haemophilia B but an alterationof other factors seems to be necessary for the occurrence ofsuch condition (eg the lack of Tissue Factor as suggestedby the PSA-2D over Tissue Factor and factor IX) Finallythe PSA over factors VIII and IX showed in a situationcharacterized by very high concentrations of these factorsa counterintuitive behavior in which the clotting time isincreased with respect to the value obtained in physiologicalconditions Indeed it would be interesting to design ad hoclaboratory experiments to verify if the BCCmodel is actuallypredictive in such extreme situations
The coagSODA software and the SBML version of theBCC model used in this work are available from the authorsupon request
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This work is partially supported by SYSBIO Centre forSystems Biology
References
[1] B B Aldridge JM BurkeDA Lauffenburger andP K SorgerldquoPhysicochemicalmodelling of cell signalling pathwaysrdquoNatureCell Biology vol 8 no 11 pp 1195ndash1203 2006
[2] I-C Chou and E O Voit ldquoRecent developments in parameterestimation and structure identification of biochemical andgenomic systemsrdquoMathematical Biosciences vol 219 no 2 pp57ndash83 2009
[3] D Besozzi P Cazzaniga D Pescini G Mauri S Colomboand EMartegani ldquoThe role of feedback control mechanisms onthe establishment of oscillatory regimes in the RascAMPPKApathway in S cerevisiaerdquo EURASIP Journal on Bioinformaticsand Systems Biology vol 2012 no 10 2012
[4] A Saltelli M Ratto T Andres et al Global Sensitivity AnalysisThe Primer Wiley-Interscience 2008
[5] A Raue C Kreutz T Maiwald et al ldquoStructural and practicalidentifiability analysis of partially observed dynamical modelsby exploiting the profile likelihoodrdquo Bioinformatics vol 25 no15 pp 1923ndash1929 2009
[6] A Drager M Kronfeld M J Ziller et al ldquoModeling metabolicnetworks in C glutamicum a comparison of rate laws incombination with various parameter optimization strategiesrdquoBMC Systems Biology vol 3 article 5 2009
BioMed Research International 19
[7] D Besozzi P Cazzaniga G Mauri D Pescini and L Van-neschi ldquoA comparison of genetic algorithms and particle swarmoptimization for parameter estimation in stochastic biochem-ical systemsrdquo in Proceedings of the 7th European ConferenceEvolutionary Computation Machine Learning and Data Miningin Bioinformatics (EvoBIO rsquo09) C Pizzuti M Ritchie and MGiacobini Eds vol 5483 of Lecture Notes in Computer Sciencepp 116ndash127 Springer 2009
[8] M S Nobile D Besozzi P Cazzaniga GMauri andD PescinildquocupSODA aGPU-basedmulti-swarmPSOmethod for param-eter estimation in stochastic biological systems exploitingdiscrete-time target seriesrdquo in Proceedings of the 10th EuropeanConference Evolutionary Computation Machine Learning andData Mining in Bioinformatics (EvoBIO rsquo12) M Giacobini LVanneschi and W Bush Eds vol 7246 of Lecture Notes inComputer Science pp 74ndash85 Springer
[9] M S Nobile D Besozzi P Cazzaniga D Pescini and GMaurildquoReverse engineering of kinetic reaction networks by meansof Cartesian Genetic Programming and Particle Swarm Opti-mizationrdquo in Proceedings of the IEEE Congress on EvolutionaryComputation (CEC rsquo13) pp 1594ndash1601 June 2013
[10] M S Nobile ldquoEvolutionary inference of biochemical inter-action networks accelerated on Graphics Processing Unitsrdquoin Proceedings of the 11th International Conference on HighPerformance Computingamp Simulation 2013 (HPCS rsquo13) pp 668ndash670 Helsinki Finland 2013
[11] K Kentzoglanakis and M Poole ldquoA swarm intelligence frame-work for reconstructing gene networks searching for bio-logically plausible architecturesrdquo IEEEACM Transactions onComputational Biology and Bioinformatics vol 9 no 2 pp 358ndash371 2012
[12] J R Koza W Mydlowec G Lanza Y Jessen and M A KeaneldquoAutomatic computational discovery of chemical reaction net-works using genetic programmingrdquo inComputationalDiscoveryof Scientific Knowledge S Dzeroski and L Todorovski Edsvol 4660 of Lecture Notes in Computer Science pp 205ndash227Springer 2007
[13] S Ando E Sakamoto and H Iba ldquoEvolutionary modeling andinference of gene networkrdquo Information Sciences vol 145 no3-4 pp 237ndash259 2002
[14] L Dematte and D Prandi ldquoGPU computing for systemsbiologyrdquo Briefings in Bioinformatics vol 11 no 3 Article IDbbq006 pp 323ndash333 2010
[15] J L Payne N A Sinnott-Armstrong and J HMoore ldquoExploit-ing graphics processing units for computational biology andbioinformaticsrdquo Interdisciplinary Sciences Computational LifeSciences vol 2 no 3 pp 213ndash220 2010
[16] M J Harvey and G De Fabritiis ldquoA survey of computationalmolecular science using graphics processing unitsrdquoWiley Inter-disciplinary Reviews Computational Molecular Science vol 2no 5 pp 734ndash742 2012
[17] Y Zhou J Liepe X Sheng M P H Stumpf and C BarnesldquoGPU accelerated biochemical network simulationrdquo Bioinfor-matics vol 27 no 6 Article ID btr015 pp 874ndash876 2011
[18] M Vigelius A Lane and B Meyer ldquoAccelerating reaction-diffusion simulations with general-purpose graphics processingunitsrdquo Bioinformatics vol 27 no 2 Article ID btq622 pp 288ndash290 2011
[19] D J Wilkinson ldquoStochastic modelling for quantitative descrip-tion of heterogeneous biological systemsrdquo Nature ReviewsGenetics vol 10 no 2 pp 122ndash133 2009
[20] J C Butcher ldquoTrees and numerical methods for ordinarydifferential equationsrdquo Numerical Algorithms vol 53 no 2-3pp 153ndash170 2010
[21] D T Gillespie ldquoStochastic simulation of chemical kineticsrdquoAnnual Review of Physical Chemistry vol 58 pp 35ndash55 2007
[22] L Petzold ldquoAutomatic selection of methods for solving stiffand nonstiff systems of ordinary differential equationsrdquo SIAMJournal of Scientific and Statistical Computing vol 4 no 1 pp136ndash148 1983
[23] Y Cao D T Gillespie and L R Petzold ldquoEfficient step sizeselection for the tau-leaping simulation methodrdquo The Journalof Chemical Physics vol 124 no 4 2006
[24] M S Nobile D Besozzi P Cazzaniga GMauri andD PescinildquocupSODA a CUDA-powered simulator of mass-action kinet-icsrdquo in Proceedings of the 12th International Conference onParallel Computing Technologies (PaCT rsquo13) V Malyshkin Edvol 7979 of Lecture Notes in Computer Science pp 344ndash357
[25] M S Nobile P Cazzaniga D Besozzi and G Mauri ldquoGPU-accelerated simulations of mass-action kinetics models withcupSODArdquoThe Journal of Supercomputing 2014
[26] M S Nobile P Cazzaniga D Besozzi D Pescini and GMauri ldquocuTauLeaping a GPU-powered tau-leaping stochasticsimulator for massive parallel analyses of biological systemsrdquoPLoS ONE vol 9 no 3 2014
[27] K Rogers Blood Physiology and CirculationTheHuman BodyBritannica Educational Publishing 2010
[28] A D Michelson Platelets Elsevier 2nd edition 2007[29] R W Colman J Hirsh V J Marder A W Clowes and J
N George Hemostasis and Thrombosis Basic Principles andClinical Practice Lippincott Williams ampWilkins 2006
[30] M S Chatterjee W S Denney H Jing and S L DiamondldquoSystems biology of coagulation initiation Kinetics of thrombingeneration in resting and activated human bloodrdquo PLoS Com-putational Biology vol 6 no 9 Article ID e1000950 2010
[31] M Bellini D Besozzi P Cazzaniga GMauri andM S NobileldquoSimulation and analysis of the blood coagulation cascadeaccelerated on GPUrdquo in Proceedings of the 22nd EuromicroInternational Conference on Parallel Distributed and Network-Based Processing (PDP) pp 590ndash593 IEEE Computer SocietyConference Publishing Services 2014
[32] T Wajima G K Isbister and S B Duffull ldquoA comprehensivemodel for the humoral coagulation network in humansrdquo Clini-cal Pharmacology and Therapeutics vol 86 no 3 pp 290ndash2982009
[33] K Lo W S Denney and S L Diamond ldquoStochastic modelingof blood coagulation initiationrdquo Pathophysiology of Haemostasisand Thrombosis vol 34 no 2-3 pp 80ndash90 2006
[34] C M Danforth T Orfeo K G Mann K E Brummel-Ziedins and S J Everse ldquoThe impact of uncertainty in a bloodcoagulation modelrdquo Mathematical Medicine and Biology vol26 no 4 Article ID dqp011 pp 323ndash336 2009
[35] S D Lewis P P Shields and J A Shafer ldquoCharacterizationof the kinetic pathway for liberation of fibrinopeptides duringassembly of fibrinrdquoThe Journal of Biological Chemistry vol 260no 18 pp 10192ndash10199 1985
[36] M F Hockin K C Jones S J Everse and K G Mann ldquoAmodel for the stoichiometric regulation of blood coagulationrdquoThe Journal of Biological Chemistry vol 277 no 21 pp 18322ndash18333 2002
[37] K Brummel-Ziedins C YVossen F R Rosendaal KUmezakiand K G Mann ldquoThe plasma hemostatic proteome thrombin
20 BioMed Research International
generation in healthy individualsrdquo Journal of Thrombosis andHaemostasis vol 3 no 7 pp 1472ndash1481 2005
[38] K E Brummel-Ziedins C Y Vossen S Butenas K G Mannand F R Rosendaal ldquoThrombin generation profiles in deepvenous thrombosisrdquo Journal of Thrombosis and Haemostasisvol 3 no 11 pp 2497ndash2505 2005
[39] S D Bungay P A Gentry and R D Gentry ldquoA mathematicalmodel of lipid-mediated thrombin generationrdquo MathematicalMedicine and Biology vol 20 no 1 pp 105ndash129 2003
[40] S Butenas T Orfeo M T Gissel K E Brummel and K GMann ldquoThe significance of circulating factor IXa in bloodrdquoThe Journal of Biological Chemistry vol 279 no 22 pp 22875ndash22882 2004
[41] D Voet and J G Voet Biochemistry Wiley 2011[42] A C Guyton and J E Hall Textbook of Medical Physiology
Saunders 2010[43] M F M James and A M Roche ldquoDose-response relationship
between plasma ionized calcium concentration and thrombe-lastographyrdquo Journal of Cardiothoracic and Vascular Anesthesiavol 18 no 5 pp 581ndash586 2004
[44] D L Nelson and M M Cox Lehninger Principles of Biochem-istry W H Freeman Company 2004
[45] R M Farber ldquoTopical perspective on massive threading andparallelismrdquo Journal of Molecular Graphics and Modelling vol30 pp 82ndash89 2011
[46] OWolkenhauerMUllahWKolch andK-HCho ldquoModelingand simulation of intracellular dynamics choosing an appropri-ate frameworkrdquo IEEETransactions onNanobioscience vol 3 no3 pp 200ndash207 2004
[47] B H Estridge A P Reynolds and N J Walters Basic MedicalLaboratory Techniques Cengage Learning 4th edition 2000
[48] T Renne A H Schmaier K F Nickel M Blomback and CMaas ldquoIn vivo roles of factor XIIrdquo Blood vol 120 no 22 pp4296ndash4303 2012
[49] P V Jenkins O Rawley O P Smith and J S ODonnellldquoElevated factor VIII levels and risk of venous thrombosisrdquoBritish Journal of Haematology vol 157 no 6 pp 653ndash663 2012
[50] C E Walsh and K M Batt ldquoHemophilia clinical gene therapybrief reviewrdquo Translational Research vol 161 no 4 pp 307ndash3122013
[51] A Van Hylckama Vlieg I K Van Der Linden R M Bertinaand F R Rosendaal ldquoHigh levels of factor IX increase the risk ofvenous thrombosisrdquo Blood vol 95 no 12 pp 3678ndash3682 2000
[52] K Brummel-Ziedins ldquoModels for thrombin generation and riskof diseaserdquo Journal ofThrombosis and Haemostasis vol 11 no 1pp 212ndash223 2013
[53] U Seligsohn and A Lubetsky ldquoGenetic susceptibility to venousthrombosisrdquoTheNew England Journal of Medicine vol 344 no16 pp 1222ndash1231 2001
[54] I Martinelli P Bucciarelli and P M Mannucci ldquoThromboticrisk factors basic pathophysiologyrdquo Critical Care Medicine vol38 no 2 pp S3ndashS9 2010
[55] K Brummel-Ziedins A Undas T Orfeo et al ldquoThrombingeneration in acute coronary syndrome and stable coronaryartery disease dependence on plasma factor compositionrdquoJournal ofThrombosis andHaemostasis vol 6 no 1 pp 104ndash1102008
[56] S Hoops R Gauges C Lee et al ldquoCOPASImdasha complexpathway simulatorrdquo Bioinformatics vol 22 no 24 pp 3067ndash3074 2006
[57] Nvidia NVIDIArsquos Next Generation CUDA Compute Architec-ture Kepler GK110 2012
[58] W Stortelder W Hemker and H Hemker ldquoMathematicalmodelling in blood coagulation simulation and parameterestimationrdquoModelling Analysis and Simulation vol 20 pp 1ndash111997
[59] S M C Orfao G Jank and K Mottaghy ldquoA new mathematicalapproach to modelling thrombin generationrdquo InternationalJournal of Artificial Organs vol 29 no 7 pp 703ndash708 2006
[60] Nvidia Nvidia CUDA C Programming Guide v50 2012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Anatomy Research International
PeptidesInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
International Journal of
Volume 2014
Zoology
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Molecular Biology International
GenomicsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
BioinformaticsAdvances in
Marine BiologyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Signal TransductionJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
BioMed Research International
Evolutionary BiologyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Biochemistry Research International
ArchaeaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Genetics Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in
Virolog y
Hindawi Publishing Corporationhttpwwwhindawicom
Nucleic AcidsJournal of
Volume 2014
Stem CellsInternational
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Enzyme Research
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Microbiology
BioMed Research International 9
Host
Kernel 1
Kernel 2
Device
Grid 1
Grid 2
Block(0 0)
Block(1 0)
Block(2 0)
Block(0 1)
Block(1 1)
Block(2 1)
Block (1 1)
Thread(0 0)
Thread(1 0)
Thread(2 0)
Thread(3 0)
Thread(4 0)
Thread(0 1)
Thread(1 1)
Thread(2 1)
Thread(3 1)
Thread(4 1)
Thread(0 2)
Thread(1 2)
Thread(2 2)
Thread(3 2)
Thread(4 2)
Global memory
Texture memory
Constant memory
Shared memory Shared memory
Block (0 0) Block (1 0)
Localmemory
Localmemory
Localmemory
Localmemory
Thread(0 0)
Thread(1 0)
Registers Registers Registers Registers
Thread(0 0)
Thread(1 0)
Grid
Figure 3 Schematic description of CUDArsquos architecture in terms of threads and memory hierarchy Left Side Threads organization a singlekernel is launched from the host (the CPU) and is executed in multiple threads on the device (the GPU) Threads can be organized in three-dimensional structures named blocks which can be in turn organized in three-dimensional grids The dimensions of blocks and grids areexplicitly defined by the programmer Right Side Memory hierarchy threads can access data from many different memories with differentscopes Registers and local memories are private for each thread Shared memory lets threads belonging to the same block communicate andhas low access latency All threads can access the global memory which suffers high latencies but it is cached since the introduction of theFermi architecture Texture and constant memory can be read from any thread and are equipped with a cache as well in this work we exploitthe constant memory Figures are taken from Nvidiarsquos CUDA programming guide [60]
devices Threads within a block are executed in groups of 32threads named warps
The GPU is equipped with different kinds of memoryIn this work we exploit the global memory (accessible fromall threads) the shared memory (accessible from threadsof the same block) the local memory (registers and arraysaccessible from owner thread) and the constant memory(cached and not modifiable) A schematic representation ofthis memory hierarchy is shown in Figure 3 right side Toachieve the best performances the sharedmemory should beexploited as much as possible however it is very limited (ie49152 bytes for each multiprocessor since the introductionof the Fermi architecture) and it introduces restrictions onthe blocksrsquo size On the other hand the global memory isvery large (thousands ofMBs) but suffers from high latenciesA solution to this problem was implemented on the Fermiarchitecture where the global memory is equipped with aL2 cache This architecture also introduced the possibility tobalance 64KB of fast on-chip memory between the sharedmemory and L1 cache using two possible configurations48KB for the shared memory and 16KB for L1 cache or16 KB for the shared memory and 48 for L1 cacheThe Keplerarchitecture used in this paper allows a third and perfectly
balanced configuration where shared memory and L1 cacheobtain the same amount of memory (32KB) See Figure 4 fora schematization of the memory architecture
The systematic analysis of models of biological systemsoften consists in the execution of large batches of simulationsOne of the standard analyses that can be executed on suchkind of models regards an intensive search within the param-eters space which requires large numbers of independentsimulations In order to reduce the computational burdenwe previously implemented on the CUDA architecture oneof the most efficient numerical integration algorithms forODEs LSODA that is able to automatically recognize stiffand nonstiff systems and to dynamically select between themost appropriate integration procedure (ie Adams methodin the absence of stiffness and the Backward DifferentiationFormulae otherwise) [22] This GPU-powered tool is calledcupSODA [24 25] and exploits CUDArsquos massive parallelismto execute different and independent simulations in eachthread thus reducing the computational time required by astandard CPU counterpart of LSODA
Besides being very efficient for the simulation of manyindependent simulations cupSODA is also user-friendlyLSODA was originally designed to solve ODEs systems
10 BioMed Research International
Thread
Shared memory L1 cache Read-onlydata cache
L2 cache
DRAM
Figure 4 Schematic description of memory hierarchies in Fermi and Kepler architectures GPUs relying on these architectures are equippedwith a two-level data cache and a read-only data cache Shared memory and L1 cache share the same on-chip 64KB memory banks theamount of memory can be reconfigured by the user according to the specific needs of the application Figure taken from Nvidiarsquos KeplerGK110 whitepaper [57]
written in the canonical form but the user is supposed tospecify the system of ODEs by implementing a custom Cfunction that is passed to the algorithmmoreover in order tospeed up the computation when dealing with stiff systems inLSODA the Jacobianmatrix associatedwith theODEs systemmust be implemented as a custom C function as well On thecontrary cupSODA was conceived as a black-box simulatorthat can be easily used without any programming skillscupSODA consists in a tool to automatically convert a genericmechanistic reaction-based model of a biological system intothe corresponding set of ODEs to comply with the mass-action kinetics [46] and to directly encode the obtainedsystem along with the corresponding Jacobian matrix as Carrays
This fully automaticmethodology can be exploited for thesimulation of models whose chemical kinetics is based on themass-action law The BCC model described in Section 21though includes a set 119878
120576of reactions that do not follow
the mass-action kinetics [30] the platelets activity is notexplicitly modeled by biochemical reactions but it is realizedby modulating the rate of the dissociation of the complexesformed on a plateletrsquos surface by means of the variable120576 which is calculated with a special equation during theintegration steps For this reason cupSODA was used asa starting point for the development of a new tool ableto compute at run-time the specific kinetics of this set ofreactions
This new CUDA-powered simulation tool named coag-SODA is specifically tailored for the simulation of the BCC
model developed in [30] In particular coagSODA realizesthe run-time calculation of the 120576 value required to correctlysimulate the activity of platelets which is determined accord-ing to the equations described in Section 21 The plateletsactivation state 120576 is calculated at each time instant by solvingthe following differential equation
119889120576
119889119905
= 119896 (120576max minus 120576) (5)
where the constant 119896 is inversely proportional to the timescale of platelets activation and is set to 0005 This isconsistent with the fact that platelets do not instantly achievetheir maximum attainable activation state (120576max) but theyreach it on a physiologically relevant timescale [30]
Dealing with the 14 reactions in the set 119878120576that are
influenced by the Hill functionH (see Section 21) the valueof [fIIalowast(119905)]must be stored on the GPU because during eachintegration step coagSODA recalculates (4) which exploitsthe value [fIIalowast(119905)] to determine a new 120576 value Since CUDArsquosarchitecture does not offer static variables the informationfor each thread has to be memorized in the global memoryThe accesses to the global memory and the computationalcosts due to these additional calculations slow down theintegration process with respect to the integration of ODEsperformed according to strictly mass-action based kinetics(as in the cupSODA simulator) Nevertheless in Section 33we show that our parallel implementation largely outper-forms a sequential counterpart of the LSODA algorithm
BioMed Research International 11
0 100 200 300 400 500 600 700
Fibr
inog
en co
ncen
trat
ion
(M)
Time (s)
30 of fibrinogen
0
1eminus6
2eminus6
3eminus6
4eminus6
5eminus6
6eminus6
7eminus6
8eminus6
9eminus6
Figure 5 Dynamics of fibrinogen in physiological condition The clotting time is defined as the time necessary to convert the 70 of thefibrinogen into fibrin
As in the case of cupSODA [24 25] coagSODA exploitsthe shared memory to improve performances by storing thecurrent state and time of the simulations in these low-latencymemory banks Despite the improvement of performancesensured by this solution it strongly affects the occupancyof GPUrsquos multiprocessors and therefore it represents thelimiting factor for the number of blocks that can be executedsimultaneously as a matter of fact coagSODA is limited to2 blocks per streaming multiprocessor on GPUs based onthe Fermi architecture with a reduced exploitation of theGPU with respect to the theoretical 8 blocks allowed by thisarchitecture
3 Results and Discussion
In this section we discuss the results of the parameter sweepanalysis (PSA) carried out on the BCC model to the aimof investigating either the prolongation or the reduction ofthe clotting time in response to perturbed values of somereaction constants and of the initial concentration of somemolecular species chosen according to their meaning withinthe whole pathway PSA was performed by generating aset of different initial conditions corresponding to differentparameterizations of the model and then automaticallyexecuting the deterministic simulations with coagSODATheuse ofGPU technology is fundamental in this type of analysisespecially for large biological systems as the BCC is becauseit drastically reduces the computational time
The sweep analysis for single parameters (PSA-1D) wasperformed considering a logarithmic sampling of numericalvalues of each parameter under investigation (reaction con-stant or initial molecular concentration) within a specifiedrange with respect to its physiological reference value Thesweep analysis over pairs of parameters (PSA-2D) was per-formed by simultaneously varying the values of two param-eters within a specified range considering a logarithmicsampling on the resulting lattice The logarithmic samplingallows uniformly spanning different orders of magnitude ofthe parameters value using a reduced and fine-grained set
of samples therefore efficiently analyzing the response of thesystem in a broad range of conditions
To determine the response of the BCC to perturbed con-ditions we chose the clotting time (CT) as output of the PSAThe CT is defined as the time necessary to convert the 70of the fibrinogen into fibrin (see Figure 5) conventionallyassumed to correspond to the time required to form the clot[47] and it is generally used in laboratory tests formonitoringthe therapywith anticoagulant drugs According to themodeldefined in [30] the reference value of CT is around 300seconds in physiological conditions We investigate herethe response of the BCC by evaluating the CT in variousconditions corresponding to different values of the reactionconstants varying over six orders of magnitude with respectto their physiological values (ie three below and three abovethe reference values if not otherwise specified) as well asto different values of the initial molecular concentrationsand varying over twelve orders of magnitude with respect totheir physiological values (ie six below and six above thereference values if not otherwise specified)
The total number of parallel simulations executed tocarry out these analyses was 100 for PSA-1D over reactionconstants 200 for PSA-1D over initial concentrations and1600 for PSA-2D
Finally we present the comparison of the performanceof the CPU and GPU to run an increasing number ofsimulations of the BCC model to prove the efficiency ofcoagSODA
31 PSA-1D of the BCC Model
311 Reaction 11990344 Thefirst PSA was performed to determine
the value of the kinetic constant for the autoactivation offactor XII (reaction 119903
44in Table 1) which corresponds to an
upstream process in the intrinsic pathway This analysis wasmotivated by two considerations Firstly by using the fullparameterization given in [30] the action of the intrinsicpathway turns out to be fundamental for the BCC in vivowhich is in contrast to experimental observations which
12 BioMed Research International
324
326
328
330
332
334
336
338
340
Clot
ting
time (
s)
1eminus
10
1eminus
9
1eminus
8
1eminus
7
1eminus
6
1eminus
5
1eminus
4
1eminus
3
k44 (sminus1)
Figure 6 Clotting time according to PSA-1D over constant 11989644of reaction 119903
44 XII rarr XIIaThe reference value of 119896
44used in [30] is 5sdot10minus4sminus1
the sweep range is [5 sdot 10minus10 5 sdot 10minus4]
280
300
320
340
360
380
400
1e4 1e5 1e6 1e7 1e8 1e9 1e10 1e11
Clot
ting
time (
s)
k27 (Mminus1sminus1)
Figure 7 Clotting time according to PSA-1D over constant 11989627of reaction 119903
27 IIa + V rarr IIa + Va The reference value is 2 sdot 107Mminus1 sminus1 the
sweep range is [2 sdot 104 2 sdot 1010]
indicate that the extrinsic pathway is the main responsibleof clot formation Secondly in [30] all constants values havea reference to experimental measurements except for theconstant of this reaction (which corresponds to entry 29 inTable 1 in [30])
Figure 6 shows the results of this PSA-1D where 11989644was
varied over six orders of magnitude considering the valuegiven in [30] as the upper limit of the sweep interval We canobserve that by decreasing the value of 119896
44with respect to
the value considered in [30] the CT increases with respect toits reference value however for values of this constant lowerthan 100 sdot 10minus6 sminus1 the CT remains unaltered and this canbe intuitively explained by the fact that in this situation thefibrinogen is mainly activated by the extrinsic pathway
Consequently we assigned the value 700 sdot 10minus6 sminus1 to11989644 achieving a CT that is comparable to the experimental
observations of the BCC in vivo This new value was used inall PSA discussed in what follows
312 Reactions 11990327
and 11990358 In the next PSA we investigated
the effect of the perturbation of the kinetics of two pivotal
reactions of the BCC model Reaction 11990327 which describes
the catalytic activation of factor V by factor IIa was chosenbecause it represents the main positive feedback within thecommon pathway reaction 119903
58 which is involved in the
activation of factor XI by binding to factor IIa was chosenbecause it represents the main positive feedback in theintrinsic pathway (Figure 1) Moreover preliminary PSA overall reaction constants of the BCC model given in Table 1evidenced that these two reactions are among the mostrelevant steps of the coagulation network
The PSA-1D over 11989627(Figure 7) shows that the CT is very
sensitive to the perturbation of the rate of this reaction whenthe reference value of its constant (ie 119896
27= 2 sdot 10
7Mminus1 sminus1)is either increased or decreased in particular when 119896
27is
very low a plateau in CT is reached since the strength ofthe positive feedback exerted by factor IIa is largely reduceda condition where the contribution of the amplification ofthe hemostatic stimulus (due by the common pathway) tothe formation of the clot is basically not effective On theother side the PSA-1D over 119896
58(Figure 8) shows that while
a decrease of the constant with respect to its reference value
BioMed Research International 13
310
320
330
340
350
360
370
1e5 1e6 1e7 1e8 1e9 1e10 1e11
Clot
ting
time (
s)
k58 (Mminus1sminus1)
Figure 8 Clotting time according to PSA-1D over constant 11989658of reaction 119903
58 IIa + XI rarr IIa-XI The reference value is 1 sdot 108Mminus1 sminus1 the
sweep range is [1 sdot 105 2 sdot 1011]
220
240
260
280
300
320
340
360
380
400
Clot
ting
time (
s)
Factor VIII (M)
Seve
re h
aem
ophi
lia
Mild
hae
mop
hilia
Prot
hrom
botic
stat
e
1eminus
16
1eminus
14
1eminus
12
1eminus
10
1eminus
8
1eminus
6
1eminus
4
1eminus
2
Figure 9 Clotting time at different initial concentrations of factor VIII The reference value is 700 sdot 10minus10M (dashed black line) the sweeprange is [700 sdot 10minus16 700 sdot 10minus4] The blue area indicates a condition in which factor VIII concentration is less than 1 of its physiologicalconcentration corresponding to a situation of severe haemophilia the red area indicates a condition in which factor VIII concentration isbetween 1 and 30 of its physiological concentration corresponding to a situation of mild haemophilia the green area indicates a conditionin which factor VIII concentration is greater than 130 of its physiological concentration corresponding to a situation of hypercoagulability
(ie 11989658= 1 sdot 10
8Mminus1 sminus1) does not have any substantialeffect an increase of its value leads to a progressive increasein the CT This increase is due to the fact that factor IIa issequestered in the formation of a complex with factor XI andhence it is no longer available as a free component in bloodto participate in other reactions especially those reactionsof the extrinsic pathway which principally lead to the clotformation in vivo This behavior highlights that the intrinsicpathway has a secondary role in blood coagulation in vivocompared with the extrinsic pathway as also evidenced byvarious experimental observations [48]
313 Factors VIII IX and II The next set of PSA-1D wasrealized by varying the initial concentrations of factors VIIIIX and II These factors were selected since both an excess
and a deficiency of their concentrations lead to diseasesrelated to blood clotting
The PSA-1D over factor VIII (Figure 9) shows thatincreasing the initial concentration of this factor resultsin decreasing the CT suggesting the possible presence ofhypercoagulable states in these perturbed conditions As amatter of fact high levels of factor VIII cause an increasedrisk of deep vein thrombosis and pulmonary embolism[49] On the other hand individuals with less than 1of the average concentration of factor VIII show a severehaemophilia A characterized by higher CT (Figure 9) andrequire infusions of plasma containing the deficient factorotherwise frequent spontaneous bleeding would occur [50]When the concentration of this factor is between 5 and 30of the average concentration individuals still risk bleeding incase of trauma [50]
14 BioMed Research International
250
300
350
400
450
500
550
600
650
700
Clot
ting
time (
s)
Factor IX (M)
1eminus
14
1eminus
12
1eminus
10
1eminus
8
1eminus
6
1eminus
4
1eminus
2
1e0
Figure 10 Clotting time at different initial concentrations of factor IXThe reference value is 900sdot10minus8M the sweep range is [900sdot10minus14 900sdot10minus2]
0
100
200
300
400
500
600
700
Clot
ting
time (
s)
Factor II (M)
1eminus
12
1eminus
10
1eminus
8
1eminus
6
1eminus
4
1eminus
2
1e0
1e2
Figure 11 Clotting time at different initial concentrations of thrombinThe reference value is 140sdot10minus6M the sweep range is [140sdot10minus12 140sdot100]
The PSA-1D over factor IX (Figure 10) shows only aslight decrease of the CT as the initial concentration offIX increases this is in contrast to recent studies thatdemonstrated how the excess of factor IX leads to anincreased risk of deep vein thrombosis [51] This result canbe explained by considering that (i) a high concentrationof factor IX is not sufficient to bring about coagulationproblems though when the concentration of other factorsis above the average value (yet not at pathological levels)prothrombotic states can be observed (ii) in this model weconsider average values as initial concentrations of factorshowever individuals are characterized by different (balanced)combinations of procoagulant and anticoagulant factor levelsthat altogether contribute to define a unique coagulationphenotype that reflects the developmental environmentalgenetic nutritional and pharmacological influences of eachindividual [52] On the contrary the lack of factor IX causes
haemophilia B characterized by higher CT with respect tothe reference value (Figure 10)
Furthermore by comparing the PSA-1D of factors VIIIand IX (Figures 9 and 10) it is clear that haemophilia A ismore serious than haemophilia B since the CT achieved inconditions of factor VIII deficiency is higher than the CTobtained in the case of factor IX deficiency
In both PSA-1D over factors VIII and IX we observedafter the initial decrease of the CT an unexpected increaseof the CT as the factor concentration increasesThis counter-intuitive behavior arises at very high concentrations of thesefactors (with respect to the average physiological levels) andto the best of our knowledge it was never observed in vivoNonetheless it would be interesting to verify by means of adhoc laboratory experiments if the model correctly describesthe behavior of the BCC even in these conditions or on thecontrary it is not predictive in these extreme situations
BioMed Research International 15
The PSA-1D over factor II (Figure 11) shows a dramaticdecrease of the CT as the initial concentration of this factorincreases (with respect to the average physiological level)This behavior resembles the effects of hypercoagulability (orthrombophilia) a disease caused by mutation G20210A inthe prothrombin gene [53] that causes an increase of theprothrombin level (factor II) in the blood flow resulting inan excessive formation of the active form of this factor thusheightening venous thrombosis risks [54] Hypercoagulabil-ity is usually treated with warfarin therapy or with otheranticoagulants with a similar effect These drugs decrease thecapacity of coagulation factors to become active preventingthe formation of unwanted thrombi
On the other hand when the initial concentration offactor II is low we achieved the effects of prothrombindeficiency a rare autosomal recessive disease that causes atendency to severe bleeding [29 55] As shown in Figure 11 aconcentration equal to 10of the physiological value of factorII (ie 14 sdot 10minus6 M) leads to clotting effects similar to severehaemophilia A
32 PSA-2D of the BCCModel We present here the results ofthe PSA-2D on the BCCmodel where couples of parameterswere varied to analyze the possible effects arising from thecombined perturbation of their values
321 Reactions 11990327
and 11990358 Figure 12 shows the effect of
the simultaneous variation of constants 11989627
and 11989658
(overthe same sweep ranges considered in the two PSA-1DSection 31) This result remarks that reaction 119903
27 involved
in the common pathway has a stronger influence on theBCC and that there is a synergic interplay between thesetwo reactions In particular when the value of 119896
27is low and
the value of 11989658
is high the CT is higher than the valuesachieved when only a single constant is changed because inthis condition both the intrinsic and the common pathwaysare simultaneously inhibited
322 Factor VIII Factor IX and Tissue Factor In the last twoPSA-2Dwe varied the initial concentration of factor VIII andTissue Factor and the initial concentration of factor IX andTissue Factor respectively
The initial concentrations of factors VIII and IX were var-ied over four orders of magnitude using their physiologicalvalues as upper limit for the sweep ranges the concentrationof Tissue Factor was varied over four orders of magnitudetwo above and two below its reference value (see Table 2)The rationale behind this choice is to observe how the BCCmodel in conditions corresponding to different states ofhaemophilia (obtained by decreasing the concentrations offactors VIII and IX) behaves with different initial concentra-tions of the Tissue Factor which is the upstream factor of theextrinsic pathway that is the most important element of theBCC
The results of these PSA-2D show that with respect tothe condition of haemophilia B in the case of haemophiliaA the amount of Tissue Factor (below its reference value)has a negligible influence on the CT as indicated by the
presence of a plateau in Figure 13 on the contrary concerninghaemophilia B a deficiency of Tissue Factor leads to anincrease of the CT especially when factor IX is present in lowconcentrations (Figure 14)
The different results achieved in the two PSA-2D are dueto the presence in the BCC model of a direct interactionbetweenTissue Factor and factor IX bymeans of theTF-fVIIacomplex (see reactions 119903
13 119903
15in Table 1) Indeed the lack
of Tissue Factor directly affects the concentration of activefactor IX which results in a strong alteration of the CT withrespect to physiological conditions
33 CPU versus GPU Performance Comparison In orderto show the relevant speedup achieved by coagSODA wepresent here the comparison of the computational effortrequired by GPU and CPU for the simulation of the BCCmodel The performances of our GPU simulator were com-pared with those obtained using the LSODA algorithmimplemented in the software COPASI [56] executing onthe CPU the same set of simulations that were run on theGPU COPASI is single-threaded and does not exploit thephysical and logical cores of the CPU therefore it representsa good benchmark as a single-node CPU-bound simulator ofbiological systems
In all simulations executed on both GPU and CPU westored 100 samples uniformly distributed in the time intervalconsidered for each simulation that is [0 700] seconds of thedynamics of all chemical species involved in the BCCmodelThe settings for the LSODA algorithm were the followingrelative error 1 sdot 10minus7 absolute error 1 sdot 10minus13 and maximumnumber of internal steps set to 20000
331 Benchmark Details The GPU used for the simulationsis a Nvidia Tesla K20c equipped with 2496 cores organized in13 streamingmultiprocessors GPU clock 706MHz and 5GBofDDR5RAM In all tests coagSODAwas compiled and exe-cuted with version 55 of CUDA libraries Even though thisGPU has compute capability 35 and is based on the GK110Kepler architecture currently coagSODA does not exploitany of the new features (eg Dynamic Parallelism Hyper-QRemote DMA) with the exception of the new ISA encodingwhich allows threads to exploit an increased number ofregisters (255 instead of 63) reducing register spilling intoglobal memory and increasing performances [57] Moreoveras described in Section 22 the Kepler architecture offers thepossibility to reconfigure the 64KB on-chip cache balancingbetween L1 cache and shared memory Since coagSODAexploits the shared memory to reduce the latencies duringthe access to the concentration values of the BCC model weopted for the following configuration 48KB for the sharedmemory 16 KB for the L1 cache
The performance of the GPU was compared against apersonal computer equippedwith a dual-core CPU Intel Corei5 frequency 23 GHz 4GB of DDR3 RAM running theoperating systemMacOSXLion 1075The software used as acoagSODA-equivalent single-threaded CPU implementationis COPASI version 48 (build 35)
16 BioMed Research International
280
300
320
340
360
380
400
420
440
1e41e6
1e91e11 1e5
1e7
1e9
1e11
280 320 360 400 440
Clot
ting
time (
s)
k 58(M
minus1 sminus1 )
k27 (Mminus1sminus1)
Figure 12 Clotting time according to PSA-2D over reaction constants 11989627and 119896
58 The sweep ranges are [2 sdot 104 2 sdot 1010] and [1 sdot 105 2 sdot 1011]
respectively
100
200
300
400
Tissue factor (M) Factor VIII
(M)
Clot
ting
time (
s)
100
150
200
250
300
350
400
450
1eminus131eminus11
1eminus91eminus14
1eminus12
1eminus10
Figure 13 Clotting time at different initial concentrations of Tissue Factor and factor VIII The sweep ranges are [500 sdot 10minus14 500 sdot 10minus10]and [700 sdot 10minus14 500 sdot 10minus10] respectively
A direct comparison of the performances between thesetwo different architectures and implementations is not aneasy task since CPUs are optimized for single-thread exe-cution and exploit a large number of optimizations (eghigher clock frequency instruction caching pipelining outof order execution and branch prediction) whereas GPUsare optimized for graphics processing and parallel exe-cution of identical code relying only on multilevel datacaching For this reason in this paper we propose anempirical comparison of the performances based on therunning times of identical batches of simulations of the BCCmodel
332 Benchmark Results In Figure 15 we show the compar-ison of the running times required to run several batchesof simulations executed to carry out the PSA-1D over thereaction constant 119896
27 The choice of comparing the perfor-
mances of the GPU and CPU implementations by executingbatches of simulations that are related to a PSA instead ofrunning 119899 independent but identical simulations (ie allcharacterized by the same parameterization of the model)is due to the fact that these results represent a more realisticscenario in the computational analysis of biological systemswhereby it is useful to investigate large search spaces ofparameters corresponding to different perturbed conditionsof themodel [26]Moreover for the evaluation of the running
time the execution of a batch of 119899 identical deterministicsimulations would be futile The figure clearly shows thatcoagSODA always performs better than the CPU counter-part In particular while the CPU performance increaseslinearly with the number of simulations the running timesare strongly reduced on the GPU in this case the overallrunning time roughly corresponds to the time required forthe execution of the slowest simulations This is due to thefact that different parameterizations may require differenttime steps for LSODA to converge and the execution ofa block terminates as long as all the threads it containsterminate In turn the execution of a kernel terminates whenall blocks terminate for this reason a single simulationwhose dynamics requires more steps than the others mayaffect the overall running time
In Table 3 we report the running times of all batchesof simulations along with the speedup achieved on theGPU In particular these results highlight that the advantageof exploiting the GPUs for the simulation of the BCCmodel becomes more evident as the number of simula-tions increases with a 181times speedup when a PSA with10000 different parameterizations is executed Therefore theGPU accelerated analysis of the BCC model with coag-SODA represents a novel relevant computational means toinvestigate the behavior of this complex biological systemunder nonphysiological conditions and could be exploited
BioMed Research International 17
0
200
400
600
Tissue factor (M) Factor IX (M
)
Clot
ting
time (
s)
0
100
200
300
400
500
600
700
1eminus12
1eminus10
1eminus8
1eminus131eminus11
1eminus9
Figure 14 Clotting time at different initial concentrations of Tissue Factor and factor IXThe sweep ranges are [500 sdot 10minus14 500 sdot 10minus10] and[900 sdot 10
minus12 900 sdot 10
minus8] respectively
5 10 50 100 500 1000 5000 10000
Runn
ing
time (
s)
Number of simulations
CPUGPU
1e2
1e3
1e4
1e5
1e6
1e7
Figure 15 Comparison of the computational time required to execute an increasing number of simulations of the BCCmodel on CPU (IntelCore i5 23 GHz) and GPU (Nvidia Tesla K20c GPU clock 706MHz) The computational time is expressed in seconds The time valuesrelated to 1000 5000 and 10000 simulations on the CPU were estimated by regression (see also Table 3)
Table 3 CPU versus GPU performance comparison
Number ofsimulations CPU time (sec) GPU time (sec) Speedup
5 3808 1278 298times10 10604 2329 456times50 51704 8058 642times100 116528 10977 1061times500 536058 17394 3081times1000 1073582lowast 19982 5373times5000 5363507lowast 30960 1732times10000 10724975lowast 58951 1819timeslowastValues estimated by regression
to efficiently determine the response of the BCC to differenttherapeutical drugs
4 Conclusions
Thanks to their high-performance computing capabilitiesand the very low costs GPUs nowadays represent a suitable
technology for the development and the application of par-allel computational methods for in silico analysis of complexbiological systems However the implementation of efficientcomputational tools able to fully exploit the large potentialityof GPUs is still challenging since good programming skillsare required to implement GPU-based algorithmic methodsand to handle specific features as an optimal usage ofmemoryor the communication bandwidth between GPU and CPUMoreover algorithms cannot be directly ported on the GPUbecause of the limited programming capabilities allowed byGPUkernels as amatter of fact they need to be redesigned tomeet the requirements of the underlying SIMD architecture
In this work we presented coagSODA a GPU-poweredsimulator specifically developed to carry out fast parallelsimulations of the BCC model coagSODA was designed tooffer a black-box solution usable by any final user in aneasy way It relies on cupSODA [24] a numerical integratorfor ODEs that we previously implemented for the GPUarchitecture based on the LSODA algorithm and capableof automatically translating a reaction-based model into aset of coupled ODEs In addition to mass-action kinet-ics coagSODA implements specific functions to compute
18 BioMed Research International
the kinetics of Hill function based reactions such as thoseinvolved in the platelets activity of the BCC model coag-SODA exploits the massive parallel capabilities of modernGPUs and our results demonstrated that it can achieve arelevant reduction of the computational time required toexecutemany concurrent and independent simulations of theBCC dynamics The mutual independence of the simulationsallows fully exploiting the underlying SIMD architecturemoreover coagSODA benefits from an additional speedupthanks to our choice of storing the state of the systeminto the low-latency shared memory (a solution that wasalready implemented in cupSODA) Since the BCC modelis large (96 reactions 71 chemical species) a large amountof shared memory was assigned to each thread stronglyreducing the theoretical occupancy of the GPU that is theratio of active warps with respect to the maximum numberof warps supported by each streaming multiprocessor of theGPU However the results of the analysis presented in thiswork performed on the BCC model show that coagSODAachieves a relevant boost with respect to a reference CPUimplementation For instance in the case of 10000 simula-tions we achieved a noticeable 181times speedup Interestinglythe performances of coagSODA are better than COPASI evenfor small batches of simulations These results indicate thatfor biological models consisting in many reactions and manyspecies our GPU implementation of LSODA becomes moreprofitable than the CPU counterpart as the number of con-current simulations increases making it suitable especiallywhen performing demanding computational analysis suchas for example parameter sweep parameter estimation orsensitivity analysis
As a test case in this paper we presented several parame-ter sweep analyses over reaction constants and initial concen-trations of factors involved in the BCC model Other com-putational analyses on mathematical models of this pathwaywere previously presented For instance in [58] a parameterestimation of the reaction constants of different modelsinvolving the activation of factor X factor V prothrombinand the inactivation of factors was performed The aim ofthis analysis was to discriminate between different modelsto unravel the mechanisms on the basis of the BCC Similaranalyses were executed on a model describing thrombingeneration in plasma since the reaction mechanism thereaction constants and initial concentrations were unknown[59] In [34] a sensitivity analysis of a model consisting of44 species over 34 chemical species was presented reactionconstants were varied between 10 and 1000 of theirreference value to the aim of identifying the most influentialfactors of the BCC
All these analyses were performed bymeans of sequentialsimulations of the models under investigation thereforein general only small batches of simulations could be run(for instance in [34] only 836 simulations were executed toexecute the sensitivity analysis of the model) On the otherhand in this paper we presented the results of different PSAefficiently executed bymeans of coagSODA overall providinguseful information regarding unknownparameters and inter-esting insights into the functioning of the BCC A thoroughsensitivity analysis of the whole BCC model consisting in
around 5 sdot 105 simulations is currently in progress on GPUsby our group exploiting the great computational efficiency ofcoagSODA
The results of our computational analyses should benow validated by means of specifically designed laboratoryexperiments In particular we identified a plausible valuefor the constant of the reaction describing the autoactivationof factor XII (since no reference values can be found inliterature) moreover the results of the PSA over factor IXsuggest that a deficiency of this factor is not enough to causesevere bleeding disorders as haemophilia B but an alterationof other factors seems to be necessary for the occurrence ofsuch condition (eg the lack of Tissue Factor as suggestedby the PSA-2D over Tissue Factor and factor IX) Finallythe PSA over factors VIII and IX showed in a situationcharacterized by very high concentrations of these factorsa counterintuitive behavior in which the clotting time isincreased with respect to the value obtained in physiologicalconditions Indeed it would be interesting to design ad hoclaboratory experiments to verify if the BCCmodel is actuallypredictive in such extreme situations
The coagSODA software and the SBML version of theBCC model used in this work are available from the authorsupon request
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This work is partially supported by SYSBIO Centre forSystems Biology
References
[1] B B Aldridge JM BurkeDA Lauffenburger andP K SorgerldquoPhysicochemicalmodelling of cell signalling pathwaysrdquoNatureCell Biology vol 8 no 11 pp 1195ndash1203 2006
[2] I-C Chou and E O Voit ldquoRecent developments in parameterestimation and structure identification of biochemical andgenomic systemsrdquoMathematical Biosciences vol 219 no 2 pp57ndash83 2009
[3] D Besozzi P Cazzaniga D Pescini G Mauri S Colomboand EMartegani ldquoThe role of feedback control mechanisms onthe establishment of oscillatory regimes in the RascAMPPKApathway in S cerevisiaerdquo EURASIP Journal on Bioinformaticsand Systems Biology vol 2012 no 10 2012
[4] A Saltelli M Ratto T Andres et al Global Sensitivity AnalysisThe Primer Wiley-Interscience 2008
[5] A Raue C Kreutz T Maiwald et al ldquoStructural and practicalidentifiability analysis of partially observed dynamical modelsby exploiting the profile likelihoodrdquo Bioinformatics vol 25 no15 pp 1923ndash1929 2009
[6] A Drager M Kronfeld M J Ziller et al ldquoModeling metabolicnetworks in C glutamicum a comparison of rate laws incombination with various parameter optimization strategiesrdquoBMC Systems Biology vol 3 article 5 2009
BioMed Research International 19
[7] D Besozzi P Cazzaniga G Mauri D Pescini and L Van-neschi ldquoA comparison of genetic algorithms and particle swarmoptimization for parameter estimation in stochastic biochem-ical systemsrdquo in Proceedings of the 7th European ConferenceEvolutionary Computation Machine Learning and Data Miningin Bioinformatics (EvoBIO rsquo09) C Pizzuti M Ritchie and MGiacobini Eds vol 5483 of Lecture Notes in Computer Sciencepp 116ndash127 Springer 2009
[8] M S Nobile D Besozzi P Cazzaniga GMauri andD PescinildquocupSODA aGPU-basedmulti-swarmPSOmethod for param-eter estimation in stochastic biological systems exploitingdiscrete-time target seriesrdquo in Proceedings of the 10th EuropeanConference Evolutionary Computation Machine Learning andData Mining in Bioinformatics (EvoBIO rsquo12) M Giacobini LVanneschi and W Bush Eds vol 7246 of Lecture Notes inComputer Science pp 74ndash85 Springer
[9] M S Nobile D Besozzi P Cazzaniga D Pescini and GMaurildquoReverse engineering of kinetic reaction networks by meansof Cartesian Genetic Programming and Particle Swarm Opti-mizationrdquo in Proceedings of the IEEE Congress on EvolutionaryComputation (CEC rsquo13) pp 1594ndash1601 June 2013
[10] M S Nobile ldquoEvolutionary inference of biochemical inter-action networks accelerated on Graphics Processing Unitsrdquoin Proceedings of the 11th International Conference on HighPerformance Computingamp Simulation 2013 (HPCS rsquo13) pp 668ndash670 Helsinki Finland 2013
[11] K Kentzoglanakis and M Poole ldquoA swarm intelligence frame-work for reconstructing gene networks searching for bio-logically plausible architecturesrdquo IEEEACM Transactions onComputational Biology and Bioinformatics vol 9 no 2 pp 358ndash371 2012
[12] J R Koza W Mydlowec G Lanza Y Jessen and M A KeaneldquoAutomatic computational discovery of chemical reaction net-works using genetic programmingrdquo inComputationalDiscoveryof Scientific Knowledge S Dzeroski and L Todorovski Edsvol 4660 of Lecture Notes in Computer Science pp 205ndash227Springer 2007
[13] S Ando E Sakamoto and H Iba ldquoEvolutionary modeling andinference of gene networkrdquo Information Sciences vol 145 no3-4 pp 237ndash259 2002
[14] L Dematte and D Prandi ldquoGPU computing for systemsbiologyrdquo Briefings in Bioinformatics vol 11 no 3 Article IDbbq006 pp 323ndash333 2010
[15] J L Payne N A Sinnott-Armstrong and J HMoore ldquoExploit-ing graphics processing units for computational biology andbioinformaticsrdquo Interdisciplinary Sciences Computational LifeSciences vol 2 no 3 pp 213ndash220 2010
[16] M J Harvey and G De Fabritiis ldquoA survey of computationalmolecular science using graphics processing unitsrdquoWiley Inter-disciplinary Reviews Computational Molecular Science vol 2no 5 pp 734ndash742 2012
[17] Y Zhou J Liepe X Sheng M P H Stumpf and C BarnesldquoGPU accelerated biochemical network simulationrdquo Bioinfor-matics vol 27 no 6 Article ID btr015 pp 874ndash876 2011
[18] M Vigelius A Lane and B Meyer ldquoAccelerating reaction-diffusion simulations with general-purpose graphics processingunitsrdquo Bioinformatics vol 27 no 2 Article ID btq622 pp 288ndash290 2011
[19] D J Wilkinson ldquoStochastic modelling for quantitative descrip-tion of heterogeneous biological systemsrdquo Nature ReviewsGenetics vol 10 no 2 pp 122ndash133 2009
[20] J C Butcher ldquoTrees and numerical methods for ordinarydifferential equationsrdquo Numerical Algorithms vol 53 no 2-3pp 153ndash170 2010
[21] D T Gillespie ldquoStochastic simulation of chemical kineticsrdquoAnnual Review of Physical Chemistry vol 58 pp 35ndash55 2007
[22] L Petzold ldquoAutomatic selection of methods for solving stiffand nonstiff systems of ordinary differential equationsrdquo SIAMJournal of Scientific and Statistical Computing vol 4 no 1 pp136ndash148 1983
[23] Y Cao D T Gillespie and L R Petzold ldquoEfficient step sizeselection for the tau-leaping simulation methodrdquo The Journalof Chemical Physics vol 124 no 4 2006
[24] M S Nobile D Besozzi P Cazzaniga GMauri andD PescinildquocupSODA a CUDA-powered simulator of mass-action kinet-icsrdquo in Proceedings of the 12th International Conference onParallel Computing Technologies (PaCT rsquo13) V Malyshkin Edvol 7979 of Lecture Notes in Computer Science pp 344ndash357
[25] M S Nobile P Cazzaniga D Besozzi and G Mauri ldquoGPU-accelerated simulations of mass-action kinetics models withcupSODArdquoThe Journal of Supercomputing 2014
[26] M S Nobile P Cazzaniga D Besozzi D Pescini and GMauri ldquocuTauLeaping a GPU-powered tau-leaping stochasticsimulator for massive parallel analyses of biological systemsrdquoPLoS ONE vol 9 no 3 2014
[27] K Rogers Blood Physiology and CirculationTheHuman BodyBritannica Educational Publishing 2010
[28] A D Michelson Platelets Elsevier 2nd edition 2007[29] R W Colman J Hirsh V J Marder A W Clowes and J
N George Hemostasis and Thrombosis Basic Principles andClinical Practice Lippincott Williams ampWilkins 2006
[30] M S Chatterjee W S Denney H Jing and S L DiamondldquoSystems biology of coagulation initiation Kinetics of thrombingeneration in resting and activated human bloodrdquo PLoS Com-putational Biology vol 6 no 9 Article ID e1000950 2010
[31] M Bellini D Besozzi P Cazzaniga GMauri andM S NobileldquoSimulation and analysis of the blood coagulation cascadeaccelerated on GPUrdquo in Proceedings of the 22nd EuromicroInternational Conference on Parallel Distributed and Network-Based Processing (PDP) pp 590ndash593 IEEE Computer SocietyConference Publishing Services 2014
[32] T Wajima G K Isbister and S B Duffull ldquoA comprehensivemodel for the humoral coagulation network in humansrdquo Clini-cal Pharmacology and Therapeutics vol 86 no 3 pp 290ndash2982009
[33] K Lo W S Denney and S L Diamond ldquoStochastic modelingof blood coagulation initiationrdquo Pathophysiology of Haemostasisand Thrombosis vol 34 no 2-3 pp 80ndash90 2006
[34] C M Danforth T Orfeo K G Mann K E Brummel-Ziedins and S J Everse ldquoThe impact of uncertainty in a bloodcoagulation modelrdquo Mathematical Medicine and Biology vol26 no 4 Article ID dqp011 pp 323ndash336 2009
[35] S D Lewis P P Shields and J A Shafer ldquoCharacterizationof the kinetic pathway for liberation of fibrinopeptides duringassembly of fibrinrdquoThe Journal of Biological Chemistry vol 260no 18 pp 10192ndash10199 1985
[36] M F Hockin K C Jones S J Everse and K G Mann ldquoAmodel for the stoichiometric regulation of blood coagulationrdquoThe Journal of Biological Chemistry vol 277 no 21 pp 18322ndash18333 2002
[37] K Brummel-Ziedins C YVossen F R Rosendaal KUmezakiand K G Mann ldquoThe plasma hemostatic proteome thrombin
20 BioMed Research International
generation in healthy individualsrdquo Journal of Thrombosis andHaemostasis vol 3 no 7 pp 1472ndash1481 2005
[38] K E Brummel-Ziedins C Y Vossen S Butenas K G Mannand F R Rosendaal ldquoThrombin generation profiles in deepvenous thrombosisrdquo Journal of Thrombosis and Haemostasisvol 3 no 11 pp 2497ndash2505 2005
[39] S D Bungay P A Gentry and R D Gentry ldquoA mathematicalmodel of lipid-mediated thrombin generationrdquo MathematicalMedicine and Biology vol 20 no 1 pp 105ndash129 2003
[40] S Butenas T Orfeo M T Gissel K E Brummel and K GMann ldquoThe significance of circulating factor IXa in bloodrdquoThe Journal of Biological Chemistry vol 279 no 22 pp 22875ndash22882 2004
[41] D Voet and J G Voet Biochemistry Wiley 2011[42] A C Guyton and J E Hall Textbook of Medical Physiology
Saunders 2010[43] M F M James and A M Roche ldquoDose-response relationship
between plasma ionized calcium concentration and thrombe-lastographyrdquo Journal of Cardiothoracic and Vascular Anesthesiavol 18 no 5 pp 581ndash586 2004
[44] D L Nelson and M M Cox Lehninger Principles of Biochem-istry W H Freeman Company 2004
[45] R M Farber ldquoTopical perspective on massive threading andparallelismrdquo Journal of Molecular Graphics and Modelling vol30 pp 82ndash89 2011
[46] OWolkenhauerMUllahWKolch andK-HCho ldquoModelingand simulation of intracellular dynamics choosing an appropri-ate frameworkrdquo IEEETransactions onNanobioscience vol 3 no3 pp 200ndash207 2004
[47] B H Estridge A P Reynolds and N J Walters Basic MedicalLaboratory Techniques Cengage Learning 4th edition 2000
[48] T Renne A H Schmaier K F Nickel M Blomback and CMaas ldquoIn vivo roles of factor XIIrdquo Blood vol 120 no 22 pp4296ndash4303 2012
[49] P V Jenkins O Rawley O P Smith and J S ODonnellldquoElevated factor VIII levels and risk of venous thrombosisrdquoBritish Journal of Haematology vol 157 no 6 pp 653ndash663 2012
[50] C E Walsh and K M Batt ldquoHemophilia clinical gene therapybrief reviewrdquo Translational Research vol 161 no 4 pp 307ndash3122013
[51] A Van Hylckama Vlieg I K Van Der Linden R M Bertinaand F R Rosendaal ldquoHigh levels of factor IX increase the risk ofvenous thrombosisrdquo Blood vol 95 no 12 pp 3678ndash3682 2000
[52] K Brummel-Ziedins ldquoModels for thrombin generation and riskof diseaserdquo Journal ofThrombosis and Haemostasis vol 11 no 1pp 212ndash223 2013
[53] U Seligsohn and A Lubetsky ldquoGenetic susceptibility to venousthrombosisrdquoTheNew England Journal of Medicine vol 344 no16 pp 1222ndash1231 2001
[54] I Martinelli P Bucciarelli and P M Mannucci ldquoThromboticrisk factors basic pathophysiologyrdquo Critical Care Medicine vol38 no 2 pp S3ndashS9 2010
[55] K Brummel-Ziedins A Undas T Orfeo et al ldquoThrombingeneration in acute coronary syndrome and stable coronaryartery disease dependence on plasma factor compositionrdquoJournal ofThrombosis andHaemostasis vol 6 no 1 pp 104ndash1102008
[56] S Hoops R Gauges C Lee et al ldquoCOPASImdasha complexpathway simulatorrdquo Bioinformatics vol 22 no 24 pp 3067ndash3074 2006
[57] Nvidia NVIDIArsquos Next Generation CUDA Compute Architec-ture Kepler GK110 2012
[58] W Stortelder W Hemker and H Hemker ldquoMathematicalmodelling in blood coagulation simulation and parameterestimationrdquoModelling Analysis and Simulation vol 20 pp 1ndash111997
[59] S M C Orfao G Jank and K Mottaghy ldquoA new mathematicalapproach to modelling thrombin generationrdquo InternationalJournal of Artificial Organs vol 29 no 7 pp 703ndash708 2006
[60] Nvidia Nvidia CUDA C Programming Guide v50 2012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Anatomy Research International
PeptidesInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
International Journal of
Volume 2014
Zoology
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Molecular Biology International
GenomicsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
BioinformaticsAdvances in
Marine BiologyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Signal TransductionJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
BioMed Research International
Evolutionary BiologyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Biochemistry Research International
ArchaeaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Genetics Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in
Virolog y
Hindawi Publishing Corporationhttpwwwhindawicom
Nucleic AcidsJournal of
Volume 2014
Stem CellsInternational
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Enzyme Research
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Microbiology
10 BioMed Research International
Thread
Shared memory L1 cache Read-onlydata cache
L2 cache
DRAM
Figure 4 Schematic description of memory hierarchies in Fermi and Kepler architectures GPUs relying on these architectures are equippedwith a two-level data cache and a read-only data cache Shared memory and L1 cache share the same on-chip 64KB memory banks theamount of memory can be reconfigured by the user according to the specific needs of the application Figure taken from Nvidiarsquos KeplerGK110 whitepaper [57]
written in the canonical form but the user is supposed tospecify the system of ODEs by implementing a custom Cfunction that is passed to the algorithmmoreover in order tospeed up the computation when dealing with stiff systems inLSODA the Jacobianmatrix associatedwith theODEs systemmust be implemented as a custom C function as well On thecontrary cupSODA was conceived as a black-box simulatorthat can be easily used without any programming skillscupSODA consists in a tool to automatically convert a genericmechanistic reaction-based model of a biological system intothe corresponding set of ODEs to comply with the mass-action kinetics [46] and to directly encode the obtainedsystem along with the corresponding Jacobian matrix as Carrays
This fully automaticmethodology can be exploited for thesimulation of models whose chemical kinetics is based on themass-action law The BCC model described in Section 21though includes a set 119878
120576of reactions that do not follow
the mass-action kinetics [30] the platelets activity is notexplicitly modeled by biochemical reactions but it is realizedby modulating the rate of the dissociation of the complexesformed on a plateletrsquos surface by means of the variable120576 which is calculated with a special equation during theintegration steps For this reason cupSODA was used asa starting point for the development of a new tool ableto compute at run-time the specific kinetics of this set ofreactions
This new CUDA-powered simulation tool named coag-SODA is specifically tailored for the simulation of the BCC
model developed in [30] In particular coagSODA realizesthe run-time calculation of the 120576 value required to correctlysimulate the activity of platelets which is determined accord-ing to the equations described in Section 21 The plateletsactivation state 120576 is calculated at each time instant by solvingthe following differential equation
119889120576
119889119905
= 119896 (120576max minus 120576) (5)
where the constant 119896 is inversely proportional to the timescale of platelets activation and is set to 0005 This isconsistent with the fact that platelets do not instantly achievetheir maximum attainable activation state (120576max) but theyreach it on a physiologically relevant timescale [30]
Dealing with the 14 reactions in the set 119878120576that are
influenced by the Hill functionH (see Section 21) the valueof [fIIalowast(119905)]must be stored on the GPU because during eachintegration step coagSODA recalculates (4) which exploitsthe value [fIIalowast(119905)] to determine a new 120576 value Since CUDArsquosarchitecture does not offer static variables the informationfor each thread has to be memorized in the global memoryThe accesses to the global memory and the computationalcosts due to these additional calculations slow down theintegration process with respect to the integration of ODEsperformed according to strictly mass-action based kinetics(as in the cupSODA simulator) Nevertheless in Section 33we show that our parallel implementation largely outper-forms a sequential counterpart of the LSODA algorithm
BioMed Research International 11
0 100 200 300 400 500 600 700
Fibr
inog
en co
ncen
trat
ion
(M)
Time (s)
30 of fibrinogen
0
1eminus6
2eminus6
3eminus6
4eminus6
5eminus6
6eminus6
7eminus6
8eminus6
9eminus6
Figure 5 Dynamics of fibrinogen in physiological condition The clotting time is defined as the time necessary to convert the 70 of thefibrinogen into fibrin
As in the case of cupSODA [24 25] coagSODA exploitsthe shared memory to improve performances by storing thecurrent state and time of the simulations in these low-latencymemory banks Despite the improvement of performancesensured by this solution it strongly affects the occupancyof GPUrsquos multiprocessors and therefore it represents thelimiting factor for the number of blocks that can be executedsimultaneously as a matter of fact coagSODA is limited to2 blocks per streaming multiprocessor on GPUs based onthe Fermi architecture with a reduced exploitation of theGPU with respect to the theoretical 8 blocks allowed by thisarchitecture
3 Results and Discussion
In this section we discuss the results of the parameter sweepanalysis (PSA) carried out on the BCC model to the aimof investigating either the prolongation or the reduction ofthe clotting time in response to perturbed values of somereaction constants and of the initial concentration of somemolecular species chosen according to their meaning withinthe whole pathway PSA was performed by generating aset of different initial conditions corresponding to differentparameterizations of the model and then automaticallyexecuting the deterministic simulations with coagSODATheuse ofGPU technology is fundamental in this type of analysisespecially for large biological systems as the BCC is becauseit drastically reduces the computational time
The sweep analysis for single parameters (PSA-1D) wasperformed considering a logarithmic sampling of numericalvalues of each parameter under investigation (reaction con-stant or initial molecular concentration) within a specifiedrange with respect to its physiological reference value Thesweep analysis over pairs of parameters (PSA-2D) was per-formed by simultaneously varying the values of two param-eters within a specified range considering a logarithmicsampling on the resulting lattice The logarithmic samplingallows uniformly spanning different orders of magnitude ofthe parameters value using a reduced and fine-grained set
of samples therefore efficiently analyzing the response of thesystem in a broad range of conditions
To determine the response of the BCC to perturbed con-ditions we chose the clotting time (CT) as output of the PSAThe CT is defined as the time necessary to convert the 70of the fibrinogen into fibrin (see Figure 5) conventionallyassumed to correspond to the time required to form the clot[47] and it is generally used in laboratory tests formonitoringthe therapywith anticoagulant drugs According to themodeldefined in [30] the reference value of CT is around 300seconds in physiological conditions We investigate herethe response of the BCC by evaluating the CT in variousconditions corresponding to different values of the reactionconstants varying over six orders of magnitude with respectto their physiological values (ie three below and three abovethe reference values if not otherwise specified) as well asto different values of the initial molecular concentrationsand varying over twelve orders of magnitude with respect totheir physiological values (ie six below and six above thereference values if not otherwise specified)
The total number of parallel simulations executed tocarry out these analyses was 100 for PSA-1D over reactionconstants 200 for PSA-1D over initial concentrations and1600 for PSA-2D
Finally we present the comparison of the performanceof the CPU and GPU to run an increasing number ofsimulations of the BCC model to prove the efficiency ofcoagSODA
31 PSA-1D of the BCC Model
311 Reaction 11990344 Thefirst PSA was performed to determine
the value of the kinetic constant for the autoactivation offactor XII (reaction 119903
44in Table 1) which corresponds to an
upstream process in the intrinsic pathway This analysis wasmotivated by two considerations Firstly by using the fullparameterization given in [30] the action of the intrinsicpathway turns out to be fundamental for the BCC in vivowhich is in contrast to experimental observations which
12 BioMed Research International
324
326
328
330
332
334
336
338
340
Clot
ting
time (
s)
1eminus
10
1eminus
9
1eminus
8
1eminus
7
1eminus
6
1eminus
5
1eminus
4
1eminus
3
k44 (sminus1)
Figure 6 Clotting time according to PSA-1D over constant 11989644of reaction 119903
44 XII rarr XIIaThe reference value of 119896
44used in [30] is 5sdot10minus4sminus1
the sweep range is [5 sdot 10minus10 5 sdot 10minus4]
280
300
320
340
360
380
400
1e4 1e5 1e6 1e7 1e8 1e9 1e10 1e11
Clot
ting
time (
s)
k27 (Mminus1sminus1)
Figure 7 Clotting time according to PSA-1D over constant 11989627of reaction 119903
27 IIa + V rarr IIa + Va The reference value is 2 sdot 107Mminus1 sminus1 the
sweep range is [2 sdot 104 2 sdot 1010]
indicate that the extrinsic pathway is the main responsibleof clot formation Secondly in [30] all constants values havea reference to experimental measurements except for theconstant of this reaction (which corresponds to entry 29 inTable 1 in [30])
Figure 6 shows the results of this PSA-1D where 11989644was
varied over six orders of magnitude considering the valuegiven in [30] as the upper limit of the sweep interval We canobserve that by decreasing the value of 119896
44with respect to
the value considered in [30] the CT increases with respect toits reference value however for values of this constant lowerthan 100 sdot 10minus6 sminus1 the CT remains unaltered and this canbe intuitively explained by the fact that in this situation thefibrinogen is mainly activated by the extrinsic pathway
Consequently we assigned the value 700 sdot 10minus6 sminus1 to11989644 achieving a CT that is comparable to the experimental
observations of the BCC in vivo This new value was used inall PSA discussed in what follows
312 Reactions 11990327
and 11990358 In the next PSA we investigated
the effect of the perturbation of the kinetics of two pivotal
reactions of the BCC model Reaction 11990327 which describes
the catalytic activation of factor V by factor IIa was chosenbecause it represents the main positive feedback within thecommon pathway reaction 119903
58 which is involved in the
activation of factor XI by binding to factor IIa was chosenbecause it represents the main positive feedback in theintrinsic pathway (Figure 1) Moreover preliminary PSA overall reaction constants of the BCC model given in Table 1evidenced that these two reactions are among the mostrelevant steps of the coagulation network
The PSA-1D over 11989627(Figure 7) shows that the CT is very
sensitive to the perturbation of the rate of this reaction whenthe reference value of its constant (ie 119896
27= 2 sdot 10
7Mminus1 sminus1)is either increased or decreased in particular when 119896
27is
very low a plateau in CT is reached since the strength ofthe positive feedback exerted by factor IIa is largely reduceda condition where the contribution of the amplification ofthe hemostatic stimulus (due by the common pathway) tothe formation of the clot is basically not effective On theother side the PSA-1D over 119896
58(Figure 8) shows that while
a decrease of the constant with respect to its reference value
BioMed Research International 13
310
320
330
340
350
360
370
1e5 1e6 1e7 1e8 1e9 1e10 1e11
Clot
ting
time (
s)
k58 (Mminus1sminus1)
Figure 8 Clotting time according to PSA-1D over constant 11989658of reaction 119903
58 IIa + XI rarr IIa-XI The reference value is 1 sdot 108Mminus1 sminus1 the
sweep range is [1 sdot 105 2 sdot 1011]
220
240
260
280
300
320
340
360
380
400
Clot
ting
time (
s)
Factor VIII (M)
Seve
re h
aem
ophi
lia
Mild
hae
mop
hilia
Prot
hrom
botic
stat
e
1eminus
16
1eminus
14
1eminus
12
1eminus
10
1eminus
8
1eminus
6
1eminus
4
1eminus
2
Figure 9 Clotting time at different initial concentrations of factor VIII The reference value is 700 sdot 10minus10M (dashed black line) the sweeprange is [700 sdot 10minus16 700 sdot 10minus4] The blue area indicates a condition in which factor VIII concentration is less than 1 of its physiologicalconcentration corresponding to a situation of severe haemophilia the red area indicates a condition in which factor VIII concentration isbetween 1 and 30 of its physiological concentration corresponding to a situation of mild haemophilia the green area indicates a conditionin which factor VIII concentration is greater than 130 of its physiological concentration corresponding to a situation of hypercoagulability
(ie 11989658= 1 sdot 10
8Mminus1 sminus1) does not have any substantialeffect an increase of its value leads to a progressive increasein the CT This increase is due to the fact that factor IIa issequestered in the formation of a complex with factor XI andhence it is no longer available as a free component in bloodto participate in other reactions especially those reactionsof the extrinsic pathway which principally lead to the clotformation in vivo This behavior highlights that the intrinsicpathway has a secondary role in blood coagulation in vivocompared with the extrinsic pathway as also evidenced byvarious experimental observations [48]
313 Factors VIII IX and II The next set of PSA-1D wasrealized by varying the initial concentrations of factors VIIIIX and II These factors were selected since both an excess
and a deficiency of their concentrations lead to diseasesrelated to blood clotting
The PSA-1D over factor VIII (Figure 9) shows thatincreasing the initial concentration of this factor resultsin decreasing the CT suggesting the possible presence ofhypercoagulable states in these perturbed conditions As amatter of fact high levels of factor VIII cause an increasedrisk of deep vein thrombosis and pulmonary embolism[49] On the other hand individuals with less than 1of the average concentration of factor VIII show a severehaemophilia A characterized by higher CT (Figure 9) andrequire infusions of plasma containing the deficient factorotherwise frequent spontaneous bleeding would occur [50]When the concentration of this factor is between 5 and 30of the average concentration individuals still risk bleeding incase of trauma [50]
14 BioMed Research International
250
300
350
400
450
500
550
600
650
700
Clot
ting
time (
s)
Factor IX (M)
1eminus
14
1eminus
12
1eminus
10
1eminus
8
1eminus
6
1eminus
4
1eminus
2
1e0
Figure 10 Clotting time at different initial concentrations of factor IXThe reference value is 900sdot10minus8M the sweep range is [900sdot10minus14 900sdot10minus2]
0
100
200
300
400
500
600
700
Clot
ting
time (
s)
Factor II (M)
1eminus
12
1eminus
10
1eminus
8
1eminus
6
1eminus
4
1eminus
2
1e0
1e2
Figure 11 Clotting time at different initial concentrations of thrombinThe reference value is 140sdot10minus6M the sweep range is [140sdot10minus12 140sdot100]
The PSA-1D over factor IX (Figure 10) shows only aslight decrease of the CT as the initial concentration offIX increases this is in contrast to recent studies thatdemonstrated how the excess of factor IX leads to anincreased risk of deep vein thrombosis [51] This result canbe explained by considering that (i) a high concentrationof factor IX is not sufficient to bring about coagulationproblems though when the concentration of other factorsis above the average value (yet not at pathological levels)prothrombotic states can be observed (ii) in this model weconsider average values as initial concentrations of factorshowever individuals are characterized by different (balanced)combinations of procoagulant and anticoagulant factor levelsthat altogether contribute to define a unique coagulationphenotype that reflects the developmental environmentalgenetic nutritional and pharmacological influences of eachindividual [52] On the contrary the lack of factor IX causes
haemophilia B characterized by higher CT with respect tothe reference value (Figure 10)
Furthermore by comparing the PSA-1D of factors VIIIand IX (Figures 9 and 10) it is clear that haemophilia A ismore serious than haemophilia B since the CT achieved inconditions of factor VIII deficiency is higher than the CTobtained in the case of factor IX deficiency
In both PSA-1D over factors VIII and IX we observedafter the initial decrease of the CT an unexpected increaseof the CT as the factor concentration increasesThis counter-intuitive behavior arises at very high concentrations of thesefactors (with respect to the average physiological levels) andto the best of our knowledge it was never observed in vivoNonetheless it would be interesting to verify by means of adhoc laboratory experiments if the model correctly describesthe behavior of the BCC even in these conditions or on thecontrary it is not predictive in these extreme situations
BioMed Research International 15
The PSA-1D over factor II (Figure 11) shows a dramaticdecrease of the CT as the initial concentration of this factorincreases (with respect to the average physiological level)This behavior resembles the effects of hypercoagulability (orthrombophilia) a disease caused by mutation G20210A inthe prothrombin gene [53] that causes an increase of theprothrombin level (factor II) in the blood flow resulting inan excessive formation of the active form of this factor thusheightening venous thrombosis risks [54] Hypercoagulabil-ity is usually treated with warfarin therapy or with otheranticoagulants with a similar effect These drugs decrease thecapacity of coagulation factors to become active preventingthe formation of unwanted thrombi
On the other hand when the initial concentration offactor II is low we achieved the effects of prothrombindeficiency a rare autosomal recessive disease that causes atendency to severe bleeding [29 55] As shown in Figure 11 aconcentration equal to 10of the physiological value of factorII (ie 14 sdot 10minus6 M) leads to clotting effects similar to severehaemophilia A
32 PSA-2D of the BCCModel We present here the results ofthe PSA-2D on the BCCmodel where couples of parameterswere varied to analyze the possible effects arising from thecombined perturbation of their values
321 Reactions 11990327
and 11990358 Figure 12 shows the effect of
the simultaneous variation of constants 11989627
and 11989658
(overthe same sweep ranges considered in the two PSA-1DSection 31) This result remarks that reaction 119903
27 involved
in the common pathway has a stronger influence on theBCC and that there is a synergic interplay between thesetwo reactions In particular when the value of 119896
27is low and
the value of 11989658
is high the CT is higher than the valuesachieved when only a single constant is changed because inthis condition both the intrinsic and the common pathwaysare simultaneously inhibited
322 Factor VIII Factor IX and Tissue Factor In the last twoPSA-2Dwe varied the initial concentration of factor VIII andTissue Factor and the initial concentration of factor IX andTissue Factor respectively
The initial concentrations of factors VIII and IX were var-ied over four orders of magnitude using their physiologicalvalues as upper limit for the sweep ranges the concentrationof Tissue Factor was varied over four orders of magnitudetwo above and two below its reference value (see Table 2)The rationale behind this choice is to observe how the BCCmodel in conditions corresponding to different states ofhaemophilia (obtained by decreasing the concentrations offactors VIII and IX) behaves with different initial concentra-tions of the Tissue Factor which is the upstream factor of theextrinsic pathway that is the most important element of theBCC
The results of these PSA-2D show that with respect tothe condition of haemophilia B in the case of haemophiliaA the amount of Tissue Factor (below its reference value)has a negligible influence on the CT as indicated by the
presence of a plateau in Figure 13 on the contrary concerninghaemophilia B a deficiency of Tissue Factor leads to anincrease of the CT especially when factor IX is present in lowconcentrations (Figure 14)
The different results achieved in the two PSA-2D are dueto the presence in the BCC model of a direct interactionbetweenTissue Factor and factor IX bymeans of theTF-fVIIacomplex (see reactions 119903
13 119903
15in Table 1) Indeed the lack
of Tissue Factor directly affects the concentration of activefactor IX which results in a strong alteration of the CT withrespect to physiological conditions
33 CPU versus GPU Performance Comparison In orderto show the relevant speedup achieved by coagSODA wepresent here the comparison of the computational effortrequired by GPU and CPU for the simulation of the BCCmodel The performances of our GPU simulator were com-pared with those obtained using the LSODA algorithmimplemented in the software COPASI [56] executing onthe CPU the same set of simulations that were run on theGPU COPASI is single-threaded and does not exploit thephysical and logical cores of the CPU therefore it representsa good benchmark as a single-node CPU-bound simulator ofbiological systems
In all simulations executed on both GPU and CPU westored 100 samples uniformly distributed in the time intervalconsidered for each simulation that is [0 700] seconds of thedynamics of all chemical species involved in the BCCmodelThe settings for the LSODA algorithm were the followingrelative error 1 sdot 10minus7 absolute error 1 sdot 10minus13 and maximumnumber of internal steps set to 20000
331 Benchmark Details The GPU used for the simulationsis a Nvidia Tesla K20c equipped with 2496 cores organized in13 streamingmultiprocessors GPU clock 706MHz and 5GBofDDR5RAM In all tests coagSODAwas compiled and exe-cuted with version 55 of CUDA libraries Even though thisGPU has compute capability 35 and is based on the GK110Kepler architecture currently coagSODA does not exploitany of the new features (eg Dynamic Parallelism Hyper-QRemote DMA) with the exception of the new ISA encodingwhich allows threads to exploit an increased number ofregisters (255 instead of 63) reducing register spilling intoglobal memory and increasing performances [57] Moreoveras described in Section 22 the Kepler architecture offers thepossibility to reconfigure the 64KB on-chip cache balancingbetween L1 cache and shared memory Since coagSODAexploits the shared memory to reduce the latencies duringthe access to the concentration values of the BCC model weopted for the following configuration 48KB for the sharedmemory 16 KB for the L1 cache
The performance of the GPU was compared against apersonal computer equippedwith a dual-core CPU Intel Corei5 frequency 23 GHz 4GB of DDR3 RAM running theoperating systemMacOSXLion 1075The software used as acoagSODA-equivalent single-threaded CPU implementationis COPASI version 48 (build 35)
16 BioMed Research International
280
300
320
340
360
380
400
420
440
1e41e6
1e91e11 1e5
1e7
1e9
1e11
280 320 360 400 440
Clot
ting
time (
s)
k 58(M
minus1 sminus1 )
k27 (Mminus1sminus1)
Figure 12 Clotting time according to PSA-2D over reaction constants 11989627and 119896
58 The sweep ranges are [2 sdot 104 2 sdot 1010] and [1 sdot 105 2 sdot 1011]
respectively
100
200
300
400
Tissue factor (M) Factor VIII
(M)
Clot
ting
time (
s)
100
150
200
250
300
350
400
450
1eminus131eminus11
1eminus91eminus14
1eminus12
1eminus10
Figure 13 Clotting time at different initial concentrations of Tissue Factor and factor VIII The sweep ranges are [500 sdot 10minus14 500 sdot 10minus10]and [700 sdot 10minus14 500 sdot 10minus10] respectively
A direct comparison of the performances between thesetwo different architectures and implementations is not aneasy task since CPUs are optimized for single-thread exe-cution and exploit a large number of optimizations (eghigher clock frequency instruction caching pipelining outof order execution and branch prediction) whereas GPUsare optimized for graphics processing and parallel exe-cution of identical code relying only on multilevel datacaching For this reason in this paper we propose anempirical comparison of the performances based on therunning times of identical batches of simulations of the BCCmodel
332 Benchmark Results In Figure 15 we show the compar-ison of the running times required to run several batchesof simulations executed to carry out the PSA-1D over thereaction constant 119896
27 The choice of comparing the perfor-
mances of the GPU and CPU implementations by executingbatches of simulations that are related to a PSA instead ofrunning 119899 independent but identical simulations (ie allcharacterized by the same parameterization of the model)is due to the fact that these results represent a more realisticscenario in the computational analysis of biological systemswhereby it is useful to investigate large search spaces ofparameters corresponding to different perturbed conditionsof themodel [26]Moreover for the evaluation of the running
time the execution of a batch of 119899 identical deterministicsimulations would be futile The figure clearly shows thatcoagSODA always performs better than the CPU counter-part In particular while the CPU performance increaseslinearly with the number of simulations the running timesare strongly reduced on the GPU in this case the overallrunning time roughly corresponds to the time required forthe execution of the slowest simulations This is due to thefact that different parameterizations may require differenttime steps for LSODA to converge and the execution ofa block terminates as long as all the threads it containsterminate In turn the execution of a kernel terminates whenall blocks terminate for this reason a single simulationwhose dynamics requires more steps than the others mayaffect the overall running time
In Table 3 we report the running times of all batchesof simulations along with the speedup achieved on theGPU In particular these results highlight that the advantageof exploiting the GPUs for the simulation of the BCCmodel becomes more evident as the number of simula-tions increases with a 181times speedup when a PSA with10000 different parameterizations is executed Therefore theGPU accelerated analysis of the BCC model with coag-SODA represents a novel relevant computational means toinvestigate the behavior of this complex biological systemunder nonphysiological conditions and could be exploited
BioMed Research International 17
0
200
400
600
Tissue factor (M) Factor IX (M
)
Clot
ting
time (
s)
0
100
200
300
400
500
600
700
1eminus12
1eminus10
1eminus8
1eminus131eminus11
1eminus9
Figure 14 Clotting time at different initial concentrations of Tissue Factor and factor IXThe sweep ranges are [500 sdot 10minus14 500 sdot 10minus10] and[900 sdot 10
minus12 900 sdot 10
minus8] respectively
5 10 50 100 500 1000 5000 10000
Runn
ing
time (
s)
Number of simulations
CPUGPU
1e2
1e3
1e4
1e5
1e6
1e7
Figure 15 Comparison of the computational time required to execute an increasing number of simulations of the BCCmodel on CPU (IntelCore i5 23 GHz) and GPU (Nvidia Tesla K20c GPU clock 706MHz) The computational time is expressed in seconds The time valuesrelated to 1000 5000 and 10000 simulations on the CPU were estimated by regression (see also Table 3)
Table 3 CPU versus GPU performance comparison
Number ofsimulations CPU time (sec) GPU time (sec) Speedup
5 3808 1278 298times10 10604 2329 456times50 51704 8058 642times100 116528 10977 1061times500 536058 17394 3081times1000 1073582lowast 19982 5373times5000 5363507lowast 30960 1732times10000 10724975lowast 58951 1819timeslowastValues estimated by regression
to efficiently determine the response of the BCC to differenttherapeutical drugs
4 Conclusions
Thanks to their high-performance computing capabilitiesand the very low costs GPUs nowadays represent a suitable
technology for the development and the application of par-allel computational methods for in silico analysis of complexbiological systems However the implementation of efficientcomputational tools able to fully exploit the large potentialityof GPUs is still challenging since good programming skillsare required to implement GPU-based algorithmic methodsand to handle specific features as an optimal usage ofmemoryor the communication bandwidth between GPU and CPUMoreover algorithms cannot be directly ported on the GPUbecause of the limited programming capabilities allowed byGPUkernels as amatter of fact they need to be redesigned tomeet the requirements of the underlying SIMD architecture
In this work we presented coagSODA a GPU-poweredsimulator specifically developed to carry out fast parallelsimulations of the BCC model coagSODA was designed tooffer a black-box solution usable by any final user in aneasy way It relies on cupSODA [24] a numerical integratorfor ODEs that we previously implemented for the GPUarchitecture based on the LSODA algorithm and capableof automatically translating a reaction-based model into aset of coupled ODEs In addition to mass-action kinet-ics coagSODA implements specific functions to compute
18 BioMed Research International
the kinetics of Hill function based reactions such as thoseinvolved in the platelets activity of the BCC model coag-SODA exploits the massive parallel capabilities of modernGPUs and our results demonstrated that it can achieve arelevant reduction of the computational time required toexecutemany concurrent and independent simulations of theBCC dynamics The mutual independence of the simulationsallows fully exploiting the underlying SIMD architecturemoreover coagSODA benefits from an additional speedupthanks to our choice of storing the state of the systeminto the low-latency shared memory (a solution that wasalready implemented in cupSODA) Since the BCC modelis large (96 reactions 71 chemical species) a large amountof shared memory was assigned to each thread stronglyreducing the theoretical occupancy of the GPU that is theratio of active warps with respect to the maximum numberof warps supported by each streaming multiprocessor of theGPU However the results of the analysis presented in thiswork performed on the BCC model show that coagSODAachieves a relevant boost with respect to a reference CPUimplementation For instance in the case of 10000 simula-tions we achieved a noticeable 181times speedup Interestinglythe performances of coagSODA are better than COPASI evenfor small batches of simulations These results indicate thatfor biological models consisting in many reactions and manyspecies our GPU implementation of LSODA becomes moreprofitable than the CPU counterpart as the number of con-current simulations increases making it suitable especiallywhen performing demanding computational analysis suchas for example parameter sweep parameter estimation orsensitivity analysis
As a test case in this paper we presented several parame-ter sweep analyses over reaction constants and initial concen-trations of factors involved in the BCC model Other com-putational analyses on mathematical models of this pathwaywere previously presented For instance in [58] a parameterestimation of the reaction constants of different modelsinvolving the activation of factor X factor V prothrombinand the inactivation of factors was performed The aim ofthis analysis was to discriminate between different modelsto unravel the mechanisms on the basis of the BCC Similaranalyses were executed on a model describing thrombingeneration in plasma since the reaction mechanism thereaction constants and initial concentrations were unknown[59] In [34] a sensitivity analysis of a model consisting of44 species over 34 chemical species was presented reactionconstants were varied between 10 and 1000 of theirreference value to the aim of identifying the most influentialfactors of the BCC
All these analyses were performed bymeans of sequentialsimulations of the models under investigation thereforein general only small batches of simulations could be run(for instance in [34] only 836 simulations were executed toexecute the sensitivity analysis of the model) On the otherhand in this paper we presented the results of different PSAefficiently executed bymeans of coagSODA overall providinguseful information regarding unknownparameters and inter-esting insights into the functioning of the BCC A thoroughsensitivity analysis of the whole BCC model consisting in
around 5 sdot 105 simulations is currently in progress on GPUsby our group exploiting the great computational efficiency ofcoagSODA
The results of our computational analyses should benow validated by means of specifically designed laboratoryexperiments In particular we identified a plausible valuefor the constant of the reaction describing the autoactivationof factor XII (since no reference values can be found inliterature) moreover the results of the PSA over factor IXsuggest that a deficiency of this factor is not enough to causesevere bleeding disorders as haemophilia B but an alterationof other factors seems to be necessary for the occurrence ofsuch condition (eg the lack of Tissue Factor as suggestedby the PSA-2D over Tissue Factor and factor IX) Finallythe PSA over factors VIII and IX showed in a situationcharacterized by very high concentrations of these factorsa counterintuitive behavior in which the clotting time isincreased with respect to the value obtained in physiologicalconditions Indeed it would be interesting to design ad hoclaboratory experiments to verify if the BCCmodel is actuallypredictive in such extreme situations
The coagSODA software and the SBML version of theBCC model used in this work are available from the authorsupon request
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This work is partially supported by SYSBIO Centre forSystems Biology
References
[1] B B Aldridge JM BurkeDA Lauffenburger andP K SorgerldquoPhysicochemicalmodelling of cell signalling pathwaysrdquoNatureCell Biology vol 8 no 11 pp 1195ndash1203 2006
[2] I-C Chou and E O Voit ldquoRecent developments in parameterestimation and structure identification of biochemical andgenomic systemsrdquoMathematical Biosciences vol 219 no 2 pp57ndash83 2009
[3] D Besozzi P Cazzaniga D Pescini G Mauri S Colomboand EMartegani ldquoThe role of feedback control mechanisms onthe establishment of oscillatory regimes in the RascAMPPKApathway in S cerevisiaerdquo EURASIP Journal on Bioinformaticsand Systems Biology vol 2012 no 10 2012
[4] A Saltelli M Ratto T Andres et al Global Sensitivity AnalysisThe Primer Wiley-Interscience 2008
[5] A Raue C Kreutz T Maiwald et al ldquoStructural and practicalidentifiability analysis of partially observed dynamical modelsby exploiting the profile likelihoodrdquo Bioinformatics vol 25 no15 pp 1923ndash1929 2009
[6] A Drager M Kronfeld M J Ziller et al ldquoModeling metabolicnetworks in C glutamicum a comparison of rate laws incombination with various parameter optimization strategiesrdquoBMC Systems Biology vol 3 article 5 2009
BioMed Research International 19
[7] D Besozzi P Cazzaniga G Mauri D Pescini and L Van-neschi ldquoA comparison of genetic algorithms and particle swarmoptimization for parameter estimation in stochastic biochem-ical systemsrdquo in Proceedings of the 7th European ConferenceEvolutionary Computation Machine Learning and Data Miningin Bioinformatics (EvoBIO rsquo09) C Pizzuti M Ritchie and MGiacobini Eds vol 5483 of Lecture Notes in Computer Sciencepp 116ndash127 Springer 2009
[8] M S Nobile D Besozzi P Cazzaniga GMauri andD PescinildquocupSODA aGPU-basedmulti-swarmPSOmethod for param-eter estimation in stochastic biological systems exploitingdiscrete-time target seriesrdquo in Proceedings of the 10th EuropeanConference Evolutionary Computation Machine Learning andData Mining in Bioinformatics (EvoBIO rsquo12) M Giacobini LVanneschi and W Bush Eds vol 7246 of Lecture Notes inComputer Science pp 74ndash85 Springer
[9] M S Nobile D Besozzi P Cazzaniga D Pescini and GMaurildquoReverse engineering of kinetic reaction networks by meansof Cartesian Genetic Programming and Particle Swarm Opti-mizationrdquo in Proceedings of the IEEE Congress on EvolutionaryComputation (CEC rsquo13) pp 1594ndash1601 June 2013
[10] M S Nobile ldquoEvolutionary inference of biochemical inter-action networks accelerated on Graphics Processing Unitsrdquoin Proceedings of the 11th International Conference on HighPerformance Computingamp Simulation 2013 (HPCS rsquo13) pp 668ndash670 Helsinki Finland 2013
[11] K Kentzoglanakis and M Poole ldquoA swarm intelligence frame-work for reconstructing gene networks searching for bio-logically plausible architecturesrdquo IEEEACM Transactions onComputational Biology and Bioinformatics vol 9 no 2 pp 358ndash371 2012
[12] J R Koza W Mydlowec G Lanza Y Jessen and M A KeaneldquoAutomatic computational discovery of chemical reaction net-works using genetic programmingrdquo inComputationalDiscoveryof Scientific Knowledge S Dzeroski and L Todorovski Edsvol 4660 of Lecture Notes in Computer Science pp 205ndash227Springer 2007
[13] S Ando E Sakamoto and H Iba ldquoEvolutionary modeling andinference of gene networkrdquo Information Sciences vol 145 no3-4 pp 237ndash259 2002
[14] L Dematte and D Prandi ldquoGPU computing for systemsbiologyrdquo Briefings in Bioinformatics vol 11 no 3 Article IDbbq006 pp 323ndash333 2010
[15] J L Payne N A Sinnott-Armstrong and J HMoore ldquoExploit-ing graphics processing units for computational biology andbioinformaticsrdquo Interdisciplinary Sciences Computational LifeSciences vol 2 no 3 pp 213ndash220 2010
[16] M J Harvey and G De Fabritiis ldquoA survey of computationalmolecular science using graphics processing unitsrdquoWiley Inter-disciplinary Reviews Computational Molecular Science vol 2no 5 pp 734ndash742 2012
[17] Y Zhou J Liepe X Sheng M P H Stumpf and C BarnesldquoGPU accelerated biochemical network simulationrdquo Bioinfor-matics vol 27 no 6 Article ID btr015 pp 874ndash876 2011
[18] M Vigelius A Lane and B Meyer ldquoAccelerating reaction-diffusion simulations with general-purpose graphics processingunitsrdquo Bioinformatics vol 27 no 2 Article ID btq622 pp 288ndash290 2011
[19] D J Wilkinson ldquoStochastic modelling for quantitative descrip-tion of heterogeneous biological systemsrdquo Nature ReviewsGenetics vol 10 no 2 pp 122ndash133 2009
[20] J C Butcher ldquoTrees and numerical methods for ordinarydifferential equationsrdquo Numerical Algorithms vol 53 no 2-3pp 153ndash170 2010
[21] D T Gillespie ldquoStochastic simulation of chemical kineticsrdquoAnnual Review of Physical Chemistry vol 58 pp 35ndash55 2007
[22] L Petzold ldquoAutomatic selection of methods for solving stiffand nonstiff systems of ordinary differential equationsrdquo SIAMJournal of Scientific and Statistical Computing vol 4 no 1 pp136ndash148 1983
[23] Y Cao D T Gillespie and L R Petzold ldquoEfficient step sizeselection for the tau-leaping simulation methodrdquo The Journalof Chemical Physics vol 124 no 4 2006
[24] M S Nobile D Besozzi P Cazzaniga GMauri andD PescinildquocupSODA a CUDA-powered simulator of mass-action kinet-icsrdquo in Proceedings of the 12th International Conference onParallel Computing Technologies (PaCT rsquo13) V Malyshkin Edvol 7979 of Lecture Notes in Computer Science pp 344ndash357
[25] M S Nobile P Cazzaniga D Besozzi and G Mauri ldquoGPU-accelerated simulations of mass-action kinetics models withcupSODArdquoThe Journal of Supercomputing 2014
[26] M S Nobile P Cazzaniga D Besozzi D Pescini and GMauri ldquocuTauLeaping a GPU-powered tau-leaping stochasticsimulator for massive parallel analyses of biological systemsrdquoPLoS ONE vol 9 no 3 2014
[27] K Rogers Blood Physiology and CirculationTheHuman BodyBritannica Educational Publishing 2010
[28] A D Michelson Platelets Elsevier 2nd edition 2007[29] R W Colman J Hirsh V J Marder A W Clowes and J
N George Hemostasis and Thrombosis Basic Principles andClinical Practice Lippincott Williams ampWilkins 2006
[30] M S Chatterjee W S Denney H Jing and S L DiamondldquoSystems biology of coagulation initiation Kinetics of thrombingeneration in resting and activated human bloodrdquo PLoS Com-putational Biology vol 6 no 9 Article ID e1000950 2010
[31] M Bellini D Besozzi P Cazzaniga GMauri andM S NobileldquoSimulation and analysis of the blood coagulation cascadeaccelerated on GPUrdquo in Proceedings of the 22nd EuromicroInternational Conference on Parallel Distributed and Network-Based Processing (PDP) pp 590ndash593 IEEE Computer SocietyConference Publishing Services 2014
[32] T Wajima G K Isbister and S B Duffull ldquoA comprehensivemodel for the humoral coagulation network in humansrdquo Clini-cal Pharmacology and Therapeutics vol 86 no 3 pp 290ndash2982009
[33] K Lo W S Denney and S L Diamond ldquoStochastic modelingof blood coagulation initiationrdquo Pathophysiology of Haemostasisand Thrombosis vol 34 no 2-3 pp 80ndash90 2006
[34] C M Danforth T Orfeo K G Mann K E Brummel-Ziedins and S J Everse ldquoThe impact of uncertainty in a bloodcoagulation modelrdquo Mathematical Medicine and Biology vol26 no 4 Article ID dqp011 pp 323ndash336 2009
[35] S D Lewis P P Shields and J A Shafer ldquoCharacterizationof the kinetic pathway for liberation of fibrinopeptides duringassembly of fibrinrdquoThe Journal of Biological Chemistry vol 260no 18 pp 10192ndash10199 1985
[36] M F Hockin K C Jones S J Everse and K G Mann ldquoAmodel for the stoichiometric regulation of blood coagulationrdquoThe Journal of Biological Chemistry vol 277 no 21 pp 18322ndash18333 2002
[37] K Brummel-Ziedins C YVossen F R Rosendaal KUmezakiand K G Mann ldquoThe plasma hemostatic proteome thrombin
20 BioMed Research International
generation in healthy individualsrdquo Journal of Thrombosis andHaemostasis vol 3 no 7 pp 1472ndash1481 2005
[38] K E Brummel-Ziedins C Y Vossen S Butenas K G Mannand F R Rosendaal ldquoThrombin generation profiles in deepvenous thrombosisrdquo Journal of Thrombosis and Haemostasisvol 3 no 11 pp 2497ndash2505 2005
[39] S D Bungay P A Gentry and R D Gentry ldquoA mathematicalmodel of lipid-mediated thrombin generationrdquo MathematicalMedicine and Biology vol 20 no 1 pp 105ndash129 2003
[40] S Butenas T Orfeo M T Gissel K E Brummel and K GMann ldquoThe significance of circulating factor IXa in bloodrdquoThe Journal of Biological Chemistry vol 279 no 22 pp 22875ndash22882 2004
[41] D Voet and J G Voet Biochemistry Wiley 2011[42] A C Guyton and J E Hall Textbook of Medical Physiology
Saunders 2010[43] M F M James and A M Roche ldquoDose-response relationship
between plasma ionized calcium concentration and thrombe-lastographyrdquo Journal of Cardiothoracic and Vascular Anesthesiavol 18 no 5 pp 581ndash586 2004
[44] D L Nelson and M M Cox Lehninger Principles of Biochem-istry W H Freeman Company 2004
[45] R M Farber ldquoTopical perspective on massive threading andparallelismrdquo Journal of Molecular Graphics and Modelling vol30 pp 82ndash89 2011
[46] OWolkenhauerMUllahWKolch andK-HCho ldquoModelingand simulation of intracellular dynamics choosing an appropri-ate frameworkrdquo IEEETransactions onNanobioscience vol 3 no3 pp 200ndash207 2004
[47] B H Estridge A P Reynolds and N J Walters Basic MedicalLaboratory Techniques Cengage Learning 4th edition 2000
[48] T Renne A H Schmaier K F Nickel M Blomback and CMaas ldquoIn vivo roles of factor XIIrdquo Blood vol 120 no 22 pp4296ndash4303 2012
[49] P V Jenkins O Rawley O P Smith and J S ODonnellldquoElevated factor VIII levels and risk of venous thrombosisrdquoBritish Journal of Haematology vol 157 no 6 pp 653ndash663 2012
[50] C E Walsh and K M Batt ldquoHemophilia clinical gene therapybrief reviewrdquo Translational Research vol 161 no 4 pp 307ndash3122013
[51] A Van Hylckama Vlieg I K Van Der Linden R M Bertinaand F R Rosendaal ldquoHigh levels of factor IX increase the risk ofvenous thrombosisrdquo Blood vol 95 no 12 pp 3678ndash3682 2000
[52] K Brummel-Ziedins ldquoModels for thrombin generation and riskof diseaserdquo Journal ofThrombosis and Haemostasis vol 11 no 1pp 212ndash223 2013
[53] U Seligsohn and A Lubetsky ldquoGenetic susceptibility to venousthrombosisrdquoTheNew England Journal of Medicine vol 344 no16 pp 1222ndash1231 2001
[54] I Martinelli P Bucciarelli and P M Mannucci ldquoThromboticrisk factors basic pathophysiologyrdquo Critical Care Medicine vol38 no 2 pp S3ndashS9 2010
[55] K Brummel-Ziedins A Undas T Orfeo et al ldquoThrombingeneration in acute coronary syndrome and stable coronaryartery disease dependence on plasma factor compositionrdquoJournal ofThrombosis andHaemostasis vol 6 no 1 pp 104ndash1102008
[56] S Hoops R Gauges C Lee et al ldquoCOPASImdasha complexpathway simulatorrdquo Bioinformatics vol 22 no 24 pp 3067ndash3074 2006
[57] Nvidia NVIDIArsquos Next Generation CUDA Compute Architec-ture Kepler GK110 2012
[58] W Stortelder W Hemker and H Hemker ldquoMathematicalmodelling in blood coagulation simulation and parameterestimationrdquoModelling Analysis and Simulation vol 20 pp 1ndash111997
[59] S M C Orfao G Jank and K Mottaghy ldquoA new mathematicalapproach to modelling thrombin generationrdquo InternationalJournal of Artificial Organs vol 29 no 7 pp 703ndash708 2006
[60] Nvidia Nvidia CUDA C Programming Guide v50 2012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Anatomy Research International
PeptidesInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
International Journal of
Volume 2014
Zoology
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Molecular Biology International
GenomicsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
BioinformaticsAdvances in
Marine BiologyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Signal TransductionJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
BioMed Research International
Evolutionary BiologyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Biochemistry Research International
ArchaeaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Genetics Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in
Virolog y
Hindawi Publishing Corporationhttpwwwhindawicom
Nucleic AcidsJournal of
Volume 2014
Stem CellsInternational
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Enzyme Research
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Microbiology
BioMed Research International 11
0 100 200 300 400 500 600 700
Fibr
inog
en co
ncen
trat
ion
(M)
Time (s)
30 of fibrinogen
0
1eminus6
2eminus6
3eminus6
4eminus6
5eminus6
6eminus6
7eminus6
8eminus6
9eminus6
Figure 5 Dynamics of fibrinogen in physiological condition The clotting time is defined as the time necessary to convert the 70 of thefibrinogen into fibrin
As in the case of cupSODA [24 25] coagSODA exploitsthe shared memory to improve performances by storing thecurrent state and time of the simulations in these low-latencymemory banks Despite the improvement of performancesensured by this solution it strongly affects the occupancyof GPUrsquos multiprocessors and therefore it represents thelimiting factor for the number of blocks that can be executedsimultaneously as a matter of fact coagSODA is limited to2 blocks per streaming multiprocessor on GPUs based onthe Fermi architecture with a reduced exploitation of theGPU with respect to the theoretical 8 blocks allowed by thisarchitecture
3 Results and Discussion
In this section we discuss the results of the parameter sweepanalysis (PSA) carried out on the BCC model to the aimof investigating either the prolongation or the reduction ofthe clotting time in response to perturbed values of somereaction constants and of the initial concentration of somemolecular species chosen according to their meaning withinthe whole pathway PSA was performed by generating aset of different initial conditions corresponding to differentparameterizations of the model and then automaticallyexecuting the deterministic simulations with coagSODATheuse ofGPU technology is fundamental in this type of analysisespecially for large biological systems as the BCC is becauseit drastically reduces the computational time
The sweep analysis for single parameters (PSA-1D) wasperformed considering a logarithmic sampling of numericalvalues of each parameter under investigation (reaction con-stant or initial molecular concentration) within a specifiedrange with respect to its physiological reference value Thesweep analysis over pairs of parameters (PSA-2D) was per-formed by simultaneously varying the values of two param-eters within a specified range considering a logarithmicsampling on the resulting lattice The logarithmic samplingallows uniformly spanning different orders of magnitude ofthe parameters value using a reduced and fine-grained set
of samples therefore efficiently analyzing the response of thesystem in a broad range of conditions
To determine the response of the BCC to perturbed con-ditions we chose the clotting time (CT) as output of the PSAThe CT is defined as the time necessary to convert the 70of the fibrinogen into fibrin (see Figure 5) conventionallyassumed to correspond to the time required to form the clot[47] and it is generally used in laboratory tests formonitoringthe therapywith anticoagulant drugs According to themodeldefined in [30] the reference value of CT is around 300seconds in physiological conditions We investigate herethe response of the BCC by evaluating the CT in variousconditions corresponding to different values of the reactionconstants varying over six orders of magnitude with respectto their physiological values (ie three below and three abovethe reference values if not otherwise specified) as well asto different values of the initial molecular concentrationsand varying over twelve orders of magnitude with respect totheir physiological values (ie six below and six above thereference values if not otherwise specified)
The total number of parallel simulations executed tocarry out these analyses was 100 for PSA-1D over reactionconstants 200 for PSA-1D over initial concentrations and1600 for PSA-2D
Finally we present the comparison of the performanceof the CPU and GPU to run an increasing number ofsimulations of the BCC model to prove the efficiency ofcoagSODA
31 PSA-1D of the BCC Model
311 Reaction 11990344 Thefirst PSA was performed to determine
the value of the kinetic constant for the autoactivation offactor XII (reaction 119903
44in Table 1) which corresponds to an
upstream process in the intrinsic pathway This analysis wasmotivated by two considerations Firstly by using the fullparameterization given in [30] the action of the intrinsicpathway turns out to be fundamental for the BCC in vivowhich is in contrast to experimental observations which
12 BioMed Research International
324
326
328
330
332
334
336
338
340
Clot
ting
time (
s)
1eminus
10
1eminus
9
1eminus
8
1eminus
7
1eminus
6
1eminus
5
1eminus
4
1eminus
3
k44 (sminus1)
Figure 6 Clotting time according to PSA-1D over constant 11989644of reaction 119903
44 XII rarr XIIaThe reference value of 119896
44used in [30] is 5sdot10minus4sminus1
the sweep range is [5 sdot 10minus10 5 sdot 10minus4]
280
300
320
340
360
380
400
1e4 1e5 1e6 1e7 1e8 1e9 1e10 1e11
Clot
ting
time (
s)
k27 (Mminus1sminus1)
Figure 7 Clotting time according to PSA-1D over constant 11989627of reaction 119903
27 IIa + V rarr IIa + Va The reference value is 2 sdot 107Mminus1 sminus1 the
sweep range is [2 sdot 104 2 sdot 1010]
indicate that the extrinsic pathway is the main responsibleof clot formation Secondly in [30] all constants values havea reference to experimental measurements except for theconstant of this reaction (which corresponds to entry 29 inTable 1 in [30])
Figure 6 shows the results of this PSA-1D where 11989644was
varied over six orders of magnitude considering the valuegiven in [30] as the upper limit of the sweep interval We canobserve that by decreasing the value of 119896
44with respect to
the value considered in [30] the CT increases with respect toits reference value however for values of this constant lowerthan 100 sdot 10minus6 sminus1 the CT remains unaltered and this canbe intuitively explained by the fact that in this situation thefibrinogen is mainly activated by the extrinsic pathway
Consequently we assigned the value 700 sdot 10minus6 sminus1 to11989644 achieving a CT that is comparable to the experimental
observations of the BCC in vivo This new value was used inall PSA discussed in what follows
312 Reactions 11990327
and 11990358 In the next PSA we investigated
the effect of the perturbation of the kinetics of two pivotal
reactions of the BCC model Reaction 11990327 which describes
the catalytic activation of factor V by factor IIa was chosenbecause it represents the main positive feedback within thecommon pathway reaction 119903
58 which is involved in the
activation of factor XI by binding to factor IIa was chosenbecause it represents the main positive feedback in theintrinsic pathway (Figure 1) Moreover preliminary PSA overall reaction constants of the BCC model given in Table 1evidenced that these two reactions are among the mostrelevant steps of the coagulation network
The PSA-1D over 11989627(Figure 7) shows that the CT is very
sensitive to the perturbation of the rate of this reaction whenthe reference value of its constant (ie 119896
27= 2 sdot 10
7Mminus1 sminus1)is either increased or decreased in particular when 119896
27is
very low a plateau in CT is reached since the strength ofthe positive feedback exerted by factor IIa is largely reduceda condition where the contribution of the amplification ofthe hemostatic stimulus (due by the common pathway) tothe formation of the clot is basically not effective On theother side the PSA-1D over 119896
58(Figure 8) shows that while
a decrease of the constant with respect to its reference value
BioMed Research International 13
310
320
330
340
350
360
370
1e5 1e6 1e7 1e8 1e9 1e10 1e11
Clot
ting
time (
s)
k58 (Mminus1sminus1)
Figure 8 Clotting time according to PSA-1D over constant 11989658of reaction 119903
58 IIa + XI rarr IIa-XI The reference value is 1 sdot 108Mminus1 sminus1 the
sweep range is [1 sdot 105 2 sdot 1011]
220
240
260
280
300
320
340
360
380
400
Clot
ting
time (
s)
Factor VIII (M)
Seve
re h
aem
ophi
lia
Mild
hae
mop
hilia
Prot
hrom
botic
stat
e
1eminus
16
1eminus
14
1eminus
12
1eminus
10
1eminus
8
1eminus
6
1eminus
4
1eminus
2
Figure 9 Clotting time at different initial concentrations of factor VIII The reference value is 700 sdot 10minus10M (dashed black line) the sweeprange is [700 sdot 10minus16 700 sdot 10minus4] The blue area indicates a condition in which factor VIII concentration is less than 1 of its physiologicalconcentration corresponding to a situation of severe haemophilia the red area indicates a condition in which factor VIII concentration isbetween 1 and 30 of its physiological concentration corresponding to a situation of mild haemophilia the green area indicates a conditionin which factor VIII concentration is greater than 130 of its physiological concentration corresponding to a situation of hypercoagulability
(ie 11989658= 1 sdot 10
8Mminus1 sminus1) does not have any substantialeffect an increase of its value leads to a progressive increasein the CT This increase is due to the fact that factor IIa issequestered in the formation of a complex with factor XI andhence it is no longer available as a free component in bloodto participate in other reactions especially those reactionsof the extrinsic pathway which principally lead to the clotformation in vivo This behavior highlights that the intrinsicpathway has a secondary role in blood coagulation in vivocompared with the extrinsic pathway as also evidenced byvarious experimental observations [48]
313 Factors VIII IX and II The next set of PSA-1D wasrealized by varying the initial concentrations of factors VIIIIX and II These factors were selected since both an excess
and a deficiency of their concentrations lead to diseasesrelated to blood clotting
The PSA-1D over factor VIII (Figure 9) shows thatincreasing the initial concentration of this factor resultsin decreasing the CT suggesting the possible presence ofhypercoagulable states in these perturbed conditions As amatter of fact high levels of factor VIII cause an increasedrisk of deep vein thrombosis and pulmonary embolism[49] On the other hand individuals with less than 1of the average concentration of factor VIII show a severehaemophilia A characterized by higher CT (Figure 9) andrequire infusions of plasma containing the deficient factorotherwise frequent spontaneous bleeding would occur [50]When the concentration of this factor is between 5 and 30of the average concentration individuals still risk bleeding incase of trauma [50]
14 BioMed Research International
250
300
350
400
450
500
550
600
650
700
Clot
ting
time (
s)
Factor IX (M)
1eminus
14
1eminus
12
1eminus
10
1eminus
8
1eminus
6
1eminus
4
1eminus
2
1e0
Figure 10 Clotting time at different initial concentrations of factor IXThe reference value is 900sdot10minus8M the sweep range is [900sdot10minus14 900sdot10minus2]
0
100
200
300
400
500
600
700
Clot
ting
time (
s)
Factor II (M)
1eminus
12
1eminus
10
1eminus
8
1eminus
6
1eminus
4
1eminus
2
1e0
1e2
Figure 11 Clotting time at different initial concentrations of thrombinThe reference value is 140sdot10minus6M the sweep range is [140sdot10minus12 140sdot100]
The PSA-1D over factor IX (Figure 10) shows only aslight decrease of the CT as the initial concentration offIX increases this is in contrast to recent studies thatdemonstrated how the excess of factor IX leads to anincreased risk of deep vein thrombosis [51] This result canbe explained by considering that (i) a high concentrationof factor IX is not sufficient to bring about coagulationproblems though when the concentration of other factorsis above the average value (yet not at pathological levels)prothrombotic states can be observed (ii) in this model weconsider average values as initial concentrations of factorshowever individuals are characterized by different (balanced)combinations of procoagulant and anticoagulant factor levelsthat altogether contribute to define a unique coagulationphenotype that reflects the developmental environmentalgenetic nutritional and pharmacological influences of eachindividual [52] On the contrary the lack of factor IX causes
haemophilia B characterized by higher CT with respect tothe reference value (Figure 10)
Furthermore by comparing the PSA-1D of factors VIIIand IX (Figures 9 and 10) it is clear that haemophilia A ismore serious than haemophilia B since the CT achieved inconditions of factor VIII deficiency is higher than the CTobtained in the case of factor IX deficiency
In both PSA-1D over factors VIII and IX we observedafter the initial decrease of the CT an unexpected increaseof the CT as the factor concentration increasesThis counter-intuitive behavior arises at very high concentrations of thesefactors (with respect to the average physiological levels) andto the best of our knowledge it was never observed in vivoNonetheless it would be interesting to verify by means of adhoc laboratory experiments if the model correctly describesthe behavior of the BCC even in these conditions or on thecontrary it is not predictive in these extreme situations
BioMed Research International 15
The PSA-1D over factor II (Figure 11) shows a dramaticdecrease of the CT as the initial concentration of this factorincreases (with respect to the average physiological level)This behavior resembles the effects of hypercoagulability (orthrombophilia) a disease caused by mutation G20210A inthe prothrombin gene [53] that causes an increase of theprothrombin level (factor II) in the blood flow resulting inan excessive formation of the active form of this factor thusheightening venous thrombosis risks [54] Hypercoagulabil-ity is usually treated with warfarin therapy or with otheranticoagulants with a similar effect These drugs decrease thecapacity of coagulation factors to become active preventingthe formation of unwanted thrombi
On the other hand when the initial concentration offactor II is low we achieved the effects of prothrombindeficiency a rare autosomal recessive disease that causes atendency to severe bleeding [29 55] As shown in Figure 11 aconcentration equal to 10of the physiological value of factorII (ie 14 sdot 10minus6 M) leads to clotting effects similar to severehaemophilia A
32 PSA-2D of the BCCModel We present here the results ofthe PSA-2D on the BCCmodel where couples of parameterswere varied to analyze the possible effects arising from thecombined perturbation of their values
321 Reactions 11990327
and 11990358 Figure 12 shows the effect of
the simultaneous variation of constants 11989627
and 11989658
(overthe same sweep ranges considered in the two PSA-1DSection 31) This result remarks that reaction 119903
27 involved
in the common pathway has a stronger influence on theBCC and that there is a synergic interplay between thesetwo reactions In particular when the value of 119896
27is low and
the value of 11989658
is high the CT is higher than the valuesachieved when only a single constant is changed because inthis condition both the intrinsic and the common pathwaysare simultaneously inhibited
322 Factor VIII Factor IX and Tissue Factor In the last twoPSA-2Dwe varied the initial concentration of factor VIII andTissue Factor and the initial concentration of factor IX andTissue Factor respectively
The initial concentrations of factors VIII and IX were var-ied over four orders of magnitude using their physiologicalvalues as upper limit for the sweep ranges the concentrationof Tissue Factor was varied over four orders of magnitudetwo above and two below its reference value (see Table 2)The rationale behind this choice is to observe how the BCCmodel in conditions corresponding to different states ofhaemophilia (obtained by decreasing the concentrations offactors VIII and IX) behaves with different initial concentra-tions of the Tissue Factor which is the upstream factor of theextrinsic pathway that is the most important element of theBCC
The results of these PSA-2D show that with respect tothe condition of haemophilia B in the case of haemophiliaA the amount of Tissue Factor (below its reference value)has a negligible influence on the CT as indicated by the
presence of a plateau in Figure 13 on the contrary concerninghaemophilia B a deficiency of Tissue Factor leads to anincrease of the CT especially when factor IX is present in lowconcentrations (Figure 14)
The different results achieved in the two PSA-2D are dueto the presence in the BCC model of a direct interactionbetweenTissue Factor and factor IX bymeans of theTF-fVIIacomplex (see reactions 119903
13 119903
15in Table 1) Indeed the lack
of Tissue Factor directly affects the concentration of activefactor IX which results in a strong alteration of the CT withrespect to physiological conditions
33 CPU versus GPU Performance Comparison In orderto show the relevant speedup achieved by coagSODA wepresent here the comparison of the computational effortrequired by GPU and CPU for the simulation of the BCCmodel The performances of our GPU simulator were com-pared with those obtained using the LSODA algorithmimplemented in the software COPASI [56] executing onthe CPU the same set of simulations that were run on theGPU COPASI is single-threaded and does not exploit thephysical and logical cores of the CPU therefore it representsa good benchmark as a single-node CPU-bound simulator ofbiological systems
In all simulations executed on both GPU and CPU westored 100 samples uniformly distributed in the time intervalconsidered for each simulation that is [0 700] seconds of thedynamics of all chemical species involved in the BCCmodelThe settings for the LSODA algorithm were the followingrelative error 1 sdot 10minus7 absolute error 1 sdot 10minus13 and maximumnumber of internal steps set to 20000
331 Benchmark Details The GPU used for the simulationsis a Nvidia Tesla K20c equipped with 2496 cores organized in13 streamingmultiprocessors GPU clock 706MHz and 5GBofDDR5RAM In all tests coagSODAwas compiled and exe-cuted with version 55 of CUDA libraries Even though thisGPU has compute capability 35 and is based on the GK110Kepler architecture currently coagSODA does not exploitany of the new features (eg Dynamic Parallelism Hyper-QRemote DMA) with the exception of the new ISA encodingwhich allows threads to exploit an increased number ofregisters (255 instead of 63) reducing register spilling intoglobal memory and increasing performances [57] Moreoveras described in Section 22 the Kepler architecture offers thepossibility to reconfigure the 64KB on-chip cache balancingbetween L1 cache and shared memory Since coagSODAexploits the shared memory to reduce the latencies duringthe access to the concentration values of the BCC model weopted for the following configuration 48KB for the sharedmemory 16 KB for the L1 cache
The performance of the GPU was compared against apersonal computer equippedwith a dual-core CPU Intel Corei5 frequency 23 GHz 4GB of DDR3 RAM running theoperating systemMacOSXLion 1075The software used as acoagSODA-equivalent single-threaded CPU implementationis COPASI version 48 (build 35)
16 BioMed Research International
280
300
320
340
360
380
400
420
440
1e41e6
1e91e11 1e5
1e7
1e9
1e11
280 320 360 400 440
Clot
ting
time (
s)
k 58(M
minus1 sminus1 )
k27 (Mminus1sminus1)
Figure 12 Clotting time according to PSA-2D over reaction constants 11989627and 119896
58 The sweep ranges are [2 sdot 104 2 sdot 1010] and [1 sdot 105 2 sdot 1011]
respectively
100
200
300
400
Tissue factor (M) Factor VIII
(M)
Clot
ting
time (
s)
100
150
200
250
300
350
400
450
1eminus131eminus11
1eminus91eminus14
1eminus12
1eminus10
Figure 13 Clotting time at different initial concentrations of Tissue Factor and factor VIII The sweep ranges are [500 sdot 10minus14 500 sdot 10minus10]and [700 sdot 10minus14 500 sdot 10minus10] respectively
A direct comparison of the performances between thesetwo different architectures and implementations is not aneasy task since CPUs are optimized for single-thread exe-cution and exploit a large number of optimizations (eghigher clock frequency instruction caching pipelining outof order execution and branch prediction) whereas GPUsare optimized for graphics processing and parallel exe-cution of identical code relying only on multilevel datacaching For this reason in this paper we propose anempirical comparison of the performances based on therunning times of identical batches of simulations of the BCCmodel
332 Benchmark Results In Figure 15 we show the compar-ison of the running times required to run several batchesof simulations executed to carry out the PSA-1D over thereaction constant 119896
27 The choice of comparing the perfor-
mances of the GPU and CPU implementations by executingbatches of simulations that are related to a PSA instead ofrunning 119899 independent but identical simulations (ie allcharacterized by the same parameterization of the model)is due to the fact that these results represent a more realisticscenario in the computational analysis of biological systemswhereby it is useful to investigate large search spaces ofparameters corresponding to different perturbed conditionsof themodel [26]Moreover for the evaluation of the running
time the execution of a batch of 119899 identical deterministicsimulations would be futile The figure clearly shows thatcoagSODA always performs better than the CPU counter-part In particular while the CPU performance increaseslinearly with the number of simulations the running timesare strongly reduced on the GPU in this case the overallrunning time roughly corresponds to the time required forthe execution of the slowest simulations This is due to thefact that different parameterizations may require differenttime steps for LSODA to converge and the execution ofa block terminates as long as all the threads it containsterminate In turn the execution of a kernel terminates whenall blocks terminate for this reason a single simulationwhose dynamics requires more steps than the others mayaffect the overall running time
In Table 3 we report the running times of all batchesof simulations along with the speedup achieved on theGPU In particular these results highlight that the advantageof exploiting the GPUs for the simulation of the BCCmodel becomes more evident as the number of simula-tions increases with a 181times speedup when a PSA with10000 different parameterizations is executed Therefore theGPU accelerated analysis of the BCC model with coag-SODA represents a novel relevant computational means toinvestigate the behavior of this complex biological systemunder nonphysiological conditions and could be exploited
BioMed Research International 17
0
200
400
600
Tissue factor (M) Factor IX (M
)
Clot
ting
time (
s)
0
100
200
300
400
500
600
700
1eminus12
1eminus10
1eminus8
1eminus131eminus11
1eminus9
Figure 14 Clotting time at different initial concentrations of Tissue Factor and factor IXThe sweep ranges are [500 sdot 10minus14 500 sdot 10minus10] and[900 sdot 10
minus12 900 sdot 10
minus8] respectively
5 10 50 100 500 1000 5000 10000
Runn
ing
time (
s)
Number of simulations
CPUGPU
1e2
1e3
1e4
1e5
1e6
1e7
Figure 15 Comparison of the computational time required to execute an increasing number of simulations of the BCCmodel on CPU (IntelCore i5 23 GHz) and GPU (Nvidia Tesla K20c GPU clock 706MHz) The computational time is expressed in seconds The time valuesrelated to 1000 5000 and 10000 simulations on the CPU were estimated by regression (see also Table 3)
Table 3 CPU versus GPU performance comparison
Number ofsimulations CPU time (sec) GPU time (sec) Speedup
5 3808 1278 298times10 10604 2329 456times50 51704 8058 642times100 116528 10977 1061times500 536058 17394 3081times1000 1073582lowast 19982 5373times5000 5363507lowast 30960 1732times10000 10724975lowast 58951 1819timeslowastValues estimated by regression
to efficiently determine the response of the BCC to differenttherapeutical drugs
4 Conclusions
Thanks to their high-performance computing capabilitiesand the very low costs GPUs nowadays represent a suitable
technology for the development and the application of par-allel computational methods for in silico analysis of complexbiological systems However the implementation of efficientcomputational tools able to fully exploit the large potentialityof GPUs is still challenging since good programming skillsare required to implement GPU-based algorithmic methodsand to handle specific features as an optimal usage ofmemoryor the communication bandwidth between GPU and CPUMoreover algorithms cannot be directly ported on the GPUbecause of the limited programming capabilities allowed byGPUkernels as amatter of fact they need to be redesigned tomeet the requirements of the underlying SIMD architecture
In this work we presented coagSODA a GPU-poweredsimulator specifically developed to carry out fast parallelsimulations of the BCC model coagSODA was designed tooffer a black-box solution usable by any final user in aneasy way It relies on cupSODA [24] a numerical integratorfor ODEs that we previously implemented for the GPUarchitecture based on the LSODA algorithm and capableof automatically translating a reaction-based model into aset of coupled ODEs In addition to mass-action kinet-ics coagSODA implements specific functions to compute
18 BioMed Research International
the kinetics of Hill function based reactions such as thoseinvolved in the platelets activity of the BCC model coag-SODA exploits the massive parallel capabilities of modernGPUs and our results demonstrated that it can achieve arelevant reduction of the computational time required toexecutemany concurrent and independent simulations of theBCC dynamics The mutual independence of the simulationsallows fully exploiting the underlying SIMD architecturemoreover coagSODA benefits from an additional speedupthanks to our choice of storing the state of the systeminto the low-latency shared memory (a solution that wasalready implemented in cupSODA) Since the BCC modelis large (96 reactions 71 chemical species) a large amountof shared memory was assigned to each thread stronglyreducing the theoretical occupancy of the GPU that is theratio of active warps with respect to the maximum numberof warps supported by each streaming multiprocessor of theGPU However the results of the analysis presented in thiswork performed on the BCC model show that coagSODAachieves a relevant boost with respect to a reference CPUimplementation For instance in the case of 10000 simula-tions we achieved a noticeable 181times speedup Interestinglythe performances of coagSODA are better than COPASI evenfor small batches of simulations These results indicate thatfor biological models consisting in many reactions and manyspecies our GPU implementation of LSODA becomes moreprofitable than the CPU counterpart as the number of con-current simulations increases making it suitable especiallywhen performing demanding computational analysis suchas for example parameter sweep parameter estimation orsensitivity analysis
As a test case in this paper we presented several parame-ter sweep analyses over reaction constants and initial concen-trations of factors involved in the BCC model Other com-putational analyses on mathematical models of this pathwaywere previously presented For instance in [58] a parameterestimation of the reaction constants of different modelsinvolving the activation of factor X factor V prothrombinand the inactivation of factors was performed The aim ofthis analysis was to discriminate between different modelsto unravel the mechanisms on the basis of the BCC Similaranalyses were executed on a model describing thrombingeneration in plasma since the reaction mechanism thereaction constants and initial concentrations were unknown[59] In [34] a sensitivity analysis of a model consisting of44 species over 34 chemical species was presented reactionconstants were varied between 10 and 1000 of theirreference value to the aim of identifying the most influentialfactors of the BCC
All these analyses were performed bymeans of sequentialsimulations of the models under investigation thereforein general only small batches of simulations could be run(for instance in [34] only 836 simulations were executed toexecute the sensitivity analysis of the model) On the otherhand in this paper we presented the results of different PSAefficiently executed bymeans of coagSODA overall providinguseful information regarding unknownparameters and inter-esting insights into the functioning of the BCC A thoroughsensitivity analysis of the whole BCC model consisting in
around 5 sdot 105 simulations is currently in progress on GPUsby our group exploiting the great computational efficiency ofcoagSODA
The results of our computational analyses should benow validated by means of specifically designed laboratoryexperiments In particular we identified a plausible valuefor the constant of the reaction describing the autoactivationof factor XII (since no reference values can be found inliterature) moreover the results of the PSA over factor IXsuggest that a deficiency of this factor is not enough to causesevere bleeding disorders as haemophilia B but an alterationof other factors seems to be necessary for the occurrence ofsuch condition (eg the lack of Tissue Factor as suggestedby the PSA-2D over Tissue Factor and factor IX) Finallythe PSA over factors VIII and IX showed in a situationcharacterized by very high concentrations of these factorsa counterintuitive behavior in which the clotting time isincreased with respect to the value obtained in physiologicalconditions Indeed it would be interesting to design ad hoclaboratory experiments to verify if the BCCmodel is actuallypredictive in such extreme situations
The coagSODA software and the SBML version of theBCC model used in this work are available from the authorsupon request
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This work is partially supported by SYSBIO Centre forSystems Biology
References
[1] B B Aldridge JM BurkeDA Lauffenburger andP K SorgerldquoPhysicochemicalmodelling of cell signalling pathwaysrdquoNatureCell Biology vol 8 no 11 pp 1195ndash1203 2006
[2] I-C Chou and E O Voit ldquoRecent developments in parameterestimation and structure identification of biochemical andgenomic systemsrdquoMathematical Biosciences vol 219 no 2 pp57ndash83 2009
[3] D Besozzi P Cazzaniga D Pescini G Mauri S Colomboand EMartegani ldquoThe role of feedback control mechanisms onthe establishment of oscillatory regimes in the RascAMPPKApathway in S cerevisiaerdquo EURASIP Journal on Bioinformaticsand Systems Biology vol 2012 no 10 2012
[4] A Saltelli M Ratto T Andres et al Global Sensitivity AnalysisThe Primer Wiley-Interscience 2008
[5] A Raue C Kreutz T Maiwald et al ldquoStructural and practicalidentifiability analysis of partially observed dynamical modelsby exploiting the profile likelihoodrdquo Bioinformatics vol 25 no15 pp 1923ndash1929 2009
[6] A Drager M Kronfeld M J Ziller et al ldquoModeling metabolicnetworks in C glutamicum a comparison of rate laws incombination with various parameter optimization strategiesrdquoBMC Systems Biology vol 3 article 5 2009
BioMed Research International 19
[7] D Besozzi P Cazzaniga G Mauri D Pescini and L Van-neschi ldquoA comparison of genetic algorithms and particle swarmoptimization for parameter estimation in stochastic biochem-ical systemsrdquo in Proceedings of the 7th European ConferenceEvolutionary Computation Machine Learning and Data Miningin Bioinformatics (EvoBIO rsquo09) C Pizzuti M Ritchie and MGiacobini Eds vol 5483 of Lecture Notes in Computer Sciencepp 116ndash127 Springer 2009
[8] M S Nobile D Besozzi P Cazzaniga GMauri andD PescinildquocupSODA aGPU-basedmulti-swarmPSOmethod for param-eter estimation in stochastic biological systems exploitingdiscrete-time target seriesrdquo in Proceedings of the 10th EuropeanConference Evolutionary Computation Machine Learning andData Mining in Bioinformatics (EvoBIO rsquo12) M Giacobini LVanneschi and W Bush Eds vol 7246 of Lecture Notes inComputer Science pp 74ndash85 Springer
[9] M S Nobile D Besozzi P Cazzaniga D Pescini and GMaurildquoReverse engineering of kinetic reaction networks by meansof Cartesian Genetic Programming and Particle Swarm Opti-mizationrdquo in Proceedings of the IEEE Congress on EvolutionaryComputation (CEC rsquo13) pp 1594ndash1601 June 2013
[10] M S Nobile ldquoEvolutionary inference of biochemical inter-action networks accelerated on Graphics Processing Unitsrdquoin Proceedings of the 11th International Conference on HighPerformance Computingamp Simulation 2013 (HPCS rsquo13) pp 668ndash670 Helsinki Finland 2013
[11] K Kentzoglanakis and M Poole ldquoA swarm intelligence frame-work for reconstructing gene networks searching for bio-logically plausible architecturesrdquo IEEEACM Transactions onComputational Biology and Bioinformatics vol 9 no 2 pp 358ndash371 2012
[12] J R Koza W Mydlowec G Lanza Y Jessen and M A KeaneldquoAutomatic computational discovery of chemical reaction net-works using genetic programmingrdquo inComputationalDiscoveryof Scientific Knowledge S Dzeroski and L Todorovski Edsvol 4660 of Lecture Notes in Computer Science pp 205ndash227Springer 2007
[13] S Ando E Sakamoto and H Iba ldquoEvolutionary modeling andinference of gene networkrdquo Information Sciences vol 145 no3-4 pp 237ndash259 2002
[14] L Dematte and D Prandi ldquoGPU computing for systemsbiologyrdquo Briefings in Bioinformatics vol 11 no 3 Article IDbbq006 pp 323ndash333 2010
[15] J L Payne N A Sinnott-Armstrong and J HMoore ldquoExploit-ing graphics processing units for computational biology andbioinformaticsrdquo Interdisciplinary Sciences Computational LifeSciences vol 2 no 3 pp 213ndash220 2010
[16] M J Harvey and G De Fabritiis ldquoA survey of computationalmolecular science using graphics processing unitsrdquoWiley Inter-disciplinary Reviews Computational Molecular Science vol 2no 5 pp 734ndash742 2012
[17] Y Zhou J Liepe X Sheng M P H Stumpf and C BarnesldquoGPU accelerated biochemical network simulationrdquo Bioinfor-matics vol 27 no 6 Article ID btr015 pp 874ndash876 2011
[18] M Vigelius A Lane and B Meyer ldquoAccelerating reaction-diffusion simulations with general-purpose graphics processingunitsrdquo Bioinformatics vol 27 no 2 Article ID btq622 pp 288ndash290 2011
[19] D J Wilkinson ldquoStochastic modelling for quantitative descrip-tion of heterogeneous biological systemsrdquo Nature ReviewsGenetics vol 10 no 2 pp 122ndash133 2009
[20] J C Butcher ldquoTrees and numerical methods for ordinarydifferential equationsrdquo Numerical Algorithms vol 53 no 2-3pp 153ndash170 2010
[21] D T Gillespie ldquoStochastic simulation of chemical kineticsrdquoAnnual Review of Physical Chemistry vol 58 pp 35ndash55 2007
[22] L Petzold ldquoAutomatic selection of methods for solving stiffand nonstiff systems of ordinary differential equationsrdquo SIAMJournal of Scientific and Statistical Computing vol 4 no 1 pp136ndash148 1983
[23] Y Cao D T Gillespie and L R Petzold ldquoEfficient step sizeselection for the tau-leaping simulation methodrdquo The Journalof Chemical Physics vol 124 no 4 2006
[24] M S Nobile D Besozzi P Cazzaniga GMauri andD PescinildquocupSODA a CUDA-powered simulator of mass-action kinet-icsrdquo in Proceedings of the 12th International Conference onParallel Computing Technologies (PaCT rsquo13) V Malyshkin Edvol 7979 of Lecture Notes in Computer Science pp 344ndash357
[25] M S Nobile P Cazzaniga D Besozzi and G Mauri ldquoGPU-accelerated simulations of mass-action kinetics models withcupSODArdquoThe Journal of Supercomputing 2014
[26] M S Nobile P Cazzaniga D Besozzi D Pescini and GMauri ldquocuTauLeaping a GPU-powered tau-leaping stochasticsimulator for massive parallel analyses of biological systemsrdquoPLoS ONE vol 9 no 3 2014
[27] K Rogers Blood Physiology and CirculationTheHuman BodyBritannica Educational Publishing 2010
[28] A D Michelson Platelets Elsevier 2nd edition 2007[29] R W Colman J Hirsh V J Marder A W Clowes and J
N George Hemostasis and Thrombosis Basic Principles andClinical Practice Lippincott Williams ampWilkins 2006
[30] M S Chatterjee W S Denney H Jing and S L DiamondldquoSystems biology of coagulation initiation Kinetics of thrombingeneration in resting and activated human bloodrdquo PLoS Com-putational Biology vol 6 no 9 Article ID e1000950 2010
[31] M Bellini D Besozzi P Cazzaniga GMauri andM S NobileldquoSimulation and analysis of the blood coagulation cascadeaccelerated on GPUrdquo in Proceedings of the 22nd EuromicroInternational Conference on Parallel Distributed and Network-Based Processing (PDP) pp 590ndash593 IEEE Computer SocietyConference Publishing Services 2014
[32] T Wajima G K Isbister and S B Duffull ldquoA comprehensivemodel for the humoral coagulation network in humansrdquo Clini-cal Pharmacology and Therapeutics vol 86 no 3 pp 290ndash2982009
[33] K Lo W S Denney and S L Diamond ldquoStochastic modelingof blood coagulation initiationrdquo Pathophysiology of Haemostasisand Thrombosis vol 34 no 2-3 pp 80ndash90 2006
[34] C M Danforth T Orfeo K G Mann K E Brummel-Ziedins and S J Everse ldquoThe impact of uncertainty in a bloodcoagulation modelrdquo Mathematical Medicine and Biology vol26 no 4 Article ID dqp011 pp 323ndash336 2009
[35] S D Lewis P P Shields and J A Shafer ldquoCharacterizationof the kinetic pathway for liberation of fibrinopeptides duringassembly of fibrinrdquoThe Journal of Biological Chemistry vol 260no 18 pp 10192ndash10199 1985
[36] M F Hockin K C Jones S J Everse and K G Mann ldquoAmodel for the stoichiometric regulation of blood coagulationrdquoThe Journal of Biological Chemistry vol 277 no 21 pp 18322ndash18333 2002
[37] K Brummel-Ziedins C YVossen F R Rosendaal KUmezakiand K G Mann ldquoThe plasma hemostatic proteome thrombin
20 BioMed Research International
generation in healthy individualsrdquo Journal of Thrombosis andHaemostasis vol 3 no 7 pp 1472ndash1481 2005
[38] K E Brummel-Ziedins C Y Vossen S Butenas K G Mannand F R Rosendaal ldquoThrombin generation profiles in deepvenous thrombosisrdquo Journal of Thrombosis and Haemostasisvol 3 no 11 pp 2497ndash2505 2005
[39] S D Bungay P A Gentry and R D Gentry ldquoA mathematicalmodel of lipid-mediated thrombin generationrdquo MathematicalMedicine and Biology vol 20 no 1 pp 105ndash129 2003
[40] S Butenas T Orfeo M T Gissel K E Brummel and K GMann ldquoThe significance of circulating factor IXa in bloodrdquoThe Journal of Biological Chemistry vol 279 no 22 pp 22875ndash22882 2004
[41] D Voet and J G Voet Biochemistry Wiley 2011[42] A C Guyton and J E Hall Textbook of Medical Physiology
Saunders 2010[43] M F M James and A M Roche ldquoDose-response relationship
between plasma ionized calcium concentration and thrombe-lastographyrdquo Journal of Cardiothoracic and Vascular Anesthesiavol 18 no 5 pp 581ndash586 2004
[44] D L Nelson and M M Cox Lehninger Principles of Biochem-istry W H Freeman Company 2004
[45] R M Farber ldquoTopical perspective on massive threading andparallelismrdquo Journal of Molecular Graphics and Modelling vol30 pp 82ndash89 2011
[46] OWolkenhauerMUllahWKolch andK-HCho ldquoModelingand simulation of intracellular dynamics choosing an appropri-ate frameworkrdquo IEEETransactions onNanobioscience vol 3 no3 pp 200ndash207 2004
[47] B H Estridge A P Reynolds and N J Walters Basic MedicalLaboratory Techniques Cengage Learning 4th edition 2000
[48] T Renne A H Schmaier K F Nickel M Blomback and CMaas ldquoIn vivo roles of factor XIIrdquo Blood vol 120 no 22 pp4296ndash4303 2012
[49] P V Jenkins O Rawley O P Smith and J S ODonnellldquoElevated factor VIII levels and risk of venous thrombosisrdquoBritish Journal of Haematology vol 157 no 6 pp 653ndash663 2012
[50] C E Walsh and K M Batt ldquoHemophilia clinical gene therapybrief reviewrdquo Translational Research vol 161 no 4 pp 307ndash3122013
[51] A Van Hylckama Vlieg I K Van Der Linden R M Bertinaand F R Rosendaal ldquoHigh levels of factor IX increase the risk ofvenous thrombosisrdquo Blood vol 95 no 12 pp 3678ndash3682 2000
[52] K Brummel-Ziedins ldquoModels for thrombin generation and riskof diseaserdquo Journal ofThrombosis and Haemostasis vol 11 no 1pp 212ndash223 2013
[53] U Seligsohn and A Lubetsky ldquoGenetic susceptibility to venousthrombosisrdquoTheNew England Journal of Medicine vol 344 no16 pp 1222ndash1231 2001
[54] I Martinelli P Bucciarelli and P M Mannucci ldquoThromboticrisk factors basic pathophysiologyrdquo Critical Care Medicine vol38 no 2 pp S3ndashS9 2010
[55] K Brummel-Ziedins A Undas T Orfeo et al ldquoThrombingeneration in acute coronary syndrome and stable coronaryartery disease dependence on plasma factor compositionrdquoJournal ofThrombosis andHaemostasis vol 6 no 1 pp 104ndash1102008
[56] S Hoops R Gauges C Lee et al ldquoCOPASImdasha complexpathway simulatorrdquo Bioinformatics vol 22 no 24 pp 3067ndash3074 2006
[57] Nvidia NVIDIArsquos Next Generation CUDA Compute Architec-ture Kepler GK110 2012
[58] W Stortelder W Hemker and H Hemker ldquoMathematicalmodelling in blood coagulation simulation and parameterestimationrdquoModelling Analysis and Simulation vol 20 pp 1ndash111997
[59] S M C Orfao G Jank and K Mottaghy ldquoA new mathematicalapproach to modelling thrombin generationrdquo InternationalJournal of Artificial Organs vol 29 no 7 pp 703ndash708 2006
[60] Nvidia Nvidia CUDA C Programming Guide v50 2012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Anatomy Research International
PeptidesInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
International Journal of
Volume 2014
Zoology
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Molecular Biology International
GenomicsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
BioinformaticsAdvances in
Marine BiologyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Signal TransductionJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
BioMed Research International
Evolutionary BiologyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Biochemistry Research International
ArchaeaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Genetics Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in
Virolog y
Hindawi Publishing Corporationhttpwwwhindawicom
Nucleic AcidsJournal of
Volume 2014
Stem CellsInternational
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Enzyme Research
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Microbiology
12 BioMed Research International
324
326
328
330
332
334
336
338
340
Clot
ting
time (
s)
1eminus
10
1eminus
9
1eminus
8
1eminus
7
1eminus
6
1eminus
5
1eminus
4
1eminus
3
k44 (sminus1)
Figure 6 Clotting time according to PSA-1D over constant 11989644of reaction 119903
44 XII rarr XIIaThe reference value of 119896
44used in [30] is 5sdot10minus4sminus1
the sweep range is [5 sdot 10minus10 5 sdot 10minus4]
280
300
320
340
360
380
400
1e4 1e5 1e6 1e7 1e8 1e9 1e10 1e11
Clot
ting
time (
s)
k27 (Mminus1sminus1)
Figure 7 Clotting time according to PSA-1D over constant 11989627of reaction 119903
27 IIa + V rarr IIa + Va The reference value is 2 sdot 107Mminus1 sminus1 the
sweep range is [2 sdot 104 2 sdot 1010]
indicate that the extrinsic pathway is the main responsibleof clot formation Secondly in [30] all constants values havea reference to experimental measurements except for theconstant of this reaction (which corresponds to entry 29 inTable 1 in [30])
Figure 6 shows the results of this PSA-1D where 11989644was
varied over six orders of magnitude considering the valuegiven in [30] as the upper limit of the sweep interval We canobserve that by decreasing the value of 119896
44with respect to
the value considered in [30] the CT increases with respect toits reference value however for values of this constant lowerthan 100 sdot 10minus6 sminus1 the CT remains unaltered and this canbe intuitively explained by the fact that in this situation thefibrinogen is mainly activated by the extrinsic pathway
Consequently we assigned the value 700 sdot 10minus6 sminus1 to11989644 achieving a CT that is comparable to the experimental
observations of the BCC in vivo This new value was used inall PSA discussed in what follows
312 Reactions 11990327
and 11990358 In the next PSA we investigated
the effect of the perturbation of the kinetics of two pivotal
reactions of the BCC model Reaction 11990327 which describes
the catalytic activation of factor V by factor IIa was chosenbecause it represents the main positive feedback within thecommon pathway reaction 119903
58 which is involved in the
activation of factor XI by binding to factor IIa was chosenbecause it represents the main positive feedback in theintrinsic pathway (Figure 1) Moreover preliminary PSA overall reaction constants of the BCC model given in Table 1evidenced that these two reactions are among the mostrelevant steps of the coagulation network
The PSA-1D over 11989627(Figure 7) shows that the CT is very
sensitive to the perturbation of the rate of this reaction whenthe reference value of its constant (ie 119896
27= 2 sdot 10
7Mminus1 sminus1)is either increased or decreased in particular when 119896
27is
very low a plateau in CT is reached since the strength ofthe positive feedback exerted by factor IIa is largely reduceda condition where the contribution of the amplification ofthe hemostatic stimulus (due by the common pathway) tothe formation of the clot is basically not effective On theother side the PSA-1D over 119896
58(Figure 8) shows that while
a decrease of the constant with respect to its reference value
BioMed Research International 13
310
320
330
340
350
360
370
1e5 1e6 1e7 1e8 1e9 1e10 1e11
Clot
ting
time (
s)
k58 (Mminus1sminus1)
Figure 8 Clotting time according to PSA-1D over constant 11989658of reaction 119903
58 IIa + XI rarr IIa-XI The reference value is 1 sdot 108Mminus1 sminus1 the
sweep range is [1 sdot 105 2 sdot 1011]
220
240
260
280
300
320
340
360
380
400
Clot
ting
time (
s)
Factor VIII (M)
Seve
re h
aem
ophi
lia
Mild
hae
mop
hilia
Prot
hrom
botic
stat
e
1eminus
16
1eminus
14
1eminus
12
1eminus
10
1eminus
8
1eminus
6
1eminus
4
1eminus
2
Figure 9 Clotting time at different initial concentrations of factor VIII The reference value is 700 sdot 10minus10M (dashed black line) the sweeprange is [700 sdot 10minus16 700 sdot 10minus4] The blue area indicates a condition in which factor VIII concentration is less than 1 of its physiologicalconcentration corresponding to a situation of severe haemophilia the red area indicates a condition in which factor VIII concentration isbetween 1 and 30 of its physiological concentration corresponding to a situation of mild haemophilia the green area indicates a conditionin which factor VIII concentration is greater than 130 of its physiological concentration corresponding to a situation of hypercoagulability
(ie 11989658= 1 sdot 10
8Mminus1 sminus1) does not have any substantialeffect an increase of its value leads to a progressive increasein the CT This increase is due to the fact that factor IIa issequestered in the formation of a complex with factor XI andhence it is no longer available as a free component in bloodto participate in other reactions especially those reactionsof the extrinsic pathway which principally lead to the clotformation in vivo This behavior highlights that the intrinsicpathway has a secondary role in blood coagulation in vivocompared with the extrinsic pathway as also evidenced byvarious experimental observations [48]
313 Factors VIII IX and II The next set of PSA-1D wasrealized by varying the initial concentrations of factors VIIIIX and II These factors were selected since both an excess
and a deficiency of their concentrations lead to diseasesrelated to blood clotting
The PSA-1D over factor VIII (Figure 9) shows thatincreasing the initial concentration of this factor resultsin decreasing the CT suggesting the possible presence ofhypercoagulable states in these perturbed conditions As amatter of fact high levels of factor VIII cause an increasedrisk of deep vein thrombosis and pulmonary embolism[49] On the other hand individuals with less than 1of the average concentration of factor VIII show a severehaemophilia A characterized by higher CT (Figure 9) andrequire infusions of plasma containing the deficient factorotherwise frequent spontaneous bleeding would occur [50]When the concentration of this factor is between 5 and 30of the average concentration individuals still risk bleeding incase of trauma [50]
14 BioMed Research International
250
300
350
400
450
500
550
600
650
700
Clot
ting
time (
s)
Factor IX (M)
1eminus
14
1eminus
12
1eminus
10
1eminus
8
1eminus
6
1eminus
4
1eminus
2
1e0
Figure 10 Clotting time at different initial concentrations of factor IXThe reference value is 900sdot10minus8M the sweep range is [900sdot10minus14 900sdot10minus2]
0
100
200
300
400
500
600
700
Clot
ting
time (
s)
Factor II (M)
1eminus
12
1eminus
10
1eminus
8
1eminus
6
1eminus
4
1eminus
2
1e0
1e2
Figure 11 Clotting time at different initial concentrations of thrombinThe reference value is 140sdot10minus6M the sweep range is [140sdot10minus12 140sdot100]
The PSA-1D over factor IX (Figure 10) shows only aslight decrease of the CT as the initial concentration offIX increases this is in contrast to recent studies thatdemonstrated how the excess of factor IX leads to anincreased risk of deep vein thrombosis [51] This result canbe explained by considering that (i) a high concentrationof factor IX is not sufficient to bring about coagulationproblems though when the concentration of other factorsis above the average value (yet not at pathological levels)prothrombotic states can be observed (ii) in this model weconsider average values as initial concentrations of factorshowever individuals are characterized by different (balanced)combinations of procoagulant and anticoagulant factor levelsthat altogether contribute to define a unique coagulationphenotype that reflects the developmental environmentalgenetic nutritional and pharmacological influences of eachindividual [52] On the contrary the lack of factor IX causes
haemophilia B characterized by higher CT with respect tothe reference value (Figure 10)
Furthermore by comparing the PSA-1D of factors VIIIand IX (Figures 9 and 10) it is clear that haemophilia A ismore serious than haemophilia B since the CT achieved inconditions of factor VIII deficiency is higher than the CTobtained in the case of factor IX deficiency
In both PSA-1D over factors VIII and IX we observedafter the initial decrease of the CT an unexpected increaseof the CT as the factor concentration increasesThis counter-intuitive behavior arises at very high concentrations of thesefactors (with respect to the average physiological levels) andto the best of our knowledge it was never observed in vivoNonetheless it would be interesting to verify by means of adhoc laboratory experiments if the model correctly describesthe behavior of the BCC even in these conditions or on thecontrary it is not predictive in these extreme situations
BioMed Research International 15
The PSA-1D over factor II (Figure 11) shows a dramaticdecrease of the CT as the initial concentration of this factorincreases (with respect to the average physiological level)This behavior resembles the effects of hypercoagulability (orthrombophilia) a disease caused by mutation G20210A inthe prothrombin gene [53] that causes an increase of theprothrombin level (factor II) in the blood flow resulting inan excessive formation of the active form of this factor thusheightening venous thrombosis risks [54] Hypercoagulabil-ity is usually treated with warfarin therapy or with otheranticoagulants with a similar effect These drugs decrease thecapacity of coagulation factors to become active preventingthe formation of unwanted thrombi
On the other hand when the initial concentration offactor II is low we achieved the effects of prothrombindeficiency a rare autosomal recessive disease that causes atendency to severe bleeding [29 55] As shown in Figure 11 aconcentration equal to 10of the physiological value of factorII (ie 14 sdot 10minus6 M) leads to clotting effects similar to severehaemophilia A
32 PSA-2D of the BCCModel We present here the results ofthe PSA-2D on the BCCmodel where couples of parameterswere varied to analyze the possible effects arising from thecombined perturbation of their values
321 Reactions 11990327
and 11990358 Figure 12 shows the effect of
the simultaneous variation of constants 11989627
and 11989658
(overthe same sweep ranges considered in the two PSA-1DSection 31) This result remarks that reaction 119903
27 involved
in the common pathway has a stronger influence on theBCC and that there is a synergic interplay between thesetwo reactions In particular when the value of 119896
27is low and
the value of 11989658
is high the CT is higher than the valuesachieved when only a single constant is changed because inthis condition both the intrinsic and the common pathwaysare simultaneously inhibited
322 Factor VIII Factor IX and Tissue Factor In the last twoPSA-2Dwe varied the initial concentration of factor VIII andTissue Factor and the initial concentration of factor IX andTissue Factor respectively
The initial concentrations of factors VIII and IX were var-ied over four orders of magnitude using their physiologicalvalues as upper limit for the sweep ranges the concentrationof Tissue Factor was varied over four orders of magnitudetwo above and two below its reference value (see Table 2)The rationale behind this choice is to observe how the BCCmodel in conditions corresponding to different states ofhaemophilia (obtained by decreasing the concentrations offactors VIII and IX) behaves with different initial concentra-tions of the Tissue Factor which is the upstream factor of theextrinsic pathway that is the most important element of theBCC
The results of these PSA-2D show that with respect tothe condition of haemophilia B in the case of haemophiliaA the amount of Tissue Factor (below its reference value)has a negligible influence on the CT as indicated by the
presence of a plateau in Figure 13 on the contrary concerninghaemophilia B a deficiency of Tissue Factor leads to anincrease of the CT especially when factor IX is present in lowconcentrations (Figure 14)
The different results achieved in the two PSA-2D are dueto the presence in the BCC model of a direct interactionbetweenTissue Factor and factor IX bymeans of theTF-fVIIacomplex (see reactions 119903
13 119903
15in Table 1) Indeed the lack
of Tissue Factor directly affects the concentration of activefactor IX which results in a strong alteration of the CT withrespect to physiological conditions
33 CPU versus GPU Performance Comparison In orderto show the relevant speedup achieved by coagSODA wepresent here the comparison of the computational effortrequired by GPU and CPU for the simulation of the BCCmodel The performances of our GPU simulator were com-pared with those obtained using the LSODA algorithmimplemented in the software COPASI [56] executing onthe CPU the same set of simulations that were run on theGPU COPASI is single-threaded and does not exploit thephysical and logical cores of the CPU therefore it representsa good benchmark as a single-node CPU-bound simulator ofbiological systems
In all simulations executed on both GPU and CPU westored 100 samples uniformly distributed in the time intervalconsidered for each simulation that is [0 700] seconds of thedynamics of all chemical species involved in the BCCmodelThe settings for the LSODA algorithm were the followingrelative error 1 sdot 10minus7 absolute error 1 sdot 10minus13 and maximumnumber of internal steps set to 20000
331 Benchmark Details The GPU used for the simulationsis a Nvidia Tesla K20c equipped with 2496 cores organized in13 streamingmultiprocessors GPU clock 706MHz and 5GBofDDR5RAM In all tests coagSODAwas compiled and exe-cuted with version 55 of CUDA libraries Even though thisGPU has compute capability 35 and is based on the GK110Kepler architecture currently coagSODA does not exploitany of the new features (eg Dynamic Parallelism Hyper-QRemote DMA) with the exception of the new ISA encodingwhich allows threads to exploit an increased number ofregisters (255 instead of 63) reducing register spilling intoglobal memory and increasing performances [57] Moreoveras described in Section 22 the Kepler architecture offers thepossibility to reconfigure the 64KB on-chip cache balancingbetween L1 cache and shared memory Since coagSODAexploits the shared memory to reduce the latencies duringthe access to the concentration values of the BCC model weopted for the following configuration 48KB for the sharedmemory 16 KB for the L1 cache
The performance of the GPU was compared against apersonal computer equippedwith a dual-core CPU Intel Corei5 frequency 23 GHz 4GB of DDR3 RAM running theoperating systemMacOSXLion 1075The software used as acoagSODA-equivalent single-threaded CPU implementationis COPASI version 48 (build 35)
16 BioMed Research International
280
300
320
340
360
380
400
420
440
1e41e6
1e91e11 1e5
1e7
1e9
1e11
280 320 360 400 440
Clot
ting
time (
s)
k 58(M
minus1 sminus1 )
k27 (Mminus1sminus1)
Figure 12 Clotting time according to PSA-2D over reaction constants 11989627and 119896
58 The sweep ranges are [2 sdot 104 2 sdot 1010] and [1 sdot 105 2 sdot 1011]
respectively
100
200
300
400
Tissue factor (M) Factor VIII
(M)
Clot
ting
time (
s)
100
150
200
250
300
350
400
450
1eminus131eminus11
1eminus91eminus14
1eminus12
1eminus10
Figure 13 Clotting time at different initial concentrations of Tissue Factor and factor VIII The sweep ranges are [500 sdot 10minus14 500 sdot 10minus10]and [700 sdot 10minus14 500 sdot 10minus10] respectively
A direct comparison of the performances between thesetwo different architectures and implementations is not aneasy task since CPUs are optimized for single-thread exe-cution and exploit a large number of optimizations (eghigher clock frequency instruction caching pipelining outof order execution and branch prediction) whereas GPUsare optimized for graphics processing and parallel exe-cution of identical code relying only on multilevel datacaching For this reason in this paper we propose anempirical comparison of the performances based on therunning times of identical batches of simulations of the BCCmodel
332 Benchmark Results In Figure 15 we show the compar-ison of the running times required to run several batchesof simulations executed to carry out the PSA-1D over thereaction constant 119896
27 The choice of comparing the perfor-
mances of the GPU and CPU implementations by executingbatches of simulations that are related to a PSA instead ofrunning 119899 independent but identical simulations (ie allcharacterized by the same parameterization of the model)is due to the fact that these results represent a more realisticscenario in the computational analysis of biological systemswhereby it is useful to investigate large search spaces ofparameters corresponding to different perturbed conditionsof themodel [26]Moreover for the evaluation of the running
time the execution of a batch of 119899 identical deterministicsimulations would be futile The figure clearly shows thatcoagSODA always performs better than the CPU counter-part In particular while the CPU performance increaseslinearly with the number of simulations the running timesare strongly reduced on the GPU in this case the overallrunning time roughly corresponds to the time required forthe execution of the slowest simulations This is due to thefact that different parameterizations may require differenttime steps for LSODA to converge and the execution ofa block terminates as long as all the threads it containsterminate In turn the execution of a kernel terminates whenall blocks terminate for this reason a single simulationwhose dynamics requires more steps than the others mayaffect the overall running time
In Table 3 we report the running times of all batchesof simulations along with the speedup achieved on theGPU In particular these results highlight that the advantageof exploiting the GPUs for the simulation of the BCCmodel becomes more evident as the number of simula-tions increases with a 181times speedup when a PSA with10000 different parameterizations is executed Therefore theGPU accelerated analysis of the BCC model with coag-SODA represents a novel relevant computational means toinvestigate the behavior of this complex biological systemunder nonphysiological conditions and could be exploited
BioMed Research International 17
0
200
400
600
Tissue factor (M) Factor IX (M
)
Clot
ting
time (
s)
0
100
200
300
400
500
600
700
1eminus12
1eminus10
1eminus8
1eminus131eminus11
1eminus9
Figure 14 Clotting time at different initial concentrations of Tissue Factor and factor IXThe sweep ranges are [500 sdot 10minus14 500 sdot 10minus10] and[900 sdot 10
minus12 900 sdot 10
minus8] respectively
5 10 50 100 500 1000 5000 10000
Runn
ing
time (
s)
Number of simulations
CPUGPU
1e2
1e3
1e4
1e5
1e6
1e7
Figure 15 Comparison of the computational time required to execute an increasing number of simulations of the BCCmodel on CPU (IntelCore i5 23 GHz) and GPU (Nvidia Tesla K20c GPU clock 706MHz) The computational time is expressed in seconds The time valuesrelated to 1000 5000 and 10000 simulations on the CPU were estimated by regression (see also Table 3)
Table 3 CPU versus GPU performance comparison
Number ofsimulations CPU time (sec) GPU time (sec) Speedup
5 3808 1278 298times10 10604 2329 456times50 51704 8058 642times100 116528 10977 1061times500 536058 17394 3081times1000 1073582lowast 19982 5373times5000 5363507lowast 30960 1732times10000 10724975lowast 58951 1819timeslowastValues estimated by regression
to efficiently determine the response of the BCC to differenttherapeutical drugs
4 Conclusions
Thanks to their high-performance computing capabilitiesand the very low costs GPUs nowadays represent a suitable
technology for the development and the application of par-allel computational methods for in silico analysis of complexbiological systems However the implementation of efficientcomputational tools able to fully exploit the large potentialityof GPUs is still challenging since good programming skillsare required to implement GPU-based algorithmic methodsand to handle specific features as an optimal usage ofmemoryor the communication bandwidth between GPU and CPUMoreover algorithms cannot be directly ported on the GPUbecause of the limited programming capabilities allowed byGPUkernels as amatter of fact they need to be redesigned tomeet the requirements of the underlying SIMD architecture
In this work we presented coagSODA a GPU-poweredsimulator specifically developed to carry out fast parallelsimulations of the BCC model coagSODA was designed tooffer a black-box solution usable by any final user in aneasy way It relies on cupSODA [24] a numerical integratorfor ODEs that we previously implemented for the GPUarchitecture based on the LSODA algorithm and capableof automatically translating a reaction-based model into aset of coupled ODEs In addition to mass-action kinet-ics coagSODA implements specific functions to compute
18 BioMed Research International
the kinetics of Hill function based reactions such as thoseinvolved in the platelets activity of the BCC model coag-SODA exploits the massive parallel capabilities of modernGPUs and our results demonstrated that it can achieve arelevant reduction of the computational time required toexecutemany concurrent and independent simulations of theBCC dynamics The mutual independence of the simulationsallows fully exploiting the underlying SIMD architecturemoreover coagSODA benefits from an additional speedupthanks to our choice of storing the state of the systeminto the low-latency shared memory (a solution that wasalready implemented in cupSODA) Since the BCC modelis large (96 reactions 71 chemical species) a large amountof shared memory was assigned to each thread stronglyreducing the theoretical occupancy of the GPU that is theratio of active warps with respect to the maximum numberof warps supported by each streaming multiprocessor of theGPU However the results of the analysis presented in thiswork performed on the BCC model show that coagSODAachieves a relevant boost with respect to a reference CPUimplementation For instance in the case of 10000 simula-tions we achieved a noticeable 181times speedup Interestinglythe performances of coagSODA are better than COPASI evenfor small batches of simulations These results indicate thatfor biological models consisting in many reactions and manyspecies our GPU implementation of LSODA becomes moreprofitable than the CPU counterpart as the number of con-current simulations increases making it suitable especiallywhen performing demanding computational analysis suchas for example parameter sweep parameter estimation orsensitivity analysis
As a test case in this paper we presented several parame-ter sweep analyses over reaction constants and initial concen-trations of factors involved in the BCC model Other com-putational analyses on mathematical models of this pathwaywere previously presented For instance in [58] a parameterestimation of the reaction constants of different modelsinvolving the activation of factor X factor V prothrombinand the inactivation of factors was performed The aim ofthis analysis was to discriminate between different modelsto unravel the mechanisms on the basis of the BCC Similaranalyses were executed on a model describing thrombingeneration in plasma since the reaction mechanism thereaction constants and initial concentrations were unknown[59] In [34] a sensitivity analysis of a model consisting of44 species over 34 chemical species was presented reactionconstants were varied between 10 and 1000 of theirreference value to the aim of identifying the most influentialfactors of the BCC
All these analyses were performed bymeans of sequentialsimulations of the models under investigation thereforein general only small batches of simulations could be run(for instance in [34] only 836 simulations were executed toexecute the sensitivity analysis of the model) On the otherhand in this paper we presented the results of different PSAefficiently executed bymeans of coagSODA overall providinguseful information regarding unknownparameters and inter-esting insights into the functioning of the BCC A thoroughsensitivity analysis of the whole BCC model consisting in
around 5 sdot 105 simulations is currently in progress on GPUsby our group exploiting the great computational efficiency ofcoagSODA
The results of our computational analyses should benow validated by means of specifically designed laboratoryexperiments In particular we identified a plausible valuefor the constant of the reaction describing the autoactivationof factor XII (since no reference values can be found inliterature) moreover the results of the PSA over factor IXsuggest that a deficiency of this factor is not enough to causesevere bleeding disorders as haemophilia B but an alterationof other factors seems to be necessary for the occurrence ofsuch condition (eg the lack of Tissue Factor as suggestedby the PSA-2D over Tissue Factor and factor IX) Finallythe PSA over factors VIII and IX showed in a situationcharacterized by very high concentrations of these factorsa counterintuitive behavior in which the clotting time isincreased with respect to the value obtained in physiologicalconditions Indeed it would be interesting to design ad hoclaboratory experiments to verify if the BCCmodel is actuallypredictive in such extreme situations
The coagSODA software and the SBML version of theBCC model used in this work are available from the authorsupon request
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This work is partially supported by SYSBIO Centre forSystems Biology
References
[1] B B Aldridge JM BurkeDA Lauffenburger andP K SorgerldquoPhysicochemicalmodelling of cell signalling pathwaysrdquoNatureCell Biology vol 8 no 11 pp 1195ndash1203 2006
[2] I-C Chou and E O Voit ldquoRecent developments in parameterestimation and structure identification of biochemical andgenomic systemsrdquoMathematical Biosciences vol 219 no 2 pp57ndash83 2009
[3] D Besozzi P Cazzaniga D Pescini G Mauri S Colomboand EMartegani ldquoThe role of feedback control mechanisms onthe establishment of oscillatory regimes in the RascAMPPKApathway in S cerevisiaerdquo EURASIP Journal on Bioinformaticsand Systems Biology vol 2012 no 10 2012
[4] A Saltelli M Ratto T Andres et al Global Sensitivity AnalysisThe Primer Wiley-Interscience 2008
[5] A Raue C Kreutz T Maiwald et al ldquoStructural and practicalidentifiability analysis of partially observed dynamical modelsby exploiting the profile likelihoodrdquo Bioinformatics vol 25 no15 pp 1923ndash1929 2009
[6] A Drager M Kronfeld M J Ziller et al ldquoModeling metabolicnetworks in C glutamicum a comparison of rate laws incombination with various parameter optimization strategiesrdquoBMC Systems Biology vol 3 article 5 2009
BioMed Research International 19
[7] D Besozzi P Cazzaniga G Mauri D Pescini and L Van-neschi ldquoA comparison of genetic algorithms and particle swarmoptimization for parameter estimation in stochastic biochem-ical systemsrdquo in Proceedings of the 7th European ConferenceEvolutionary Computation Machine Learning and Data Miningin Bioinformatics (EvoBIO rsquo09) C Pizzuti M Ritchie and MGiacobini Eds vol 5483 of Lecture Notes in Computer Sciencepp 116ndash127 Springer 2009
[8] M S Nobile D Besozzi P Cazzaniga GMauri andD PescinildquocupSODA aGPU-basedmulti-swarmPSOmethod for param-eter estimation in stochastic biological systems exploitingdiscrete-time target seriesrdquo in Proceedings of the 10th EuropeanConference Evolutionary Computation Machine Learning andData Mining in Bioinformatics (EvoBIO rsquo12) M Giacobini LVanneschi and W Bush Eds vol 7246 of Lecture Notes inComputer Science pp 74ndash85 Springer
[9] M S Nobile D Besozzi P Cazzaniga D Pescini and GMaurildquoReverse engineering of kinetic reaction networks by meansof Cartesian Genetic Programming and Particle Swarm Opti-mizationrdquo in Proceedings of the IEEE Congress on EvolutionaryComputation (CEC rsquo13) pp 1594ndash1601 June 2013
[10] M S Nobile ldquoEvolutionary inference of biochemical inter-action networks accelerated on Graphics Processing Unitsrdquoin Proceedings of the 11th International Conference on HighPerformance Computingamp Simulation 2013 (HPCS rsquo13) pp 668ndash670 Helsinki Finland 2013
[11] K Kentzoglanakis and M Poole ldquoA swarm intelligence frame-work for reconstructing gene networks searching for bio-logically plausible architecturesrdquo IEEEACM Transactions onComputational Biology and Bioinformatics vol 9 no 2 pp 358ndash371 2012
[12] J R Koza W Mydlowec G Lanza Y Jessen and M A KeaneldquoAutomatic computational discovery of chemical reaction net-works using genetic programmingrdquo inComputationalDiscoveryof Scientific Knowledge S Dzeroski and L Todorovski Edsvol 4660 of Lecture Notes in Computer Science pp 205ndash227Springer 2007
[13] S Ando E Sakamoto and H Iba ldquoEvolutionary modeling andinference of gene networkrdquo Information Sciences vol 145 no3-4 pp 237ndash259 2002
[14] L Dematte and D Prandi ldquoGPU computing for systemsbiologyrdquo Briefings in Bioinformatics vol 11 no 3 Article IDbbq006 pp 323ndash333 2010
[15] J L Payne N A Sinnott-Armstrong and J HMoore ldquoExploit-ing graphics processing units for computational biology andbioinformaticsrdquo Interdisciplinary Sciences Computational LifeSciences vol 2 no 3 pp 213ndash220 2010
[16] M J Harvey and G De Fabritiis ldquoA survey of computationalmolecular science using graphics processing unitsrdquoWiley Inter-disciplinary Reviews Computational Molecular Science vol 2no 5 pp 734ndash742 2012
[17] Y Zhou J Liepe X Sheng M P H Stumpf and C BarnesldquoGPU accelerated biochemical network simulationrdquo Bioinfor-matics vol 27 no 6 Article ID btr015 pp 874ndash876 2011
[18] M Vigelius A Lane and B Meyer ldquoAccelerating reaction-diffusion simulations with general-purpose graphics processingunitsrdquo Bioinformatics vol 27 no 2 Article ID btq622 pp 288ndash290 2011
[19] D J Wilkinson ldquoStochastic modelling for quantitative descrip-tion of heterogeneous biological systemsrdquo Nature ReviewsGenetics vol 10 no 2 pp 122ndash133 2009
[20] J C Butcher ldquoTrees and numerical methods for ordinarydifferential equationsrdquo Numerical Algorithms vol 53 no 2-3pp 153ndash170 2010
[21] D T Gillespie ldquoStochastic simulation of chemical kineticsrdquoAnnual Review of Physical Chemistry vol 58 pp 35ndash55 2007
[22] L Petzold ldquoAutomatic selection of methods for solving stiffand nonstiff systems of ordinary differential equationsrdquo SIAMJournal of Scientific and Statistical Computing vol 4 no 1 pp136ndash148 1983
[23] Y Cao D T Gillespie and L R Petzold ldquoEfficient step sizeselection for the tau-leaping simulation methodrdquo The Journalof Chemical Physics vol 124 no 4 2006
[24] M S Nobile D Besozzi P Cazzaniga GMauri andD PescinildquocupSODA a CUDA-powered simulator of mass-action kinet-icsrdquo in Proceedings of the 12th International Conference onParallel Computing Technologies (PaCT rsquo13) V Malyshkin Edvol 7979 of Lecture Notes in Computer Science pp 344ndash357
[25] M S Nobile P Cazzaniga D Besozzi and G Mauri ldquoGPU-accelerated simulations of mass-action kinetics models withcupSODArdquoThe Journal of Supercomputing 2014
[26] M S Nobile P Cazzaniga D Besozzi D Pescini and GMauri ldquocuTauLeaping a GPU-powered tau-leaping stochasticsimulator for massive parallel analyses of biological systemsrdquoPLoS ONE vol 9 no 3 2014
[27] K Rogers Blood Physiology and CirculationTheHuman BodyBritannica Educational Publishing 2010
[28] A D Michelson Platelets Elsevier 2nd edition 2007[29] R W Colman J Hirsh V J Marder A W Clowes and J
N George Hemostasis and Thrombosis Basic Principles andClinical Practice Lippincott Williams ampWilkins 2006
[30] M S Chatterjee W S Denney H Jing and S L DiamondldquoSystems biology of coagulation initiation Kinetics of thrombingeneration in resting and activated human bloodrdquo PLoS Com-putational Biology vol 6 no 9 Article ID e1000950 2010
[31] M Bellini D Besozzi P Cazzaniga GMauri andM S NobileldquoSimulation and analysis of the blood coagulation cascadeaccelerated on GPUrdquo in Proceedings of the 22nd EuromicroInternational Conference on Parallel Distributed and Network-Based Processing (PDP) pp 590ndash593 IEEE Computer SocietyConference Publishing Services 2014
[32] T Wajima G K Isbister and S B Duffull ldquoA comprehensivemodel for the humoral coagulation network in humansrdquo Clini-cal Pharmacology and Therapeutics vol 86 no 3 pp 290ndash2982009
[33] K Lo W S Denney and S L Diamond ldquoStochastic modelingof blood coagulation initiationrdquo Pathophysiology of Haemostasisand Thrombosis vol 34 no 2-3 pp 80ndash90 2006
[34] C M Danforth T Orfeo K G Mann K E Brummel-Ziedins and S J Everse ldquoThe impact of uncertainty in a bloodcoagulation modelrdquo Mathematical Medicine and Biology vol26 no 4 Article ID dqp011 pp 323ndash336 2009
[35] S D Lewis P P Shields and J A Shafer ldquoCharacterizationof the kinetic pathway for liberation of fibrinopeptides duringassembly of fibrinrdquoThe Journal of Biological Chemistry vol 260no 18 pp 10192ndash10199 1985
[36] M F Hockin K C Jones S J Everse and K G Mann ldquoAmodel for the stoichiometric regulation of blood coagulationrdquoThe Journal of Biological Chemistry vol 277 no 21 pp 18322ndash18333 2002
[37] K Brummel-Ziedins C YVossen F R Rosendaal KUmezakiand K G Mann ldquoThe plasma hemostatic proteome thrombin
20 BioMed Research International
generation in healthy individualsrdquo Journal of Thrombosis andHaemostasis vol 3 no 7 pp 1472ndash1481 2005
[38] K E Brummel-Ziedins C Y Vossen S Butenas K G Mannand F R Rosendaal ldquoThrombin generation profiles in deepvenous thrombosisrdquo Journal of Thrombosis and Haemostasisvol 3 no 11 pp 2497ndash2505 2005
[39] S D Bungay P A Gentry and R D Gentry ldquoA mathematicalmodel of lipid-mediated thrombin generationrdquo MathematicalMedicine and Biology vol 20 no 1 pp 105ndash129 2003
[40] S Butenas T Orfeo M T Gissel K E Brummel and K GMann ldquoThe significance of circulating factor IXa in bloodrdquoThe Journal of Biological Chemistry vol 279 no 22 pp 22875ndash22882 2004
[41] D Voet and J G Voet Biochemistry Wiley 2011[42] A C Guyton and J E Hall Textbook of Medical Physiology
Saunders 2010[43] M F M James and A M Roche ldquoDose-response relationship
between plasma ionized calcium concentration and thrombe-lastographyrdquo Journal of Cardiothoracic and Vascular Anesthesiavol 18 no 5 pp 581ndash586 2004
[44] D L Nelson and M M Cox Lehninger Principles of Biochem-istry W H Freeman Company 2004
[45] R M Farber ldquoTopical perspective on massive threading andparallelismrdquo Journal of Molecular Graphics and Modelling vol30 pp 82ndash89 2011
[46] OWolkenhauerMUllahWKolch andK-HCho ldquoModelingand simulation of intracellular dynamics choosing an appropri-ate frameworkrdquo IEEETransactions onNanobioscience vol 3 no3 pp 200ndash207 2004
[47] B H Estridge A P Reynolds and N J Walters Basic MedicalLaboratory Techniques Cengage Learning 4th edition 2000
[48] T Renne A H Schmaier K F Nickel M Blomback and CMaas ldquoIn vivo roles of factor XIIrdquo Blood vol 120 no 22 pp4296ndash4303 2012
[49] P V Jenkins O Rawley O P Smith and J S ODonnellldquoElevated factor VIII levels and risk of venous thrombosisrdquoBritish Journal of Haematology vol 157 no 6 pp 653ndash663 2012
[50] C E Walsh and K M Batt ldquoHemophilia clinical gene therapybrief reviewrdquo Translational Research vol 161 no 4 pp 307ndash3122013
[51] A Van Hylckama Vlieg I K Van Der Linden R M Bertinaand F R Rosendaal ldquoHigh levels of factor IX increase the risk ofvenous thrombosisrdquo Blood vol 95 no 12 pp 3678ndash3682 2000
[52] K Brummel-Ziedins ldquoModels for thrombin generation and riskof diseaserdquo Journal ofThrombosis and Haemostasis vol 11 no 1pp 212ndash223 2013
[53] U Seligsohn and A Lubetsky ldquoGenetic susceptibility to venousthrombosisrdquoTheNew England Journal of Medicine vol 344 no16 pp 1222ndash1231 2001
[54] I Martinelli P Bucciarelli and P M Mannucci ldquoThromboticrisk factors basic pathophysiologyrdquo Critical Care Medicine vol38 no 2 pp S3ndashS9 2010
[55] K Brummel-Ziedins A Undas T Orfeo et al ldquoThrombingeneration in acute coronary syndrome and stable coronaryartery disease dependence on plasma factor compositionrdquoJournal ofThrombosis andHaemostasis vol 6 no 1 pp 104ndash1102008
[56] S Hoops R Gauges C Lee et al ldquoCOPASImdasha complexpathway simulatorrdquo Bioinformatics vol 22 no 24 pp 3067ndash3074 2006
[57] Nvidia NVIDIArsquos Next Generation CUDA Compute Architec-ture Kepler GK110 2012
[58] W Stortelder W Hemker and H Hemker ldquoMathematicalmodelling in blood coagulation simulation and parameterestimationrdquoModelling Analysis and Simulation vol 20 pp 1ndash111997
[59] S M C Orfao G Jank and K Mottaghy ldquoA new mathematicalapproach to modelling thrombin generationrdquo InternationalJournal of Artificial Organs vol 29 no 7 pp 703ndash708 2006
[60] Nvidia Nvidia CUDA C Programming Guide v50 2012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Anatomy Research International
PeptidesInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
International Journal of
Volume 2014
Zoology
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Molecular Biology International
GenomicsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
BioinformaticsAdvances in
Marine BiologyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Signal TransductionJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
BioMed Research International
Evolutionary BiologyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Biochemistry Research International
ArchaeaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Genetics Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in
Virolog y
Hindawi Publishing Corporationhttpwwwhindawicom
Nucleic AcidsJournal of
Volume 2014
Stem CellsInternational
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Enzyme Research
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Microbiology
BioMed Research International 13
310
320
330
340
350
360
370
1e5 1e6 1e7 1e8 1e9 1e10 1e11
Clot
ting
time (
s)
k58 (Mminus1sminus1)
Figure 8 Clotting time according to PSA-1D over constant 11989658of reaction 119903
58 IIa + XI rarr IIa-XI The reference value is 1 sdot 108Mminus1 sminus1 the
sweep range is [1 sdot 105 2 sdot 1011]
220
240
260
280
300
320
340
360
380
400
Clot
ting
time (
s)
Factor VIII (M)
Seve
re h
aem
ophi
lia
Mild
hae
mop
hilia
Prot
hrom
botic
stat
e
1eminus
16
1eminus
14
1eminus
12
1eminus
10
1eminus
8
1eminus
6
1eminus
4
1eminus
2
Figure 9 Clotting time at different initial concentrations of factor VIII The reference value is 700 sdot 10minus10M (dashed black line) the sweeprange is [700 sdot 10minus16 700 sdot 10minus4] The blue area indicates a condition in which factor VIII concentration is less than 1 of its physiologicalconcentration corresponding to a situation of severe haemophilia the red area indicates a condition in which factor VIII concentration isbetween 1 and 30 of its physiological concentration corresponding to a situation of mild haemophilia the green area indicates a conditionin which factor VIII concentration is greater than 130 of its physiological concentration corresponding to a situation of hypercoagulability
(ie 11989658= 1 sdot 10
8Mminus1 sminus1) does not have any substantialeffect an increase of its value leads to a progressive increasein the CT This increase is due to the fact that factor IIa issequestered in the formation of a complex with factor XI andhence it is no longer available as a free component in bloodto participate in other reactions especially those reactionsof the extrinsic pathway which principally lead to the clotformation in vivo This behavior highlights that the intrinsicpathway has a secondary role in blood coagulation in vivocompared with the extrinsic pathway as also evidenced byvarious experimental observations [48]
313 Factors VIII IX and II The next set of PSA-1D wasrealized by varying the initial concentrations of factors VIIIIX and II These factors were selected since both an excess
and a deficiency of their concentrations lead to diseasesrelated to blood clotting
The PSA-1D over factor VIII (Figure 9) shows thatincreasing the initial concentration of this factor resultsin decreasing the CT suggesting the possible presence ofhypercoagulable states in these perturbed conditions As amatter of fact high levels of factor VIII cause an increasedrisk of deep vein thrombosis and pulmonary embolism[49] On the other hand individuals with less than 1of the average concentration of factor VIII show a severehaemophilia A characterized by higher CT (Figure 9) andrequire infusions of plasma containing the deficient factorotherwise frequent spontaneous bleeding would occur [50]When the concentration of this factor is between 5 and 30of the average concentration individuals still risk bleeding incase of trauma [50]
14 BioMed Research International
250
300
350
400
450
500
550
600
650
700
Clot
ting
time (
s)
Factor IX (M)
1eminus
14
1eminus
12
1eminus
10
1eminus
8
1eminus
6
1eminus
4
1eminus
2
1e0
Figure 10 Clotting time at different initial concentrations of factor IXThe reference value is 900sdot10minus8M the sweep range is [900sdot10minus14 900sdot10minus2]
0
100
200
300
400
500
600
700
Clot
ting
time (
s)
Factor II (M)
1eminus
12
1eminus
10
1eminus
8
1eminus
6
1eminus
4
1eminus
2
1e0
1e2
Figure 11 Clotting time at different initial concentrations of thrombinThe reference value is 140sdot10minus6M the sweep range is [140sdot10minus12 140sdot100]
The PSA-1D over factor IX (Figure 10) shows only aslight decrease of the CT as the initial concentration offIX increases this is in contrast to recent studies thatdemonstrated how the excess of factor IX leads to anincreased risk of deep vein thrombosis [51] This result canbe explained by considering that (i) a high concentrationof factor IX is not sufficient to bring about coagulationproblems though when the concentration of other factorsis above the average value (yet not at pathological levels)prothrombotic states can be observed (ii) in this model weconsider average values as initial concentrations of factorshowever individuals are characterized by different (balanced)combinations of procoagulant and anticoagulant factor levelsthat altogether contribute to define a unique coagulationphenotype that reflects the developmental environmentalgenetic nutritional and pharmacological influences of eachindividual [52] On the contrary the lack of factor IX causes
haemophilia B characterized by higher CT with respect tothe reference value (Figure 10)
Furthermore by comparing the PSA-1D of factors VIIIand IX (Figures 9 and 10) it is clear that haemophilia A ismore serious than haemophilia B since the CT achieved inconditions of factor VIII deficiency is higher than the CTobtained in the case of factor IX deficiency
In both PSA-1D over factors VIII and IX we observedafter the initial decrease of the CT an unexpected increaseof the CT as the factor concentration increasesThis counter-intuitive behavior arises at very high concentrations of thesefactors (with respect to the average physiological levels) andto the best of our knowledge it was never observed in vivoNonetheless it would be interesting to verify by means of adhoc laboratory experiments if the model correctly describesthe behavior of the BCC even in these conditions or on thecontrary it is not predictive in these extreme situations
BioMed Research International 15
The PSA-1D over factor II (Figure 11) shows a dramaticdecrease of the CT as the initial concentration of this factorincreases (with respect to the average physiological level)This behavior resembles the effects of hypercoagulability (orthrombophilia) a disease caused by mutation G20210A inthe prothrombin gene [53] that causes an increase of theprothrombin level (factor II) in the blood flow resulting inan excessive formation of the active form of this factor thusheightening venous thrombosis risks [54] Hypercoagulabil-ity is usually treated with warfarin therapy or with otheranticoagulants with a similar effect These drugs decrease thecapacity of coagulation factors to become active preventingthe formation of unwanted thrombi
On the other hand when the initial concentration offactor II is low we achieved the effects of prothrombindeficiency a rare autosomal recessive disease that causes atendency to severe bleeding [29 55] As shown in Figure 11 aconcentration equal to 10of the physiological value of factorII (ie 14 sdot 10minus6 M) leads to clotting effects similar to severehaemophilia A
32 PSA-2D of the BCCModel We present here the results ofthe PSA-2D on the BCCmodel where couples of parameterswere varied to analyze the possible effects arising from thecombined perturbation of their values
321 Reactions 11990327
and 11990358 Figure 12 shows the effect of
the simultaneous variation of constants 11989627
and 11989658
(overthe same sweep ranges considered in the two PSA-1DSection 31) This result remarks that reaction 119903
27 involved
in the common pathway has a stronger influence on theBCC and that there is a synergic interplay between thesetwo reactions In particular when the value of 119896
27is low and
the value of 11989658
is high the CT is higher than the valuesachieved when only a single constant is changed because inthis condition both the intrinsic and the common pathwaysare simultaneously inhibited
322 Factor VIII Factor IX and Tissue Factor In the last twoPSA-2Dwe varied the initial concentration of factor VIII andTissue Factor and the initial concentration of factor IX andTissue Factor respectively
The initial concentrations of factors VIII and IX were var-ied over four orders of magnitude using their physiologicalvalues as upper limit for the sweep ranges the concentrationof Tissue Factor was varied over four orders of magnitudetwo above and two below its reference value (see Table 2)The rationale behind this choice is to observe how the BCCmodel in conditions corresponding to different states ofhaemophilia (obtained by decreasing the concentrations offactors VIII and IX) behaves with different initial concentra-tions of the Tissue Factor which is the upstream factor of theextrinsic pathway that is the most important element of theBCC
The results of these PSA-2D show that with respect tothe condition of haemophilia B in the case of haemophiliaA the amount of Tissue Factor (below its reference value)has a negligible influence on the CT as indicated by the
presence of a plateau in Figure 13 on the contrary concerninghaemophilia B a deficiency of Tissue Factor leads to anincrease of the CT especially when factor IX is present in lowconcentrations (Figure 14)
The different results achieved in the two PSA-2D are dueto the presence in the BCC model of a direct interactionbetweenTissue Factor and factor IX bymeans of theTF-fVIIacomplex (see reactions 119903
13 119903
15in Table 1) Indeed the lack
of Tissue Factor directly affects the concentration of activefactor IX which results in a strong alteration of the CT withrespect to physiological conditions
33 CPU versus GPU Performance Comparison In orderto show the relevant speedup achieved by coagSODA wepresent here the comparison of the computational effortrequired by GPU and CPU for the simulation of the BCCmodel The performances of our GPU simulator were com-pared with those obtained using the LSODA algorithmimplemented in the software COPASI [56] executing onthe CPU the same set of simulations that were run on theGPU COPASI is single-threaded and does not exploit thephysical and logical cores of the CPU therefore it representsa good benchmark as a single-node CPU-bound simulator ofbiological systems
In all simulations executed on both GPU and CPU westored 100 samples uniformly distributed in the time intervalconsidered for each simulation that is [0 700] seconds of thedynamics of all chemical species involved in the BCCmodelThe settings for the LSODA algorithm were the followingrelative error 1 sdot 10minus7 absolute error 1 sdot 10minus13 and maximumnumber of internal steps set to 20000
331 Benchmark Details The GPU used for the simulationsis a Nvidia Tesla K20c equipped with 2496 cores organized in13 streamingmultiprocessors GPU clock 706MHz and 5GBofDDR5RAM In all tests coagSODAwas compiled and exe-cuted with version 55 of CUDA libraries Even though thisGPU has compute capability 35 and is based on the GK110Kepler architecture currently coagSODA does not exploitany of the new features (eg Dynamic Parallelism Hyper-QRemote DMA) with the exception of the new ISA encodingwhich allows threads to exploit an increased number ofregisters (255 instead of 63) reducing register spilling intoglobal memory and increasing performances [57] Moreoveras described in Section 22 the Kepler architecture offers thepossibility to reconfigure the 64KB on-chip cache balancingbetween L1 cache and shared memory Since coagSODAexploits the shared memory to reduce the latencies duringthe access to the concentration values of the BCC model weopted for the following configuration 48KB for the sharedmemory 16 KB for the L1 cache
The performance of the GPU was compared against apersonal computer equippedwith a dual-core CPU Intel Corei5 frequency 23 GHz 4GB of DDR3 RAM running theoperating systemMacOSXLion 1075The software used as acoagSODA-equivalent single-threaded CPU implementationis COPASI version 48 (build 35)
16 BioMed Research International
280
300
320
340
360
380
400
420
440
1e41e6
1e91e11 1e5
1e7
1e9
1e11
280 320 360 400 440
Clot
ting
time (
s)
k 58(M
minus1 sminus1 )
k27 (Mminus1sminus1)
Figure 12 Clotting time according to PSA-2D over reaction constants 11989627and 119896
58 The sweep ranges are [2 sdot 104 2 sdot 1010] and [1 sdot 105 2 sdot 1011]
respectively
100
200
300
400
Tissue factor (M) Factor VIII
(M)
Clot
ting
time (
s)
100
150
200
250
300
350
400
450
1eminus131eminus11
1eminus91eminus14
1eminus12
1eminus10
Figure 13 Clotting time at different initial concentrations of Tissue Factor and factor VIII The sweep ranges are [500 sdot 10minus14 500 sdot 10minus10]and [700 sdot 10minus14 500 sdot 10minus10] respectively
A direct comparison of the performances between thesetwo different architectures and implementations is not aneasy task since CPUs are optimized for single-thread exe-cution and exploit a large number of optimizations (eghigher clock frequency instruction caching pipelining outof order execution and branch prediction) whereas GPUsare optimized for graphics processing and parallel exe-cution of identical code relying only on multilevel datacaching For this reason in this paper we propose anempirical comparison of the performances based on therunning times of identical batches of simulations of the BCCmodel
332 Benchmark Results In Figure 15 we show the compar-ison of the running times required to run several batchesof simulations executed to carry out the PSA-1D over thereaction constant 119896
27 The choice of comparing the perfor-
mances of the GPU and CPU implementations by executingbatches of simulations that are related to a PSA instead ofrunning 119899 independent but identical simulations (ie allcharacterized by the same parameterization of the model)is due to the fact that these results represent a more realisticscenario in the computational analysis of biological systemswhereby it is useful to investigate large search spaces ofparameters corresponding to different perturbed conditionsof themodel [26]Moreover for the evaluation of the running
time the execution of a batch of 119899 identical deterministicsimulations would be futile The figure clearly shows thatcoagSODA always performs better than the CPU counter-part In particular while the CPU performance increaseslinearly with the number of simulations the running timesare strongly reduced on the GPU in this case the overallrunning time roughly corresponds to the time required forthe execution of the slowest simulations This is due to thefact that different parameterizations may require differenttime steps for LSODA to converge and the execution ofa block terminates as long as all the threads it containsterminate In turn the execution of a kernel terminates whenall blocks terminate for this reason a single simulationwhose dynamics requires more steps than the others mayaffect the overall running time
In Table 3 we report the running times of all batchesof simulations along with the speedup achieved on theGPU In particular these results highlight that the advantageof exploiting the GPUs for the simulation of the BCCmodel becomes more evident as the number of simula-tions increases with a 181times speedup when a PSA with10000 different parameterizations is executed Therefore theGPU accelerated analysis of the BCC model with coag-SODA represents a novel relevant computational means toinvestigate the behavior of this complex biological systemunder nonphysiological conditions and could be exploited
BioMed Research International 17
0
200
400
600
Tissue factor (M) Factor IX (M
)
Clot
ting
time (
s)
0
100
200
300
400
500
600
700
1eminus12
1eminus10
1eminus8
1eminus131eminus11
1eminus9
Figure 14 Clotting time at different initial concentrations of Tissue Factor and factor IXThe sweep ranges are [500 sdot 10minus14 500 sdot 10minus10] and[900 sdot 10
minus12 900 sdot 10
minus8] respectively
5 10 50 100 500 1000 5000 10000
Runn
ing
time (
s)
Number of simulations
CPUGPU
1e2
1e3
1e4
1e5
1e6
1e7
Figure 15 Comparison of the computational time required to execute an increasing number of simulations of the BCCmodel on CPU (IntelCore i5 23 GHz) and GPU (Nvidia Tesla K20c GPU clock 706MHz) The computational time is expressed in seconds The time valuesrelated to 1000 5000 and 10000 simulations on the CPU were estimated by regression (see also Table 3)
Table 3 CPU versus GPU performance comparison
Number ofsimulations CPU time (sec) GPU time (sec) Speedup
5 3808 1278 298times10 10604 2329 456times50 51704 8058 642times100 116528 10977 1061times500 536058 17394 3081times1000 1073582lowast 19982 5373times5000 5363507lowast 30960 1732times10000 10724975lowast 58951 1819timeslowastValues estimated by regression
to efficiently determine the response of the BCC to differenttherapeutical drugs
4 Conclusions
Thanks to their high-performance computing capabilitiesand the very low costs GPUs nowadays represent a suitable
technology for the development and the application of par-allel computational methods for in silico analysis of complexbiological systems However the implementation of efficientcomputational tools able to fully exploit the large potentialityof GPUs is still challenging since good programming skillsare required to implement GPU-based algorithmic methodsand to handle specific features as an optimal usage ofmemoryor the communication bandwidth between GPU and CPUMoreover algorithms cannot be directly ported on the GPUbecause of the limited programming capabilities allowed byGPUkernels as amatter of fact they need to be redesigned tomeet the requirements of the underlying SIMD architecture
In this work we presented coagSODA a GPU-poweredsimulator specifically developed to carry out fast parallelsimulations of the BCC model coagSODA was designed tooffer a black-box solution usable by any final user in aneasy way It relies on cupSODA [24] a numerical integratorfor ODEs that we previously implemented for the GPUarchitecture based on the LSODA algorithm and capableof automatically translating a reaction-based model into aset of coupled ODEs In addition to mass-action kinet-ics coagSODA implements specific functions to compute
18 BioMed Research International
the kinetics of Hill function based reactions such as thoseinvolved in the platelets activity of the BCC model coag-SODA exploits the massive parallel capabilities of modernGPUs and our results demonstrated that it can achieve arelevant reduction of the computational time required toexecutemany concurrent and independent simulations of theBCC dynamics The mutual independence of the simulationsallows fully exploiting the underlying SIMD architecturemoreover coagSODA benefits from an additional speedupthanks to our choice of storing the state of the systeminto the low-latency shared memory (a solution that wasalready implemented in cupSODA) Since the BCC modelis large (96 reactions 71 chemical species) a large amountof shared memory was assigned to each thread stronglyreducing the theoretical occupancy of the GPU that is theratio of active warps with respect to the maximum numberof warps supported by each streaming multiprocessor of theGPU However the results of the analysis presented in thiswork performed on the BCC model show that coagSODAachieves a relevant boost with respect to a reference CPUimplementation For instance in the case of 10000 simula-tions we achieved a noticeable 181times speedup Interestinglythe performances of coagSODA are better than COPASI evenfor small batches of simulations These results indicate thatfor biological models consisting in many reactions and manyspecies our GPU implementation of LSODA becomes moreprofitable than the CPU counterpart as the number of con-current simulations increases making it suitable especiallywhen performing demanding computational analysis suchas for example parameter sweep parameter estimation orsensitivity analysis
As a test case in this paper we presented several parame-ter sweep analyses over reaction constants and initial concen-trations of factors involved in the BCC model Other com-putational analyses on mathematical models of this pathwaywere previously presented For instance in [58] a parameterestimation of the reaction constants of different modelsinvolving the activation of factor X factor V prothrombinand the inactivation of factors was performed The aim ofthis analysis was to discriminate between different modelsto unravel the mechanisms on the basis of the BCC Similaranalyses were executed on a model describing thrombingeneration in plasma since the reaction mechanism thereaction constants and initial concentrations were unknown[59] In [34] a sensitivity analysis of a model consisting of44 species over 34 chemical species was presented reactionconstants were varied between 10 and 1000 of theirreference value to the aim of identifying the most influentialfactors of the BCC
All these analyses were performed bymeans of sequentialsimulations of the models under investigation thereforein general only small batches of simulations could be run(for instance in [34] only 836 simulations were executed toexecute the sensitivity analysis of the model) On the otherhand in this paper we presented the results of different PSAefficiently executed bymeans of coagSODA overall providinguseful information regarding unknownparameters and inter-esting insights into the functioning of the BCC A thoroughsensitivity analysis of the whole BCC model consisting in
around 5 sdot 105 simulations is currently in progress on GPUsby our group exploiting the great computational efficiency ofcoagSODA
The results of our computational analyses should benow validated by means of specifically designed laboratoryexperiments In particular we identified a plausible valuefor the constant of the reaction describing the autoactivationof factor XII (since no reference values can be found inliterature) moreover the results of the PSA over factor IXsuggest that a deficiency of this factor is not enough to causesevere bleeding disorders as haemophilia B but an alterationof other factors seems to be necessary for the occurrence ofsuch condition (eg the lack of Tissue Factor as suggestedby the PSA-2D over Tissue Factor and factor IX) Finallythe PSA over factors VIII and IX showed in a situationcharacterized by very high concentrations of these factorsa counterintuitive behavior in which the clotting time isincreased with respect to the value obtained in physiologicalconditions Indeed it would be interesting to design ad hoclaboratory experiments to verify if the BCCmodel is actuallypredictive in such extreme situations
The coagSODA software and the SBML version of theBCC model used in this work are available from the authorsupon request
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This work is partially supported by SYSBIO Centre forSystems Biology
References
[1] B B Aldridge JM BurkeDA Lauffenburger andP K SorgerldquoPhysicochemicalmodelling of cell signalling pathwaysrdquoNatureCell Biology vol 8 no 11 pp 1195ndash1203 2006
[2] I-C Chou and E O Voit ldquoRecent developments in parameterestimation and structure identification of biochemical andgenomic systemsrdquoMathematical Biosciences vol 219 no 2 pp57ndash83 2009
[3] D Besozzi P Cazzaniga D Pescini G Mauri S Colomboand EMartegani ldquoThe role of feedback control mechanisms onthe establishment of oscillatory regimes in the RascAMPPKApathway in S cerevisiaerdquo EURASIP Journal on Bioinformaticsand Systems Biology vol 2012 no 10 2012
[4] A Saltelli M Ratto T Andres et al Global Sensitivity AnalysisThe Primer Wiley-Interscience 2008
[5] A Raue C Kreutz T Maiwald et al ldquoStructural and practicalidentifiability analysis of partially observed dynamical modelsby exploiting the profile likelihoodrdquo Bioinformatics vol 25 no15 pp 1923ndash1929 2009
[6] A Drager M Kronfeld M J Ziller et al ldquoModeling metabolicnetworks in C glutamicum a comparison of rate laws incombination with various parameter optimization strategiesrdquoBMC Systems Biology vol 3 article 5 2009
BioMed Research International 19
[7] D Besozzi P Cazzaniga G Mauri D Pescini and L Van-neschi ldquoA comparison of genetic algorithms and particle swarmoptimization for parameter estimation in stochastic biochem-ical systemsrdquo in Proceedings of the 7th European ConferenceEvolutionary Computation Machine Learning and Data Miningin Bioinformatics (EvoBIO rsquo09) C Pizzuti M Ritchie and MGiacobini Eds vol 5483 of Lecture Notes in Computer Sciencepp 116ndash127 Springer 2009
[8] M S Nobile D Besozzi P Cazzaniga GMauri andD PescinildquocupSODA aGPU-basedmulti-swarmPSOmethod for param-eter estimation in stochastic biological systems exploitingdiscrete-time target seriesrdquo in Proceedings of the 10th EuropeanConference Evolutionary Computation Machine Learning andData Mining in Bioinformatics (EvoBIO rsquo12) M Giacobini LVanneschi and W Bush Eds vol 7246 of Lecture Notes inComputer Science pp 74ndash85 Springer
[9] M S Nobile D Besozzi P Cazzaniga D Pescini and GMaurildquoReverse engineering of kinetic reaction networks by meansof Cartesian Genetic Programming and Particle Swarm Opti-mizationrdquo in Proceedings of the IEEE Congress on EvolutionaryComputation (CEC rsquo13) pp 1594ndash1601 June 2013
[10] M S Nobile ldquoEvolutionary inference of biochemical inter-action networks accelerated on Graphics Processing Unitsrdquoin Proceedings of the 11th International Conference on HighPerformance Computingamp Simulation 2013 (HPCS rsquo13) pp 668ndash670 Helsinki Finland 2013
[11] K Kentzoglanakis and M Poole ldquoA swarm intelligence frame-work for reconstructing gene networks searching for bio-logically plausible architecturesrdquo IEEEACM Transactions onComputational Biology and Bioinformatics vol 9 no 2 pp 358ndash371 2012
[12] J R Koza W Mydlowec G Lanza Y Jessen and M A KeaneldquoAutomatic computational discovery of chemical reaction net-works using genetic programmingrdquo inComputationalDiscoveryof Scientific Knowledge S Dzeroski and L Todorovski Edsvol 4660 of Lecture Notes in Computer Science pp 205ndash227Springer 2007
[13] S Ando E Sakamoto and H Iba ldquoEvolutionary modeling andinference of gene networkrdquo Information Sciences vol 145 no3-4 pp 237ndash259 2002
[14] L Dematte and D Prandi ldquoGPU computing for systemsbiologyrdquo Briefings in Bioinformatics vol 11 no 3 Article IDbbq006 pp 323ndash333 2010
[15] J L Payne N A Sinnott-Armstrong and J HMoore ldquoExploit-ing graphics processing units for computational biology andbioinformaticsrdquo Interdisciplinary Sciences Computational LifeSciences vol 2 no 3 pp 213ndash220 2010
[16] M J Harvey and G De Fabritiis ldquoA survey of computationalmolecular science using graphics processing unitsrdquoWiley Inter-disciplinary Reviews Computational Molecular Science vol 2no 5 pp 734ndash742 2012
[17] Y Zhou J Liepe X Sheng M P H Stumpf and C BarnesldquoGPU accelerated biochemical network simulationrdquo Bioinfor-matics vol 27 no 6 Article ID btr015 pp 874ndash876 2011
[18] M Vigelius A Lane and B Meyer ldquoAccelerating reaction-diffusion simulations with general-purpose graphics processingunitsrdquo Bioinformatics vol 27 no 2 Article ID btq622 pp 288ndash290 2011
[19] D J Wilkinson ldquoStochastic modelling for quantitative descrip-tion of heterogeneous biological systemsrdquo Nature ReviewsGenetics vol 10 no 2 pp 122ndash133 2009
[20] J C Butcher ldquoTrees and numerical methods for ordinarydifferential equationsrdquo Numerical Algorithms vol 53 no 2-3pp 153ndash170 2010
[21] D T Gillespie ldquoStochastic simulation of chemical kineticsrdquoAnnual Review of Physical Chemistry vol 58 pp 35ndash55 2007
[22] L Petzold ldquoAutomatic selection of methods for solving stiffand nonstiff systems of ordinary differential equationsrdquo SIAMJournal of Scientific and Statistical Computing vol 4 no 1 pp136ndash148 1983
[23] Y Cao D T Gillespie and L R Petzold ldquoEfficient step sizeselection for the tau-leaping simulation methodrdquo The Journalof Chemical Physics vol 124 no 4 2006
[24] M S Nobile D Besozzi P Cazzaniga GMauri andD PescinildquocupSODA a CUDA-powered simulator of mass-action kinet-icsrdquo in Proceedings of the 12th International Conference onParallel Computing Technologies (PaCT rsquo13) V Malyshkin Edvol 7979 of Lecture Notes in Computer Science pp 344ndash357
[25] M S Nobile P Cazzaniga D Besozzi and G Mauri ldquoGPU-accelerated simulations of mass-action kinetics models withcupSODArdquoThe Journal of Supercomputing 2014
[26] M S Nobile P Cazzaniga D Besozzi D Pescini and GMauri ldquocuTauLeaping a GPU-powered tau-leaping stochasticsimulator for massive parallel analyses of biological systemsrdquoPLoS ONE vol 9 no 3 2014
[27] K Rogers Blood Physiology and CirculationTheHuman BodyBritannica Educational Publishing 2010
[28] A D Michelson Platelets Elsevier 2nd edition 2007[29] R W Colman J Hirsh V J Marder A W Clowes and J
N George Hemostasis and Thrombosis Basic Principles andClinical Practice Lippincott Williams ampWilkins 2006
[30] M S Chatterjee W S Denney H Jing and S L DiamondldquoSystems biology of coagulation initiation Kinetics of thrombingeneration in resting and activated human bloodrdquo PLoS Com-putational Biology vol 6 no 9 Article ID e1000950 2010
[31] M Bellini D Besozzi P Cazzaniga GMauri andM S NobileldquoSimulation and analysis of the blood coagulation cascadeaccelerated on GPUrdquo in Proceedings of the 22nd EuromicroInternational Conference on Parallel Distributed and Network-Based Processing (PDP) pp 590ndash593 IEEE Computer SocietyConference Publishing Services 2014
[32] T Wajima G K Isbister and S B Duffull ldquoA comprehensivemodel for the humoral coagulation network in humansrdquo Clini-cal Pharmacology and Therapeutics vol 86 no 3 pp 290ndash2982009
[33] K Lo W S Denney and S L Diamond ldquoStochastic modelingof blood coagulation initiationrdquo Pathophysiology of Haemostasisand Thrombosis vol 34 no 2-3 pp 80ndash90 2006
[34] C M Danforth T Orfeo K G Mann K E Brummel-Ziedins and S J Everse ldquoThe impact of uncertainty in a bloodcoagulation modelrdquo Mathematical Medicine and Biology vol26 no 4 Article ID dqp011 pp 323ndash336 2009
[35] S D Lewis P P Shields and J A Shafer ldquoCharacterizationof the kinetic pathway for liberation of fibrinopeptides duringassembly of fibrinrdquoThe Journal of Biological Chemistry vol 260no 18 pp 10192ndash10199 1985
[36] M F Hockin K C Jones S J Everse and K G Mann ldquoAmodel for the stoichiometric regulation of blood coagulationrdquoThe Journal of Biological Chemistry vol 277 no 21 pp 18322ndash18333 2002
[37] K Brummel-Ziedins C YVossen F R Rosendaal KUmezakiand K G Mann ldquoThe plasma hemostatic proteome thrombin
20 BioMed Research International
generation in healthy individualsrdquo Journal of Thrombosis andHaemostasis vol 3 no 7 pp 1472ndash1481 2005
[38] K E Brummel-Ziedins C Y Vossen S Butenas K G Mannand F R Rosendaal ldquoThrombin generation profiles in deepvenous thrombosisrdquo Journal of Thrombosis and Haemostasisvol 3 no 11 pp 2497ndash2505 2005
[39] S D Bungay P A Gentry and R D Gentry ldquoA mathematicalmodel of lipid-mediated thrombin generationrdquo MathematicalMedicine and Biology vol 20 no 1 pp 105ndash129 2003
[40] S Butenas T Orfeo M T Gissel K E Brummel and K GMann ldquoThe significance of circulating factor IXa in bloodrdquoThe Journal of Biological Chemistry vol 279 no 22 pp 22875ndash22882 2004
[41] D Voet and J G Voet Biochemistry Wiley 2011[42] A C Guyton and J E Hall Textbook of Medical Physiology
Saunders 2010[43] M F M James and A M Roche ldquoDose-response relationship
between plasma ionized calcium concentration and thrombe-lastographyrdquo Journal of Cardiothoracic and Vascular Anesthesiavol 18 no 5 pp 581ndash586 2004
[44] D L Nelson and M M Cox Lehninger Principles of Biochem-istry W H Freeman Company 2004
[45] R M Farber ldquoTopical perspective on massive threading andparallelismrdquo Journal of Molecular Graphics and Modelling vol30 pp 82ndash89 2011
[46] OWolkenhauerMUllahWKolch andK-HCho ldquoModelingand simulation of intracellular dynamics choosing an appropri-ate frameworkrdquo IEEETransactions onNanobioscience vol 3 no3 pp 200ndash207 2004
[47] B H Estridge A P Reynolds and N J Walters Basic MedicalLaboratory Techniques Cengage Learning 4th edition 2000
[48] T Renne A H Schmaier K F Nickel M Blomback and CMaas ldquoIn vivo roles of factor XIIrdquo Blood vol 120 no 22 pp4296ndash4303 2012
[49] P V Jenkins O Rawley O P Smith and J S ODonnellldquoElevated factor VIII levels and risk of venous thrombosisrdquoBritish Journal of Haematology vol 157 no 6 pp 653ndash663 2012
[50] C E Walsh and K M Batt ldquoHemophilia clinical gene therapybrief reviewrdquo Translational Research vol 161 no 4 pp 307ndash3122013
[51] A Van Hylckama Vlieg I K Van Der Linden R M Bertinaand F R Rosendaal ldquoHigh levels of factor IX increase the risk ofvenous thrombosisrdquo Blood vol 95 no 12 pp 3678ndash3682 2000
[52] K Brummel-Ziedins ldquoModels for thrombin generation and riskof diseaserdquo Journal ofThrombosis and Haemostasis vol 11 no 1pp 212ndash223 2013
[53] U Seligsohn and A Lubetsky ldquoGenetic susceptibility to venousthrombosisrdquoTheNew England Journal of Medicine vol 344 no16 pp 1222ndash1231 2001
[54] I Martinelli P Bucciarelli and P M Mannucci ldquoThromboticrisk factors basic pathophysiologyrdquo Critical Care Medicine vol38 no 2 pp S3ndashS9 2010
[55] K Brummel-Ziedins A Undas T Orfeo et al ldquoThrombingeneration in acute coronary syndrome and stable coronaryartery disease dependence on plasma factor compositionrdquoJournal ofThrombosis andHaemostasis vol 6 no 1 pp 104ndash1102008
[56] S Hoops R Gauges C Lee et al ldquoCOPASImdasha complexpathway simulatorrdquo Bioinformatics vol 22 no 24 pp 3067ndash3074 2006
[57] Nvidia NVIDIArsquos Next Generation CUDA Compute Architec-ture Kepler GK110 2012
[58] W Stortelder W Hemker and H Hemker ldquoMathematicalmodelling in blood coagulation simulation and parameterestimationrdquoModelling Analysis and Simulation vol 20 pp 1ndash111997
[59] S M C Orfao G Jank and K Mottaghy ldquoA new mathematicalapproach to modelling thrombin generationrdquo InternationalJournal of Artificial Organs vol 29 no 7 pp 703ndash708 2006
[60] Nvidia Nvidia CUDA C Programming Guide v50 2012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Anatomy Research International
PeptidesInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
International Journal of
Volume 2014
Zoology
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Molecular Biology International
GenomicsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
BioinformaticsAdvances in
Marine BiologyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Signal TransductionJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
BioMed Research International
Evolutionary BiologyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Biochemistry Research International
ArchaeaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Genetics Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in
Virolog y
Hindawi Publishing Corporationhttpwwwhindawicom
Nucleic AcidsJournal of
Volume 2014
Stem CellsInternational
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Enzyme Research
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Microbiology
14 BioMed Research International
250
300
350
400
450
500
550
600
650
700
Clot
ting
time (
s)
Factor IX (M)
1eminus
14
1eminus
12
1eminus
10
1eminus
8
1eminus
6
1eminus
4
1eminus
2
1e0
Figure 10 Clotting time at different initial concentrations of factor IXThe reference value is 900sdot10minus8M the sweep range is [900sdot10minus14 900sdot10minus2]
0
100
200
300
400
500
600
700
Clot
ting
time (
s)
Factor II (M)
1eminus
12
1eminus
10
1eminus
8
1eminus
6
1eminus
4
1eminus
2
1e0
1e2
Figure 11 Clotting time at different initial concentrations of thrombinThe reference value is 140sdot10minus6M the sweep range is [140sdot10minus12 140sdot100]
The PSA-1D over factor IX (Figure 10) shows only aslight decrease of the CT as the initial concentration offIX increases this is in contrast to recent studies thatdemonstrated how the excess of factor IX leads to anincreased risk of deep vein thrombosis [51] This result canbe explained by considering that (i) a high concentrationof factor IX is not sufficient to bring about coagulationproblems though when the concentration of other factorsis above the average value (yet not at pathological levels)prothrombotic states can be observed (ii) in this model weconsider average values as initial concentrations of factorshowever individuals are characterized by different (balanced)combinations of procoagulant and anticoagulant factor levelsthat altogether contribute to define a unique coagulationphenotype that reflects the developmental environmentalgenetic nutritional and pharmacological influences of eachindividual [52] On the contrary the lack of factor IX causes
haemophilia B characterized by higher CT with respect tothe reference value (Figure 10)
Furthermore by comparing the PSA-1D of factors VIIIand IX (Figures 9 and 10) it is clear that haemophilia A ismore serious than haemophilia B since the CT achieved inconditions of factor VIII deficiency is higher than the CTobtained in the case of factor IX deficiency
In both PSA-1D over factors VIII and IX we observedafter the initial decrease of the CT an unexpected increaseof the CT as the factor concentration increasesThis counter-intuitive behavior arises at very high concentrations of thesefactors (with respect to the average physiological levels) andto the best of our knowledge it was never observed in vivoNonetheless it would be interesting to verify by means of adhoc laboratory experiments if the model correctly describesthe behavior of the BCC even in these conditions or on thecontrary it is not predictive in these extreme situations
BioMed Research International 15
The PSA-1D over factor II (Figure 11) shows a dramaticdecrease of the CT as the initial concentration of this factorincreases (with respect to the average physiological level)This behavior resembles the effects of hypercoagulability (orthrombophilia) a disease caused by mutation G20210A inthe prothrombin gene [53] that causes an increase of theprothrombin level (factor II) in the blood flow resulting inan excessive formation of the active form of this factor thusheightening venous thrombosis risks [54] Hypercoagulabil-ity is usually treated with warfarin therapy or with otheranticoagulants with a similar effect These drugs decrease thecapacity of coagulation factors to become active preventingthe formation of unwanted thrombi
On the other hand when the initial concentration offactor II is low we achieved the effects of prothrombindeficiency a rare autosomal recessive disease that causes atendency to severe bleeding [29 55] As shown in Figure 11 aconcentration equal to 10of the physiological value of factorII (ie 14 sdot 10minus6 M) leads to clotting effects similar to severehaemophilia A
32 PSA-2D of the BCCModel We present here the results ofthe PSA-2D on the BCCmodel where couples of parameterswere varied to analyze the possible effects arising from thecombined perturbation of their values
321 Reactions 11990327
and 11990358 Figure 12 shows the effect of
the simultaneous variation of constants 11989627
and 11989658
(overthe same sweep ranges considered in the two PSA-1DSection 31) This result remarks that reaction 119903
27 involved
in the common pathway has a stronger influence on theBCC and that there is a synergic interplay between thesetwo reactions In particular when the value of 119896
27is low and
the value of 11989658
is high the CT is higher than the valuesachieved when only a single constant is changed because inthis condition both the intrinsic and the common pathwaysare simultaneously inhibited
322 Factor VIII Factor IX and Tissue Factor In the last twoPSA-2Dwe varied the initial concentration of factor VIII andTissue Factor and the initial concentration of factor IX andTissue Factor respectively
The initial concentrations of factors VIII and IX were var-ied over four orders of magnitude using their physiologicalvalues as upper limit for the sweep ranges the concentrationof Tissue Factor was varied over four orders of magnitudetwo above and two below its reference value (see Table 2)The rationale behind this choice is to observe how the BCCmodel in conditions corresponding to different states ofhaemophilia (obtained by decreasing the concentrations offactors VIII and IX) behaves with different initial concentra-tions of the Tissue Factor which is the upstream factor of theextrinsic pathway that is the most important element of theBCC
The results of these PSA-2D show that with respect tothe condition of haemophilia B in the case of haemophiliaA the amount of Tissue Factor (below its reference value)has a negligible influence on the CT as indicated by the
presence of a plateau in Figure 13 on the contrary concerninghaemophilia B a deficiency of Tissue Factor leads to anincrease of the CT especially when factor IX is present in lowconcentrations (Figure 14)
The different results achieved in the two PSA-2D are dueto the presence in the BCC model of a direct interactionbetweenTissue Factor and factor IX bymeans of theTF-fVIIacomplex (see reactions 119903
13 119903
15in Table 1) Indeed the lack
of Tissue Factor directly affects the concentration of activefactor IX which results in a strong alteration of the CT withrespect to physiological conditions
33 CPU versus GPU Performance Comparison In orderto show the relevant speedup achieved by coagSODA wepresent here the comparison of the computational effortrequired by GPU and CPU for the simulation of the BCCmodel The performances of our GPU simulator were com-pared with those obtained using the LSODA algorithmimplemented in the software COPASI [56] executing onthe CPU the same set of simulations that were run on theGPU COPASI is single-threaded and does not exploit thephysical and logical cores of the CPU therefore it representsa good benchmark as a single-node CPU-bound simulator ofbiological systems
In all simulations executed on both GPU and CPU westored 100 samples uniformly distributed in the time intervalconsidered for each simulation that is [0 700] seconds of thedynamics of all chemical species involved in the BCCmodelThe settings for the LSODA algorithm were the followingrelative error 1 sdot 10minus7 absolute error 1 sdot 10minus13 and maximumnumber of internal steps set to 20000
331 Benchmark Details The GPU used for the simulationsis a Nvidia Tesla K20c equipped with 2496 cores organized in13 streamingmultiprocessors GPU clock 706MHz and 5GBofDDR5RAM In all tests coagSODAwas compiled and exe-cuted with version 55 of CUDA libraries Even though thisGPU has compute capability 35 and is based on the GK110Kepler architecture currently coagSODA does not exploitany of the new features (eg Dynamic Parallelism Hyper-QRemote DMA) with the exception of the new ISA encodingwhich allows threads to exploit an increased number ofregisters (255 instead of 63) reducing register spilling intoglobal memory and increasing performances [57] Moreoveras described in Section 22 the Kepler architecture offers thepossibility to reconfigure the 64KB on-chip cache balancingbetween L1 cache and shared memory Since coagSODAexploits the shared memory to reduce the latencies duringthe access to the concentration values of the BCC model weopted for the following configuration 48KB for the sharedmemory 16 KB for the L1 cache
The performance of the GPU was compared against apersonal computer equippedwith a dual-core CPU Intel Corei5 frequency 23 GHz 4GB of DDR3 RAM running theoperating systemMacOSXLion 1075The software used as acoagSODA-equivalent single-threaded CPU implementationis COPASI version 48 (build 35)
16 BioMed Research International
280
300
320
340
360
380
400
420
440
1e41e6
1e91e11 1e5
1e7
1e9
1e11
280 320 360 400 440
Clot
ting
time (
s)
k 58(M
minus1 sminus1 )
k27 (Mminus1sminus1)
Figure 12 Clotting time according to PSA-2D over reaction constants 11989627and 119896
58 The sweep ranges are [2 sdot 104 2 sdot 1010] and [1 sdot 105 2 sdot 1011]
respectively
100
200
300
400
Tissue factor (M) Factor VIII
(M)
Clot
ting
time (
s)
100
150
200
250
300
350
400
450
1eminus131eminus11
1eminus91eminus14
1eminus12
1eminus10
Figure 13 Clotting time at different initial concentrations of Tissue Factor and factor VIII The sweep ranges are [500 sdot 10minus14 500 sdot 10minus10]and [700 sdot 10minus14 500 sdot 10minus10] respectively
A direct comparison of the performances between thesetwo different architectures and implementations is not aneasy task since CPUs are optimized for single-thread exe-cution and exploit a large number of optimizations (eghigher clock frequency instruction caching pipelining outof order execution and branch prediction) whereas GPUsare optimized for graphics processing and parallel exe-cution of identical code relying only on multilevel datacaching For this reason in this paper we propose anempirical comparison of the performances based on therunning times of identical batches of simulations of the BCCmodel
332 Benchmark Results In Figure 15 we show the compar-ison of the running times required to run several batchesof simulations executed to carry out the PSA-1D over thereaction constant 119896
27 The choice of comparing the perfor-
mances of the GPU and CPU implementations by executingbatches of simulations that are related to a PSA instead ofrunning 119899 independent but identical simulations (ie allcharacterized by the same parameterization of the model)is due to the fact that these results represent a more realisticscenario in the computational analysis of biological systemswhereby it is useful to investigate large search spaces ofparameters corresponding to different perturbed conditionsof themodel [26]Moreover for the evaluation of the running
time the execution of a batch of 119899 identical deterministicsimulations would be futile The figure clearly shows thatcoagSODA always performs better than the CPU counter-part In particular while the CPU performance increaseslinearly with the number of simulations the running timesare strongly reduced on the GPU in this case the overallrunning time roughly corresponds to the time required forthe execution of the slowest simulations This is due to thefact that different parameterizations may require differenttime steps for LSODA to converge and the execution ofa block terminates as long as all the threads it containsterminate In turn the execution of a kernel terminates whenall blocks terminate for this reason a single simulationwhose dynamics requires more steps than the others mayaffect the overall running time
In Table 3 we report the running times of all batchesof simulations along with the speedup achieved on theGPU In particular these results highlight that the advantageof exploiting the GPUs for the simulation of the BCCmodel becomes more evident as the number of simula-tions increases with a 181times speedup when a PSA with10000 different parameterizations is executed Therefore theGPU accelerated analysis of the BCC model with coag-SODA represents a novel relevant computational means toinvestigate the behavior of this complex biological systemunder nonphysiological conditions and could be exploited
BioMed Research International 17
0
200
400
600
Tissue factor (M) Factor IX (M
)
Clot
ting
time (
s)
0
100
200
300
400
500
600
700
1eminus12
1eminus10
1eminus8
1eminus131eminus11
1eminus9
Figure 14 Clotting time at different initial concentrations of Tissue Factor and factor IXThe sweep ranges are [500 sdot 10minus14 500 sdot 10minus10] and[900 sdot 10
minus12 900 sdot 10
minus8] respectively
5 10 50 100 500 1000 5000 10000
Runn
ing
time (
s)
Number of simulations
CPUGPU
1e2
1e3
1e4
1e5
1e6
1e7
Figure 15 Comparison of the computational time required to execute an increasing number of simulations of the BCCmodel on CPU (IntelCore i5 23 GHz) and GPU (Nvidia Tesla K20c GPU clock 706MHz) The computational time is expressed in seconds The time valuesrelated to 1000 5000 and 10000 simulations on the CPU were estimated by regression (see also Table 3)
Table 3 CPU versus GPU performance comparison
Number ofsimulations CPU time (sec) GPU time (sec) Speedup
5 3808 1278 298times10 10604 2329 456times50 51704 8058 642times100 116528 10977 1061times500 536058 17394 3081times1000 1073582lowast 19982 5373times5000 5363507lowast 30960 1732times10000 10724975lowast 58951 1819timeslowastValues estimated by regression
to efficiently determine the response of the BCC to differenttherapeutical drugs
4 Conclusions
Thanks to their high-performance computing capabilitiesand the very low costs GPUs nowadays represent a suitable
technology for the development and the application of par-allel computational methods for in silico analysis of complexbiological systems However the implementation of efficientcomputational tools able to fully exploit the large potentialityof GPUs is still challenging since good programming skillsare required to implement GPU-based algorithmic methodsand to handle specific features as an optimal usage ofmemoryor the communication bandwidth between GPU and CPUMoreover algorithms cannot be directly ported on the GPUbecause of the limited programming capabilities allowed byGPUkernels as amatter of fact they need to be redesigned tomeet the requirements of the underlying SIMD architecture
In this work we presented coagSODA a GPU-poweredsimulator specifically developed to carry out fast parallelsimulations of the BCC model coagSODA was designed tooffer a black-box solution usable by any final user in aneasy way It relies on cupSODA [24] a numerical integratorfor ODEs that we previously implemented for the GPUarchitecture based on the LSODA algorithm and capableof automatically translating a reaction-based model into aset of coupled ODEs In addition to mass-action kinet-ics coagSODA implements specific functions to compute
18 BioMed Research International
the kinetics of Hill function based reactions such as thoseinvolved in the platelets activity of the BCC model coag-SODA exploits the massive parallel capabilities of modernGPUs and our results demonstrated that it can achieve arelevant reduction of the computational time required toexecutemany concurrent and independent simulations of theBCC dynamics The mutual independence of the simulationsallows fully exploiting the underlying SIMD architecturemoreover coagSODA benefits from an additional speedupthanks to our choice of storing the state of the systeminto the low-latency shared memory (a solution that wasalready implemented in cupSODA) Since the BCC modelis large (96 reactions 71 chemical species) a large amountof shared memory was assigned to each thread stronglyreducing the theoretical occupancy of the GPU that is theratio of active warps with respect to the maximum numberof warps supported by each streaming multiprocessor of theGPU However the results of the analysis presented in thiswork performed on the BCC model show that coagSODAachieves a relevant boost with respect to a reference CPUimplementation For instance in the case of 10000 simula-tions we achieved a noticeable 181times speedup Interestinglythe performances of coagSODA are better than COPASI evenfor small batches of simulations These results indicate thatfor biological models consisting in many reactions and manyspecies our GPU implementation of LSODA becomes moreprofitable than the CPU counterpart as the number of con-current simulations increases making it suitable especiallywhen performing demanding computational analysis suchas for example parameter sweep parameter estimation orsensitivity analysis
As a test case in this paper we presented several parame-ter sweep analyses over reaction constants and initial concen-trations of factors involved in the BCC model Other com-putational analyses on mathematical models of this pathwaywere previously presented For instance in [58] a parameterestimation of the reaction constants of different modelsinvolving the activation of factor X factor V prothrombinand the inactivation of factors was performed The aim ofthis analysis was to discriminate between different modelsto unravel the mechanisms on the basis of the BCC Similaranalyses were executed on a model describing thrombingeneration in plasma since the reaction mechanism thereaction constants and initial concentrations were unknown[59] In [34] a sensitivity analysis of a model consisting of44 species over 34 chemical species was presented reactionconstants were varied between 10 and 1000 of theirreference value to the aim of identifying the most influentialfactors of the BCC
All these analyses were performed bymeans of sequentialsimulations of the models under investigation thereforein general only small batches of simulations could be run(for instance in [34] only 836 simulations were executed toexecute the sensitivity analysis of the model) On the otherhand in this paper we presented the results of different PSAefficiently executed bymeans of coagSODA overall providinguseful information regarding unknownparameters and inter-esting insights into the functioning of the BCC A thoroughsensitivity analysis of the whole BCC model consisting in
around 5 sdot 105 simulations is currently in progress on GPUsby our group exploiting the great computational efficiency ofcoagSODA
The results of our computational analyses should benow validated by means of specifically designed laboratoryexperiments In particular we identified a plausible valuefor the constant of the reaction describing the autoactivationof factor XII (since no reference values can be found inliterature) moreover the results of the PSA over factor IXsuggest that a deficiency of this factor is not enough to causesevere bleeding disorders as haemophilia B but an alterationof other factors seems to be necessary for the occurrence ofsuch condition (eg the lack of Tissue Factor as suggestedby the PSA-2D over Tissue Factor and factor IX) Finallythe PSA over factors VIII and IX showed in a situationcharacterized by very high concentrations of these factorsa counterintuitive behavior in which the clotting time isincreased with respect to the value obtained in physiologicalconditions Indeed it would be interesting to design ad hoclaboratory experiments to verify if the BCCmodel is actuallypredictive in such extreme situations
The coagSODA software and the SBML version of theBCC model used in this work are available from the authorsupon request
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This work is partially supported by SYSBIO Centre forSystems Biology
References
[1] B B Aldridge JM BurkeDA Lauffenburger andP K SorgerldquoPhysicochemicalmodelling of cell signalling pathwaysrdquoNatureCell Biology vol 8 no 11 pp 1195ndash1203 2006
[2] I-C Chou and E O Voit ldquoRecent developments in parameterestimation and structure identification of biochemical andgenomic systemsrdquoMathematical Biosciences vol 219 no 2 pp57ndash83 2009
[3] D Besozzi P Cazzaniga D Pescini G Mauri S Colomboand EMartegani ldquoThe role of feedback control mechanisms onthe establishment of oscillatory regimes in the RascAMPPKApathway in S cerevisiaerdquo EURASIP Journal on Bioinformaticsand Systems Biology vol 2012 no 10 2012
[4] A Saltelli M Ratto T Andres et al Global Sensitivity AnalysisThe Primer Wiley-Interscience 2008
[5] A Raue C Kreutz T Maiwald et al ldquoStructural and practicalidentifiability analysis of partially observed dynamical modelsby exploiting the profile likelihoodrdquo Bioinformatics vol 25 no15 pp 1923ndash1929 2009
[6] A Drager M Kronfeld M J Ziller et al ldquoModeling metabolicnetworks in C glutamicum a comparison of rate laws incombination with various parameter optimization strategiesrdquoBMC Systems Biology vol 3 article 5 2009
BioMed Research International 19
[7] D Besozzi P Cazzaniga G Mauri D Pescini and L Van-neschi ldquoA comparison of genetic algorithms and particle swarmoptimization for parameter estimation in stochastic biochem-ical systemsrdquo in Proceedings of the 7th European ConferenceEvolutionary Computation Machine Learning and Data Miningin Bioinformatics (EvoBIO rsquo09) C Pizzuti M Ritchie and MGiacobini Eds vol 5483 of Lecture Notes in Computer Sciencepp 116ndash127 Springer 2009
[8] M S Nobile D Besozzi P Cazzaniga GMauri andD PescinildquocupSODA aGPU-basedmulti-swarmPSOmethod for param-eter estimation in stochastic biological systems exploitingdiscrete-time target seriesrdquo in Proceedings of the 10th EuropeanConference Evolutionary Computation Machine Learning andData Mining in Bioinformatics (EvoBIO rsquo12) M Giacobini LVanneschi and W Bush Eds vol 7246 of Lecture Notes inComputer Science pp 74ndash85 Springer
[9] M S Nobile D Besozzi P Cazzaniga D Pescini and GMaurildquoReverse engineering of kinetic reaction networks by meansof Cartesian Genetic Programming and Particle Swarm Opti-mizationrdquo in Proceedings of the IEEE Congress on EvolutionaryComputation (CEC rsquo13) pp 1594ndash1601 June 2013
[10] M S Nobile ldquoEvolutionary inference of biochemical inter-action networks accelerated on Graphics Processing Unitsrdquoin Proceedings of the 11th International Conference on HighPerformance Computingamp Simulation 2013 (HPCS rsquo13) pp 668ndash670 Helsinki Finland 2013
[11] K Kentzoglanakis and M Poole ldquoA swarm intelligence frame-work for reconstructing gene networks searching for bio-logically plausible architecturesrdquo IEEEACM Transactions onComputational Biology and Bioinformatics vol 9 no 2 pp 358ndash371 2012
[12] J R Koza W Mydlowec G Lanza Y Jessen and M A KeaneldquoAutomatic computational discovery of chemical reaction net-works using genetic programmingrdquo inComputationalDiscoveryof Scientific Knowledge S Dzeroski and L Todorovski Edsvol 4660 of Lecture Notes in Computer Science pp 205ndash227Springer 2007
[13] S Ando E Sakamoto and H Iba ldquoEvolutionary modeling andinference of gene networkrdquo Information Sciences vol 145 no3-4 pp 237ndash259 2002
[14] L Dematte and D Prandi ldquoGPU computing for systemsbiologyrdquo Briefings in Bioinformatics vol 11 no 3 Article IDbbq006 pp 323ndash333 2010
[15] J L Payne N A Sinnott-Armstrong and J HMoore ldquoExploit-ing graphics processing units for computational biology andbioinformaticsrdquo Interdisciplinary Sciences Computational LifeSciences vol 2 no 3 pp 213ndash220 2010
[16] M J Harvey and G De Fabritiis ldquoA survey of computationalmolecular science using graphics processing unitsrdquoWiley Inter-disciplinary Reviews Computational Molecular Science vol 2no 5 pp 734ndash742 2012
[17] Y Zhou J Liepe X Sheng M P H Stumpf and C BarnesldquoGPU accelerated biochemical network simulationrdquo Bioinfor-matics vol 27 no 6 Article ID btr015 pp 874ndash876 2011
[18] M Vigelius A Lane and B Meyer ldquoAccelerating reaction-diffusion simulations with general-purpose graphics processingunitsrdquo Bioinformatics vol 27 no 2 Article ID btq622 pp 288ndash290 2011
[19] D J Wilkinson ldquoStochastic modelling for quantitative descrip-tion of heterogeneous biological systemsrdquo Nature ReviewsGenetics vol 10 no 2 pp 122ndash133 2009
[20] J C Butcher ldquoTrees and numerical methods for ordinarydifferential equationsrdquo Numerical Algorithms vol 53 no 2-3pp 153ndash170 2010
[21] D T Gillespie ldquoStochastic simulation of chemical kineticsrdquoAnnual Review of Physical Chemistry vol 58 pp 35ndash55 2007
[22] L Petzold ldquoAutomatic selection of methods for solving stiffand nonstiff systems of ordinary differential equationsrdquo SIAMJournal of Scientific and Statistical Computing vol 4 no 1 pp136ndash148 1983
[23] Y Cao D T Gillespie and L R Petzold ldquoEfficient step sizeselection for the tau-leaping simulation methodrdquo The Journalof Chemical Physics vol 124 no 4 2006
[24] M S Nobile D Besozzi P Cazzaniga GMauri andD PescinildquocupSODA a CUDA-powered simulator of mass-action kinet-icsrdquo in Proceedings of the 12th International Conference onParallel Computing Technologies (PaCT rsquo13) V Malyshkin Edvol 7979 of Lecture Notes in Computer Science pp 344ndash357
[25] M S Nobile P Cazzaniga D Besozzi and G Mauri ldquoGPU-accelerated simulations of mass-action kinetics models withcupSODArdquoThe Journal of Supercomputing 2014
[26] M S Nobile P Cazzaniga D Besozzi D Pescini and GMauri ldquocuTauLeaping a GPU-powered tau-leaping stochasticsimulator for massive parallel analyses of biological systemsrdquoPLoS ONE vol 9 no 3 2014
[27] K Rogers Blood Physiology and CirculationTheHuman BodyBritannica Educational Publishing 2010
[28] A D Michelson Platelets Elsevier 2nd edition 2007[29] R W Colman J Hirsh V J Marder A W Clowes and J
N George Hemostasis and Thrombosis Basic Principles andClinical Practice Lippincott Williams ampWilkins 2006
[30] M S Chatterjee W S Denney H Jing and S L DiamondldquoSystems biology of coagulation initiation Kinetics of thrombingeneration in resting and activated human bloodrdquo PLoS Com-putational Biology vol 6 no 9 Article ID e1000950 2010
[31] M Bellini D Besozzi P Cazzaniga GMauri andM S NobileldquoSimulation and analysis of the blood coagulation cascadeaccelerated on GPUrdquo in Proceedings of the 22nd EuromicroInternational Conference on Parallel Distributed and Network-Based Processing (PDP) pp 590ndash593 IEEE Computer SocietyConference Publishing Services 2014
[32] T Wajima G K Isbister and S B Duffull ldquoA comprehensivemodel for the humoral coagulation network in humansrdquo Clini-cal Pharmacology and Therapeutics vol 86 no 3 pp 290ndash2982009
[33] K Lo W S Denney and S L Diamond ldquoStochastic modelingof blood coagulation initiationrdquo Pathophysiology of Haemostasisand Thrombosis vol 34 no 2-3 pp 80ndash90 2006
[34] C M Danforth T Orfeo K G Mann K E Brummel-Ziedins and S J Everse ldquoThe impact of uncertainty in a bloodcoagulation modelrdquo Mathematical Medicine and Biology vol26 no 4 Article ID dqp011 pp 323ndash336 2009
[35] S D Lewis P P Shields and J A Shafer ldquoCharacterizationof the kinetic pathway for liberation of fibrinopeptides duringassembly of fibrinrdquoThe Journal of Biological Chemistry vol 260no 18 pp 10192ndash10199 1985
[36] M F Hockin K C Jones S J Everse and K G Mann ldquoAmodel for the stoichiometric regulation of blood coagulationrdquoThe Journal of Biological Chemistry vol 277 no 21 pp 18322ndash18333 2002
[37] K Brummel-Ziedins C YVossen F R Rosendaal KUmezakiand K G Mann ldquoThe plasma hemostatic proteome thrombin
20 BioMed Research International
generation in healthy individualsrdquo Journal of Thrombosis andHaemostasis vol 3 no 7 pp 1472ndash1481 2005
[38] K E Brummel-Ziedins C Y Vossen S Butenas K G Mannand F R Rosendaal ldquoThrombin generation profiles in deepvenous thrombosisrdquo Journal of Thrombosis and Haemostasisvol 3 no 11 pp 2497ndash2505 2005
[39] S D Bungay P A Gentry and R D Gentry ldquoA mathematicalmodel of lipid-mediated thrombin generationrdquo MathematicalMedicine and Biology vol 20 no 1 pp 105ndash129 2003
[40] S Butenas T Orfeo M T Gissel K E Brummel and K GMann ldquoThe significance of circulating factor IXa in bloodrdquoThe Journal of Biological Chemistry vol 279 no 22 pp 22875ndash22882 2004
[41] D Voet and J G Voet Biochemistry Wiley 2011[42] A C Guyton and J E Hall Textbook of Medical Physiology
Saunders 2010[43] M F M James and A M Roche ldquoDose-response relationship
between plasma ionized calcium concentration and thrombe-lastographyrdquo Journal of Cardiothoracic and Vascular Anesthesiavol 18 no 5 pp 581ndash586 2004
[44] D L Nelson and M M Cox Lehninger Principles of Biochem-istry W H Freeman Company 2004
[45] R M Farber ldquoTopical perspective on massive threading andparallelismrdquo Journal of Molecular Graphics and Modelling vol30 pp 82ndash89 2011
[46] OWolkenhauerMUllahWKolch andK-HCho ldquoModelingand simulation of intracellular dynamics choosing an appropri-ate frameworkrdquo IEEETransactions onNanobioscience vol 3 no3 pp 200ndash207 2004
[47] B H Estridge A P Reynolds and N J Walters Basic MedicalLaboratory Techniques Cengage Learning 4th edition 2000
[48] T Renne A H Schmaier K F Nickel M Blomback and CMaas ldquoIn vivo roles of factor XIIrdquo Blood vol 120 no 22 pp4296ndash4303 2012
[49] P V Jenkins O Rawley O P Smith and J S ODonnellldquoElevated factor VIII levels and risk of venous thrombosisrdquoBritish Journal of Haematology vol 157 no 6 pp 653ndash663 2012
[50] C E Walsh and K M Batt ldquoHemophilia clinical gene therapybrief reviewrdquo Translational Research vol 161 no 4 pp 307ndash3122013
[51] A Van Hylckama Vlieg I K Van Der Linden R M Bertinaand F R Rosendaal ldquoHigh levels of factor IX increase the risk ofvenous thrombosisrdquo Blood vol 95 no 12 pp 3678ndash3682 2000
[52] K Brummel-Ziedins ldquoModels for thrombin generation and riskof diseaserdquo Journal ofThrombosis and Haemostasis vol 11 no 1pp 212ndash223 2013
[53] U Seligsohn and A Lubetsky ldquoGenetic susceptibility to venousthrombosisrdquoTheNew England Journal of Medicine vol 344 no16 pp 1222ndash1231 2001
[54] I Martinelli P Bucciarelli and P M Mannucci ldquoThromboticrisk factors basic pathophysiologyrdquo Critical Care Medicine vol38 no 2 pp S3ndashS9 2010
[55] K Brummel-Ziedins A Undas T Orfeo et al ldquoThrombingeneration in acute coronary syndrome and stable coronaryartery disease dependence on plasma factor compositionrdquoJournal ofThrombosis andHaemostasis vol 6 no 1 pp 104ndash1102008
[56] S Hoops R Gauges C Lee et al ldquoCOPASImdasha complexpathway simulatorrdquo Bioinformatics vol 22 no 24 pp 3067ndash3074 2006
[57] Nvidia NVIDIArsquos Next Generation CUDA Compute Architec-ture Kepler GK110 2012
[58] W Stortelder W Hemker and H Hemker ldquoMathematicalmodelling in blood coagulation simulation and parameterestimationrdquoModelling Analysis and Simulation vol 20 pp 1ndash111997
[59] S M C Orfao G Jank and K Mottaghy ldquoA new mathematicalapproach to modelling thrombin generationrdquo InternationalJournal of Artificial Organs vol 29 no 7 pp 703ndash708 2006
[60] Nvidia Nvidia CUDA C Programming Guide v50 2012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Anatomy Research International
PeptidesInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
International Journal of
Volume 2014
Zoology
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Molecular Biology International
GenomicsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
BioinformaticsAdvances in
Marine BiologyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Signal TransductionJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
BioMed Research International
Evolutionary BiologyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Biochemistry Research International
ArchaeaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Genetics Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in
Virolog y
Hindawi Publishing Corporationhttpwwwhindawicom
Nucleic AcidsJournal of
Volume 2014
Stem CellsInternational
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Enzyme Research
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Microbiology
BioMed Research International 15
The PSA-1D over factor II (Figure 11) shows a dramaticdecrease of the CT as the initial concentration of this factorincreases (with respect to the average physiological level)This behavior resembles the effects of hypercoagulability (orthrombophilia) a disease caused by mutation G20210A inthe prothrombin gene [53] that causes an increase of theprothrombin level (factor II) in the blood flow resulting inan excessive formation of the active form of this factor thusheightening venous thrombosis risks [54] Hypercoagulabil-ity is usually treated with warfarin therapy or with otheranticoagulants with a similar effect These drugs decrease thecapacity of coagulation factors to become active preventingthe formation of unwanted thrombi
On the other hand when the initial concentration offactor II is low we achieved the effects of prothrombindeficiency a rare autosomal recessive disease that causes atendency to severe bleeding [29 55] As shown in Figure 11 aconcentration equal to 10of the physiological value of factorII (ie 14 sdot 10minus6 M) leads to clotting effects similar to severehaemophilia A
32 PSA-2D of the BCCModel We present here the results ofthe PSA-2D on the BCCmodel where couples of parameterswere varied to analyze the possible effects arising from thecombined perturbation of their values
321 Reactions 11990327
and 11990358 Figure 12 shows the effect of
the simultaneous variation of constants 11989627
and 11989658
(overthe same sweep ranges considered in the two PSA-1DSection 31) This result remarks that reaction 119903
27 involved
in the common pathway has a stronger influence on theBCC and that there is a synergic interplay between thesetwo reactions In particular when the value of 119896
27is low and
the value of 11989658
is high the CT is higher than the valuesachieved when only a single constant is changed because inthis condition both the intrinsic and the common pathwaysare simultaneously inhibited
322 Factor VIII Factor IX and Tissue Factor In the last twoPSA-2Dwe varied the initial concentration of factor VIII andTissue Factor and the initial concentration of factor IX andTissue Factor respectively
The initial concentrations of factors VIII and IX were var-ied over four orders of magnitude using their physiologicalvalues as upper limit for the sweep ranges the concentrationof Tissue Factor was varied over four orders of magnitudetwo above and two below its reference value (see Table 2)The rationale behind this choice is to observe how the BCCmodel in conditions corresponding to different states ofhaemophilia (obtained by decreasing the concentrations offactors VIII and IX) behaves with different initial concentra-tions of the Tissue Factor which is the upstream factor of theextrinsic pathway that is the most important element of theBCC
The results of these PSA-2D show that with respect tothe condition of haemophilia B in the case of haemophiliaA the amount of Tissue Factor (below its reference value)has a negligible influence on the CT as indicated by the
presence of a plateau in Figure 13 on the contrary concerninghaemophilia B a deficiency of Tissue Factor leads to anincrease of the CT especially when factor IX is present in lowconcentrations (Figure 14)
The different results achieved in the two PSA-2D are dueto the presence in the BCC model of a direct interactionbetweenTissue Factor and factor IX bymeans of theTF-fVIIacomplex (see reactions 119903
13 119903
15in Table 1) Indeed the lack
of Tissue Factor directly affects the concentration of activefactor IX which results in a strong alteration of the CT withrespect to physiological conditions
33 CPU versus GPU Performance Comparison In orderto show the relevant speedup achieved by coagSODA wepresent here the comparison of the computational effortrequired by GPU and CPU for the simulation of the BCCmodel The performances of our GPU simulator were com-pared with those obtained using the LSODA algorithmimplemented in the software COPASI [56] executing onthe CPU the same set of simulations that were run on theGPU COPASI is single-threaded and does not exploit thephysical and logical cores of the CPU therefore it representsa good benchmark as a single-node CPU-bound simulator ofbiological systems
In all simulations executed on both GPU and CPU westored 100 samples uniformly distributed in the time intervalconsidered for each simulation that is [0 700] seconds of thedynamics of all chemical species involved in the BCCmodelThe settings for the LSODA algorithm were the followingrelative error 1 sdot 10minus7 absolute error 1 sdot 10minus13 and maximumnumber of internal steps set to 20000
331 Benchmark Details The GPU used for the simulationsis a Nvidia Tesla K20c equipped with 2496 cores organized in13 streamingmultiprocessors GPU clock 706MHz and 5GBofDDR5RAM In all tests coagSODAwas compiled and exe-cuted with version 55 of CUDA libraries Even though thisGPU has compute capability 35 and is based on the GK110Kepler architecture currently coagSODA does not exploitany of the new features (eg Dynamic Parallelism Hyper-QRemote DMA) with the exception of the new ISA encodingwhich allows threads to exploit an increased number ofregisters (255 instead of 63) reducing register spilling intoglobal memory and increasing performances [57] Moreoveras described in Section 22 the Kepler architecture offers thepossibility to reconfigure the 64KB on-chip cache balancingbetween L1 cache and shared memory Since coagSODAexploits the shared memory to reduce the latencies duringthe access to the concentration values of the BCC model weopted for the following configuration 48KB for the sharedmemory 16 KB for the L1 cache
The performance of the GPU was compared against apersonal computer equippedwith a dual-core CPU Intel Corei5 frequency 23 GHz 4GB of DDR3 RAM running theoperating systemMacOSXLion 1075The software used as acoagSODA-equivalent single-threaded CPU implementationis COPASI version 48 (build 35)
16 BioMed Research International
280
300
320
340
360
380
400
420
440
1e41e6
1e91e11 1e5
1e7
1e9
1e11
280 320 360 400 440
Clot
ting
time (
s)
k 58(M
minus1 sminus1 )
k27 (Mminus1sminus1)
Figure 12 Clotting time according to PSA-2D over reaction constants 11989627and 119896
58 The sweep ranges are [2 sdot 104 2 sdot 1010] and [1 sdot 105 2 sdot 1011]
respectively
100
200
300
400
Tissue factor (M) Factor VIII
(M)
Clot
ting
time (
s)
100
150
200
250
300
350
400
450
1eminus131eminus11
1eminus91eminus14
1eminus12
1eminus10
Figure 13 Clotting time at different initial concentrations of Tissue Factor and factor VIII The sweep ranges are [500 sdot 10minus14 500 sdot 10minus10]and [700 sdot 10minus14 500 sdot 10minus10] respectively
A direct comparison of the performances between thesetwo different architectures and implementations is not aneasy task since CPUs are optimized for single-thread exe-cution and exploit a large number of optimizations (eghigher clock frequency instruction caching pipelining outof order execution and branch prediction) whereas GPUsare optimized for graphics processing and parallel exe-cution of identical code relying only on multilevel datacaching For this reason in this paper we propose anempirical comparison of the performances based on therunning times of identical batches of simulations of the BCCmodel
332 Benchmark Results In Figure 15 we show the compar-ison of the running times required to run several batchesof simulations executed to carry out the PSA-1D over thereaction constant 119896
27 The choice of comparing the perfor-
mances of the GPU and CPU implementations by executingbatches of simulations that are related to a PSA instead ofrunning 119899 independent but identical simulations (ie allcharacterized by the same parameterization of the model)is due to the fact that these results represent a more realisticscenario in the computational analysis of biological systemswhereby it is useful to investigate large search spaces ofparameters corresponding to different perturbed conditionsof themodel [26]Moreover for the evaluation of the running
time the execution of a batch of 119899 identical deterministicsimulations would be futile The figure clearly shows thatcoagSODA always performs better than the CPU counter-part In particular while the CPU performance increaseslinearly with the number of simulations the running timesare strongly reduced on the GPU in this case the overallrunning time roughly corresponds to the time required forthe execution of the slowest simulations This is due to thefact that different parameterizations may require differenttime steps for LSODA to converge and the execution ofa block terminates as long as all the threads it containsterminate In turn the execution of a kernel terminates whenall blocks terminate for this reason a single simulationwhose dynamics requires more steps than the others mayaffect the overall running time
In Table 3 we report the running times of all batchesof simulations along with the speedup achieved on theGPU In particular these results highlight that the advantageof exploiting the GPUs for the simulation of the BCCmodel becomes more evident as the number of simula-tions increases with a 181times speedup when a PSA with10000 different parameterizations is executed Therefore theGPU accelerated analysis of the BCC model with coag-SODA represents a novel relevant computational means toinvestigate the behavior of this complex biological systemunder nonphysiological conditions and could be exploited
BioMed Research International 17
0
200
400
600
Tissue factor (M) Factor IX (M
)
Clot
ting
time (
s)
0
100
200
300
400
500
600
700
1eminus12
1eminus10
1eminus8
1eminus131eminus11
1eminus9
Figure 14 Clotting time at different initial concentrations of Tissue Factor and factor IXThe sweep ranges are [500 sdot 10minus14 500 sdot 10minus10] and[900 sdot 10
minus12 900 sdot 10
minus8] respectively
5 10 50 100 500 1000 5000 10000
Runn
ing
time (
s)
Number of simulations
CPUGPU
1e2
1e3
1e4
1e5
1e6
1e7
Figure 15 Comparison of the computational time required to execute an increasing number of simulations of the BCCmodel on CPU (IntelCore i5 23 GHz) and GPU (Nvidia Tesla K20c GPU clock 706MHz) The computational time is expressed in seconds The time valuesrelated to 1000 5000 and 10000 simulations on the CPU were estimated by regression (see also Table 3)
Table 3 CPU versus GPU performance comparison
Number ofsimulations CPU time (sec) GPU time (sec) Speedup
5 3808 1278 298times10 10604 2329 456times50 51704 8058 642times100 116528 10977 1061times500 536058 17394 3081times1000 1073582lowast 19982 5373times5000 5363507lowast 30960 1732times10000 10724975lowast 58951 1819timeslowastValues estimated by regression
to efficiently determine the response of the BCC to differenttherapeutical drugs
4 Conclusions
Thanks to their high-performance computing capabilitiesand the very low costs GPUs nowadays represent a suitable
technology for the development and the application of par-allel computational methods for in silico analysis of complexbiological systems However the implementation of efficientcomputational tools able to fully exploit the large potentialityof GPUs is still challenging since good programming skillsare required to implement GPU-based algorithmic methodsand to handle specific features as an optimal usage ofmemoryor the communication bandwidth between GPU and CPUMoreover algorithms cannot be directly ported on the GPUbecause of the limited programming capabilities allowed byGPUkernels as amatter of fact they need to be redesigned tomeet the requirements of the underlying SIMD architecture
In this work we presented coagSODA a GPU-poweredsimulator specifically developed to carry out fast parallelsimulations of the BCC model coagSODA was designed tooffer a black-box solution usable by any final user in aneasy way It relies on cupSODA [24] a numerical integratorfor ODEs that we previously implemented for the GPUarchitecture based on the LSODA algorithm and capableof automatically translating a reaction-based model into aset of coupled ODEs In addition to mass-action kinet-ics coagSODA implements specific functions to compute
18 BioMed Research International
the kinetics of Hill function based reactions such as thoseinvolved in the platelets activity of the BCC model coag-SODA exploits the massive parallel capabilities of modernGPUs and our results demonstrated that it can achieve arelevant reduction of the computational time required toexecutemany concurrent and independent simulations of theBCC dynamics The mutual independence of the simulationsallows fully exploiting the underlying SIMD architecturemoreover coagSODA benefits from an additional speedupthanks to our choice of storing the state of the systeminto the low-latency shared memory (a solution that wasalready implemented in cupSODA) Since the BCC modelis large (96 reactions 71 chemical species) a large amountof shared memory was assigned to each thread stronglyreducing the theoretical occupancy of the GPU that is theratio of active warps with respect to the maximum numberof warps supported by each streaming multiprocessor of theGPU However the results of the analysis presented in thiswork performed on the BCC model show that coagSODAachieves a relevant boost with respect to a reference CPUimplementation For instance in the case of 10000 simula-tions we achieved a noticeable 181times speedup Interestinglythe performances of coagSODA are better than COPASI evenfor small batches of simulations These results indicate thatfor biological models consisting in many reactions and manyspecies our GPU implementation of LSODA becomes moreprofitable than the CPU counterpart as the number of con-current simulations increases making it suitable especiallywhen performing demanding computational analysis suchas for example parameter sweep parameter estimation orsensitivity analysis
As a test case in this paper we presented several parame-ter sweep analyses over reaction constants and initial concen-trations of factors involved in the BCC model Other com-putational analyses on mathematical models of this pathwaywere previously presented For instance in [58] a parameterestimation of the reaction constants of different modelsinvolving the activation of factor X factor V prothrombinand the inactivation of factors was performed The aim ofthis analysis was to discriminate between different modelsto unravel the mechanisms on the basis of the BCC Similaranalyses were executed on a model describing thrombingeneration in plasma since the reaction mechanism thereaction constants and initial concentrations were unknown[59] In [34] a sensitivity analysis of a model consisting of44 species over 34 chemical species was presented reactionconstants were varied between 10 and 1000 of theirreference value to the aim of identifying the most influentialfactors of the BCC
All these analyses were performed bymeans of sequentialsimulations of the models under investigation thereforein general only small batches of simulations could be run(for instance in [34] only 836 simulations were executed toexecute the sensitivity analysis of the model) On the otherhand in this paper we presented the results of different PSAefficiently executed bymeans of coagSODA overall providinguseful information regarding unknownparameters and inter-esting insights into the functioning of the BCC A thoroughsensitivity analysis of the whole BCC model consisting in
around 5 sdot 105 simulations is currently in progress on GPUsby our group exploiting the great computational efficiency ofcoagSODA
The results of our computational analyses should benow validated by means of specifically designed laboratoryexperiments In particular we identified a plausible valuefor the constant of the reaction describing the autoactivationof factor XII (since no reference values can be found inliterature) moreover the results of the PSA over factor IXsuggest that a deficiency of this factor is not enough to causesevere bleeding disorders as haemophilia B but an alterationof other factors seems to be necessary for the occurrence ofsuch condition (eg the lack of Tissue Factor as suggestedby the PSA-2D over Tissue Factor and factor IX) Finallythe PSA over factors VIII and IX showed in a situationcharacterized by very high concentrations of these factorsa counterintuitive behavior in which the clotting time isincreased with respect to the value obtained in physiologicalconditions Indeed it would be interesting to design ad hoclaboratory experiments to verify if the BCCmodel is actuallypredictive in such extreme situations
The coagSODA software and the SBML version of theBCC model used in this work are available from the authorsupon request
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This work is partially supported by SYSBIO Centre forSystems Biology
References
[1] B B Aldridge JM BurkeDA Lauffenburger andP K SorgerldquoPhysicochemicalmodelling of cell signalling pathwaysrdquoNatureCell Biology vol 8 no 11 pp 1195ndash1203 2006
[2] I-C Chou and E O Voit ldquoRecent developments in parameterestimation and structure identification of biochemical andgenomic systemsrdquoMathematical Biosciences vol 219 no 2 pp57ndash83 2009
[3] D Besozzi P Cazzaniga D Pescini G Mauri S Colomboand EMartegani ldquoThe role of feedback control mechanisms onthe establishment of oscillatory regimes in the RascAMPPKApathway in S cerevisiaerdquo EURASIP Journal on Bioinformaticsand Systems Biology vol 2012 no 10 2012
[4] A Saltelli M Ratto T Andres et al Global Sensitivity AnalysisThe Primer Wiley-Interscience 2008
[5] A Raue C Kreutz T Maiwald et al ldquoStructural and practicalidentifiability analysis of partially observed dynamical modelsby exploiting the profile likelihoodrdquo Bioinformatics vol 25 no15 pp 1923ndash1929 2009
[6] A Drager M Kronfeld M J Ziller et al ldquoModeling metabolicnetworks in C glutamicum a comparison of rate laws incombination with various parameter optimization strategiesrdquoBMC Systems Biology vol 3 article 5 2009
BioMed Research International 19
[7] D Besozzi P Cazzaniga G Mauri D Pescini and L Van-neschi ldquoA comparison of genetic algorithms and particle swarmoptimization for parameter estimation in stochastic biochem-ical systemsrdquo in Proceedings of the 7th European ConferenceEvolutionary Computation Machine Learning and Data Miningin Bioinformatics (EvoBIO rsquo09) C Pizzuti M Ritchie and MGiacobini Eds vol 5483 of Lecture Notes in Computer Sciencepp 116ndash127 Springer 2009
[8] M S Nobile D Besozzi P Cazzaniga GMauri andD PescinildquocupSODA aGPU-basedmulti-swarmPSOmethod for param-eter estimation in stochastic biological systems exploitingdiscrete-time target seriesrdquo in Proceedings of the 10th EuropeanConference Evolutionary Computation Machine Learning andData Mining in Bioinformatics (EvoBIO rsquo12) M Giacobini LVanneschi and W Bush Eds vol 7246 of Lecture Notes inComputer Science pp 74ndash85 Springer
[9] M S Nobile D Besozzi P Cazzaniga D Pescini and GMaurildquoReverse engineering of kinetic reaction networks by meansof Cartesian Genetic Programming and Particle Swarm Opti-mizationrdquo in Proceedings of the IEEE Congress on EvolutionaryComputation (CEC rsquo13) pp 1594ndash1601 June 2013
[10] M S Nobile ldquoEvolutionary inference of biochemical inter-action networks accelerated on Graphics Processing Unitsrdquoin Proceedings of the 11th International Conference on HighPerformance Computingamp Simulation 2013 (HPCS rsquo13) pp 668ndash670 Helsinki Finland 2013
[11] K Kentzoglanakis and M Poole ldquoA swarm intelligence frame-work for reconstructing gene networks searching for bio-logically plausible architecturesrdquo IEEEACM Transactions onComputational Biology and Bioinformatics vol 9 no 2 pp 358ndash371 2012
[12] J R Koza W Mydlowec G Lanza Y Jessen and M A KeaneldquoAutomatic computational discovery of chemical reaction net-works using genetic programmingrdquo inComputationalDiscoveryof Scientific Knowledge S Dzeroski and L Todorovski Edsvol 4660 of Lecture Notes in Computer Science pp 205ndash227Springer 2007
[13] S Ando E Sakamoto and H Iba ldquoEvolutionary modeling andinference of gene networkrdquo Information Sciences vol 145 no3-4 pp 237ndash259 2002
[14] L Dematte and D Prandi ldquoGPU computing for systemsbiologyrdquo Briefings in Bioinformatics vol 11 no 3 Article IDbbq006 pp 323ndash333 2010
[15] J L Payne N A Sinnott-Armstrong and J HMoore ldquoExploit-ing graphics processing units for computational biology andbioinformaticsrdquo Interdisciplinary Sciences Computational LifeSciences vol 2 no 3 pp 213ndash220 2010
[16] M J Harvey and G De Fabritiis ldquoA survey of computationalmolecular science using graphics processing unitsrdquoWiley Inter-disciplinary Reviews Computational Molecular Science vol 2no 5 pp 734ndash742 2012
[17] Y Zhou J Liepe X Sheng M P H Stumpf and C BarnesldquoGPU accelerated biochemical network simulationrdquo Bioinfor-matics vol 27 no 6 Article ID btr015 pp 874ndash876 2011
[18] M Vigelius A Lane and B Meyer ldquoAccelerating reaction-diffusion simulations with general-purpose graphics processingunitsrdquo Bioinformatics vol 27 no 2 Article ID btq622 pp 288ndash290 2011
[19] D J Wilkinson ldquoStochastic modelling for quantitative descrip-tion of heterogeneous biological systemsrdquo Nature ReviewsGenetics vol 10 no 2 pp 122ndash133 2009
[20] J C Butcher ldquoTrees and numerical methods for ordinarydifferential equationsrdquo Numerical Algorithms vol 53 no 2-3pp 153ndash170 2010
[21] D T Gillespie ldquoStochastic simulation of chemical kineticsrdquoAnnual Review of Physical Chemistry vol 58 pp 35ndash55 2007
[22] L Petzold ldquoAutomatic selection of methods for solving stiffand nonstiff systems of ordinary differential equationsrdquo SIAMJournal of Scientific and Statistical Computing vol 4 no 1 pp136ndash148 1983
[23] Y Cao D T Gillespie and L R Petzold ldquoEfficient step sizeselection for the tau-leaping simulation methodrdquo The Journalof Chemical Physics vol 124 no 4 2006
[24] M S Nobile D Besozzi P Cazzaniga GMauri andD PescinildquocupSODA a CUDA-powered simulator of mass-action kinet-icsrdquo in Proceedings of the 12th International Conference onParallel Computing Technologies (PaCT rsquo13) V Malyshkin Edvol 7979 of Lecture Notes in Computer Science pp 344ndash357
[25] M S Nobile P Cazzaniga D Besozzi and G Mauri ldquoGPU-accelerated simulations of mass-action kinetics models withcupSODArdquoThe Journal of Supercomputing 2014
[26] M S Nobile P Cazzaniga D Besozzi D Pescini and GMauri ldquocuTauLeaping a GPU-powered tau-leaping stochasticsimulator for massive parallel analyses of biological systemsrdquoPLoS ONE vol 9 no 3 2014
[27] K Rogers Blood Physiology and CirculationTheHuman BodyBritannica Educational Publishing 2010
[28] A D Michelson Platelets Elsevier 2nd edition 2007[29] R W Colman J Hirsh V J Marder A W Clowes and J
N George Hemostasis and Thrombosis Basic Principles andClinical Practice Lippincott Williams ampWilkins 2006
[30] M S Chatterjee W S Denney H Jing and S L DiamondldquoSystems biology of coagulation initiation Kinetics of thrombingeneration in resting and activated human bloodrdquo PLoS Com-putational Biology vol 6 no 9 Article ID e1000950 2010
[31] M Bellini D Besozzi P Cazzaniga GMauri andM S NobileldquoSimulation and analysis of the blood coagulation cascadeaccelerated on GPUrdquo in Proceedings of the 22nd EuromicroInternational Conference on Parallel Distributed and Network-Based Processing (PDP) pp 590ndash593 IEEE Computer SocietyConference Publishing Services 2014
[32] T Wajima G K Isbister and S B Duffull ldquoA comprehensivemodel for the humoral coagulation network in humansrdquo Clini-cal Pharmacology and Therapeutics vol 86 no 3 pp 290ndash2982009
[33] K Lo W S Denney and S L Diamond ldquoStochastic modelingof blood coagulation initiationrdquo Pathophysiology of Haemostasisand Thrombosis vol 34 no 2-3 pp 80ndash90 2006
[34] C M Danforth T Orfeo K G Mann K E Brummel-Ziedins and S J Everse ldquoThe impact of uncertainty in a bloodcoagulation modelrdquo Mathematical Medicine and Biology vol26 no 4 Article ID dqp011 pp 323ndash336 2009
[35] S D Lewis P P Shields and J A Shafer ldquoCharacterizationof the kinetic pathway for liberation of fibrinopeptides duringassembly of fibrinrdquoThe Journal of Biological Chemistry vol 260no 18 pp 10192ndash10199 1985
[36] M F Hockin K C Jones S J Everse and K G Mann ldquoAmodel for the stoichiometric regulation of blood coagulationrdquoThe Journal of Biological Chemistry vol 277 no 21 pp 18322ndash18333 2002
[37] K Brummel-Ziedins C YVossen F R Rosendaal KUmezakiand K G Mann ldquoThe plasma hemostatic proteome thrombin
20 BioMed Research International
generation in healthy individualsrdquo Journal of Thrombosis andHaemostasis vol 3 no 7 pp 1472ndash1481 2005
[38] K E Brummel-Ziedins C Y Vossen S Butenas K G Mannand F R Rosendaal ldquoThrombin generation profiles in deepvenous thrombosisrdquo Journal of Thrombosis and Haemostasisvol 3 no 11 pp 2497ndash2505 2005
[39] S D Bungay P A Gentry and R D Gentry ldquoA mathematicalmodel of lipid-mediated thrombin generationrdquo MathematicalMedicine and Biology vol 20 no 1 pp 105ndash129 2003
[40] S Butenas T Orfeo M T Gissel K E Brummel and K GMann ldquoThe significance of circulating factor IXa in bloodrdquoThe Journal of Biological Chemistry vol 279 no 22 pp 22875ndash22882 2004
[41] D Voet and J G Voet Biochemistry Wiley 2011[42] A C Guyton and J E Hall Textbook of Medical Physiology
Saunders 2010[43] M F M James and A M Roche ldquoDose-response relationship
between plasma ionized calcium concentration and thrombe-lastographyrdquo Journal of Cardiothoracic and Vascular Anesthesiavol 18 no 5 pp 581ndash586 2004
[44] D L Nelson and M M Cox Lehninger Principles of Biochem-istry W H Freeman Company 2004
[45] R M Farber ldquoTopical perspective on massive threading andparallelismrdquo Journal of Molecular Graphics and Modelling vol30 pp 82ndash89 2011
[46] OWolkenhauerMUllahWKolch andK-HCho ldquoModelingand simulation of intracellular dynamics choosing an appropri-ate frameworkrdquo IEEETransactions onNanobioscience vol 3 no3 pp 200ndash207 2004
[47] B H Estridge A P Reynolds and N J Walters Basic MedicalLaboratory Techniques Cengage Learning 4th edition 2000
[48] T Renne A H Schmaier K F Nickel M Blomback and CMaas ldquoIn vivo roles of factor XIIrdquo Blood vol 120 no 22 pp4296ndash4303 2012
[49] P V Jenkins O Rawley O P Smith and J S ODonnellldquoElevated factor VIII levels and risk of venous thrombosisrdquoBritish Journal of Haematology vol 157 no 6 pp 653ndash663 2012
[50] C E Walsh and K M Batt ldquoHemophilia clinical gene therapybrief reviewrdquo Translational Research vol 161 no 4 pp 307ndash3122013
[51] A Van Hylckama Vlieg I K Van Der Linden R M Bertinaand F R Rosendaal ldquoHigh levels of factor IX increase the risk ofvenous thrombosisrdquo Blood vol 95 no 12 pp 3678ndash3682 2000
[52] K Brummel-Ziedins ldquoModels for thrombin generation and riskof diseaserdquo Journal ofThrombosis and Haemostasis vol 11 no 1pp 212ndash223 2013
[53] U Seligsohn and A Lubetsky ldquoGenetic susceptibility to venousthrombosisrdquoTheNew England Journal of Medicine vol 344 no16 pp 1222ndash1231 2001
[54] I Martinelli P Bucciarelli and P M Mannucci ldquoThromboticrisk factors basic pathophysiologyrdquo Critical Care Medicine vol38 no 2 pp S3ndashS9 2010
[55] K Brummel-Ziedins A Undas T Orfeo et al ldquoThrombingeneration in acute coronary syndrome and stable coronaryartery disease dependence on plasma factor compositionrdquoJournal ofThrombosis andHaemostasis vol 6 no 1 pp 104ndash1102008
[56] S Hoops R Gauges C Lee et al ldquoCOPASImdasha complexpathway simulatorrdquo Bioinformatics vol 22 no 24 pp 3067ndash3074 2006
[57] Nvidia NVIDIArsquos Next Generation CUDA Compute Architec-ture Kepler GK110 2012
[58] W Stortelder W Hemker and H Hemker ldquoMathematicalmodelling in blood coagulation simulation and parameterestimationrdquoModelling Analysis and Simulation vol 20 pp 1ndash111997
[59] S M C Orfao G Jank and K Mottaghy ldquoA new mathematicalapproach to modelling thrombin generationrdquo InternationalJournal of Artificial Organs vol 29 no 7 pp 703ndash708 2006
[60] Nvidia Nvidia CUDA C Programming Guide v50 2012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Anatomy Research International
PeptidesInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
International Journal of
Volume 2014
Zoology
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Molecular Biology International
GenomicsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
BioinformaticsAdvances in
Marine BiologyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Signal TransductionJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
BioMed Research International
Evolutionary BiologyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Biochemistry Research International
ArchaeaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Genetics Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in
Virolog y
Hindawi Publishing Corporationhttpwwwhindawicom
Nucleic AcidsJournal of
Volume 2014
Stem CellsInternational
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Enzyme Research
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Microbiology
16 BioMed Research International
280
300
320
340
360
380
400
420
440
1e41e6
1e91e11 1e5
1e7
1e9
1e11
280 320 360 400 440
Clot
ting
time (
s)
k 58(M
minus1 sminus1 )
k27 (Mminus1sminus1)
Figure 12 Clotting time according to PSA-2D over reaction constants 11989627and 119896
58 The sweep ranges are [2 sdot 104 2 sdot 1010] and [1 sdot 105 2 sdot 1011]
respectively
100
200
300
400
Tissue factor (M) Factor VIII
(M)
Clot
ting
time (
s)
100
150
200
250
300
350
400
450
1eminus131eminus11
1eminus91eminus14
1eminus12
1eminus10
Figure 13 Clotting time at different initial concentrations of Tissue Factor and factor VIII The sweep ranges are [500 sdot 10minus14 500 sdot 10minus10]and [700 sdot 10minus14 500 sdot 10minus10] respectively
A direct comparison of the performances between thesetwo different architectures and implementations is not aneasy task since CPUs are optimized for single-thread exe-cution and exploit a large number of optimizations (eghigher clock frequency instruction caching pipelining outof order execution and branch prediction) whereas GPUsare optimized for graphics processing and parallel exe-cution of identical code relying only on multilevel datacaching For this reason in this paper we propose anempirical comparison of the performances based on therunning times of identical batches of simulations of the BCCmodel
332 Benchmark Results In Figure 15 we show the compar-ison of the running times required to run several batchesof simulations executed to carry out the PSA-1D over thereaction constant 119896
27 The choice of comparing the perfor-
mances of the GPU and CPU implementations by executingbatches of simulations that are related to a PSA instead ofrunning 119899 independent but identical simulations (ie allcharacterized by the same parameterization of the model)is due to the fact that these results represent a more realisticscenario in the computational analysis of biological systemswhereby it is useful to investigate large search spaces ofparameters corresponding to different perturbed conditionsof themodel [26]Moreover for the evaluation of the running
time the execution of a batch of 119899 identical deterministicsimulations would be futile The figure clearly shows thatcoagSODA always performs better than the CPU counter-part In particular while the CPU performance increaseslinearly with the number of simulations the running timesare strongly reduced on the GPU in this case the overallrunning time roughly corresponds to the time required forthe execution of the slowest simulations This is due to thefact that different parameterizations may require differenttime steps for LSODA to converge and the execution ofa block terminates as long as all the threads it containsterminate In turn the execution of a kernel terminates whenall blocks terminate for this reason a single simulationwhose dynamics requires more steps than the others mayaffect the overall running time
In Table 3 we report the running times of all batchesof simulations along with the speedup achieved on theGPU In particular these results highlight that the advantageof exploiting the GPUs for the simulation of the BCCmodel becomes more evident as the number of simula-tions increases with a 181times speedup when a PSA with10000 different parameterizations is executed Therefore theGPU accelerated analysis of the BCC model with coag-SODA represents a novel relevant computational means toinvestigate the behavior of this complex biological systemunder nonphysiological conditions and could be exploited
BioMed Research International 17
0
200
400
600
Tissue factor (M) Factor IX (M
)
Clot
ting
time (
s)
0
100
200
300
400
500
600
700
1eminus12
1eminus10
1eminus8
1eminus131eminus11
1eminus9
Figure 14 Clotting time at different initial concentrations of Tissue Factor and factor IXThe sweep ranges are [500 sdot 10minus14 500 sdot 10minus10] and[900 sdot 10
minus12 900 sdot 10
minus8] respectively
5 10 50 100 500 1000 5000 10000
Runn
ing
time (
s)
Number of simulations
CPUGPU
1e2
1e3
1e4
1e5
1e6
1e7
Figure 15 Comparison of the computational time required to execute an increasing number of simulations of the BCCmodel on CPU (IntelCore i5 23 GHz) and GPU (Nvidia Tesla K20c GPU clock 706MHz) The computational time is expressed in seconds The time valuesrelated to 1000 5000 and 10000 simulations on the CPU were estimated by regression (see also Table 3)
Table 3 CPU versus GPU performance comparison
Number ofsimulations CPU time (sec) GPU time (sec) Speedup
5 3808 1278 298times10 10604 2329 456times50 51704 8058 642times100 116528 10977 1061times500 536058 17394 3081times1000 1073582lowast 19982 5373times5000 5363507lowast 30960 1732times10000 10724975lowast 58951 1819timeslowastValues estimated by regression
to efficiently determine the response of the BCC to differenttherapeutical drugs
4 Conclusions
Thanks to their high-performance computing capabilitiesand the very low costs GPUs nowadays represent a suitable
technology for the development and the application of par-allel computational methods for in silico analysis of complexbiological systems However the implementation of efficientcomputational tools able to fully exploit the large potentialityof GPUs is still challenging since good programming skillsare required to implement GPU-based algorithmic methodsand to handle specific features as an optimal usage ofmemoryor the communication bandwidth between GPU and CPUMoreover algorithms cannot be directly ported on the GPUbecause of the limited programming capabilities allowed byGPUkernels as amatter of fact they need to be redesigned tomeet the requirements of the underlying SIMD architecture
In this work we presented coagSODA a GPU-poweredsimulator specifically developed to carry out fast parallelsimulations of the BCC model coagSODA was designed tooffer a black-box solution usable by any final user in aneasy way It relies on cupSODA [24] a numerical integratorfor ODEs that we previously implemented for the GPUarchitecture based on the LSODA algorithm and capableof automatically translating a reaction-based model into aset of coupled ODEs In addition to mass-action kinet-ics coagSODA implements specific functions to compute
18 BioMed Research International
the kinetics of Hill function based reactions such as thoseinvolved in the platelets activity of the BCC model coag-SODA exploits the massive parallel capabilities of modernGPUs and our results demonstrated that it can achieve arelevant reduction of the computational time required toexecutemany concurrent and independent simulations of theBCC dynamics The mutual independence of the simulationsallows fully exploiting the underlying SIMD architecturemoreover coagSODA benefits from an additional speedupthanks to our choice of storing the state of the systeminto the low-latency shared memory (a solution that wasalready implemented in cupSODA) Since the BCC modelis large (96 reactions 71 chemical species) a large amountof shared memory was assigned to each thread stronglyreducing the theoretical occupancy of the GPU that is theratio of active warps with respect to the maximum numberof warps supported by each streaming multiprocessor of theGPU However the results of the analysis presented in thiswork performed on the BCC model show that coagSODAachieves a relevant boost with respect to a reference CPUimplementation For instance in the case of 10000 simula-tions we achieved a noticeable 181times speedup Interestinglythe performances of coagSODA are better than COPASI evenfor small batches of simulations These results indicate thatfor biological models consisting in many reactions and manyspecies our GPU implementation of LSODA becomes moreprofitable than the CPU counterpart as the number of con-current simulations increases making it suitable especiallywhen performing demanding computational analysis suchas for example parameter sweep parameter estimation orsensitivity analysis
As a test case in this paper we presented several parame-ter sweep analyses over reaction constants and initial concen-trations of factors involved in the BCC model Other com-putational analyses on mathematical models of this pathwaywere previously presented For instance in [58] a parameterestimation of the reaction constants of different modelsinvolving the activation of factor X factor V prothrombinand the inactivation of factors was performed The aim ofthis analysis was to discriminate between different modelsto unravel the mechanisms on the basis of the BCC Similaranalyses were executed on a model describing thrombingeneration in plasma since the reaction mechanism thereaction constants and initial concentrations were unknown[59] In [34] a sensitivity analysis of a model consisting of44 species over 34 chemical species was presented reactionconstants were varied between 10 and 1000 of theirreference value to the aim of identifying the most influentialfactors of the BCC
All these analyses were performed bymeans of sequentialsimulations of the models under investigation thereforein general only small batches of simulations could be run(for instance in [34] only 836 simulations were executed toexecute the sensitivity analysis of the model) On the otherhand in this paper we presented the results of different PSAefficiently executed bymeans of coagSODA overall providinguseful information regarding unknownparameters and inter-esting insights into the functioning of the BCC A thoroughsensitivity analysis of the whole BCC model consisting in
around 5 sdot 105 simulations is currently in progress on GPUsby our group exploiting the great computational efficiency ofcoagSODA
The results of our computational analyses should benow validated by means of specifically designed laboratoryexperiments In particular we identified a plausible valuefor the constant of the reaction describing the autoactivationof factor XII (since no reference values can be found inliterature) moreover the results of the PSA over factor IXsuggest that a deficiency of this factor is not enough to causesevere bleeding disorders as haemophilia B but an alterationof other factors seems to be necessary for the occurrence ofsuch condition (eg the lack of Tissue Factor as suggestedby the PSA-2D over Tissue Factor and factor IX) Finallythe PSA over factors VIII and IX showed in a situationcharacterized by very high concentrations of these factorsa counterintuitive behavior in which the clotting time isincreased with respect to the value obtained in physiologicalconditions Indeed it would be interesting to design ad hoclaboratory experiments to verify if the BCCmodel is actuallypredictive in such extreme situations
The coagSODA software and the SBML version of theBCC model used in this work are available from the authorsupon request
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This work is partially supported by SYSBIO Centre forSystems Biology
References
[1] B B Aldridge JM BurkeDA Lauffenburger andP K SorgerldquoPhysicochemicalmodelling of cell signalling pathwaysrdquoNatureCell Biology vol 8 no 11 pp 1195ndash1203 2006
[2] I-C Chou and E O Voit ldquoRecent developments in parameterestimation and structure identification of biochemical andgenomic systemsrdquoMathematical Biosciences vol 219 no 2 pp57ndash83 2009
[3] D Besozzi P Cazzaniga D Pescini G Mauri S Colomboand EMartegani ldquoThe role of feedback control mechanisms onthe establishment of oscillatory regimes in the RascAMPPKApathway in S cerevisiaerdquo EURASIP Journal on Bioinformaticsand Systems Biology vol 2012 no 10 2012
[4] A Saltelli M Ratto T Andres et al Global Sensitivity AnalysisThe Primer Wiley-Interscience 2008
[5] A Raue C Kreutz T Maiwald et al ldquoStructural and practicalidentifiability analysis of partially observed dynamical modelsby exploiting the profile likelihoodrdquo Bioinformatics vol 25 no15 pp 1923ndash1929 2009
[6] A Drager M Kronfeld M J Ziller et al ldquoModeling metabolicnetworks in C glutamicum a comparison of rate laws incombination with various parameter optimization strategiesrdquoBMC Systems Biology vol 3 article 5 2009
BioMed Research International 19
[7] D Besozzi P Cazzaniga G Mauri D Pescini and L Van-neschi ldquoA comparison of genetic algorithms and particle swarmoptimization for parameter estimation in stochastic biochem-ical systemsrdquo in Proceedings of the 7th European ConferenceEvolutionary Computation Machine Learning and Data Miningin Bioinformatics (EvoBIO rsquo09) C Pizzuti M Ritchie and MGiacobini Eds vol 5483 of Lecture Notes in Computer Sciencepp 116ndash127 Springer 2009
[8] M S Nobile D Besozzi P Cazzaniga GMauri andD PescinildquocupSODA aGPU-basedmulti-swarmPSOmethod for param-eter estimation in stochastic biological systems exploitingdiscrete-time target seriesrdquo in Proceedings of the 10th EuropeanConference Evolutionary Computation Machine Learning andData Mining in Bioinformatics (EvoBIO rsquo12) M Giacobini LVanneschi and W Bush Eds vol 7246 of Lecture Notes inComputer Science pp 74ndash85 Springer
[9] M S Nobile D Besozzi P Cazzaniga D Pescini and GMaurildquoReverse engineering of kinetic reaction networks by meansof Cartesian Genetic Programming and Particle Swarm Opti-mizationrdquo in Proceedings of the IEEE Congress on EvolutionaryComputation (CEC rsquo13) pp 1594ndash1601 June 2013
[10] M S Nobile ldquoEvolutionary inference of biochemical inter-action networks accelerated on Graphics Processing Unitsrdquoin Proceedings of the 11th International Conference on HighPerformance Computingamp Simulation 2013 (HPCS rsquo13) pp 668ndash670 Helsinki Finland 2013
[11] K Kentzoglanakis and M Poole ldquoA swarm intelligence frame-work for reconstructing gene networks searching for bio-logically plausible architecturesrdquo IEEEACM Transactions onComputational Biology and Bioinformatics vol 9 no 2 pp 358ndash371 2012
[12] J R Koza W Mydlowec G Lanza Y Jessen and M A KeaneldquoAutomatic computational discovery of chemical reaction net-works using genetic programmingrdquo inComputationalDiscoveryof Scientific Knowledge S Dzeroski and L Todorovski Edsvol 4660 of Lecture Notes in Computer Science pp 205ndash227Springer 2007
[13] S Ando E Sakamoto and H Iba ldquoEvolutionary modeling andinference of gene networkrdquo Information Sciences vol 145 no3-4 pp 237ndash259 2002
[14] L Dematte and D Prandi ldquoGPU computing for systemsbiologyrdquo Briefings in Bioinformatics vol 11 no 3 Article IDbbq006 pp 323ndash333 2010
[15] J L Payne N A Sinnott-Armstrong and J HMoore ldquoExploit-ing graphics processing units for computational biology andbioinformaticsrdquo Interdisciplinary Sciences Computational LifeSciences vol 2 no 3 pp 213ndash220 2010
[16] M J Harvey and G De Fabritiis ldquoA survey of computationalmolecular science using graphics processing unitsrdquoWiley Inter-disciplinary Reviews Computational Molecular Science vol 2no 5 pp 734ndash742 2012
[17] Y Zhou J Liepe X Sheng M P H Stumpf and C BarnesldquoGPU accelerated biochemical network simulationrdquo Bioinfor-matics vol 27 no 6 Article ID btr015 pp 874ndash876 2011
[18] M Vigelius A Lane and B Meyer ldquoAccelerating reaction-diffusion simulations with general-purpose graphics processingunitsrdquo Bioinformatics vol 27 no 2 Article ID btq622 pp 288ndash290 2011
[19] D J Wilkinson ldquoStochastic modelling for quantitative descrip-tion of heterogeneous biological systemsrdquo Nature ReviewsGenetics vol 10 no 2 pp 122ndash133 2009
[20] J C Butcher ldquoTrees and numerical methods for ordinarydifferential equationsrdquo Numerical Algorithms vol 53 no 2-3pp 153ndash170 2010
[21] D T Gillespie ldquoStochastic simulation of chemical kineticsrdquoAnnual Review of Physical Chemistry vol 58 pp 35ndash55 2007
[22] L Petzold ldquoAutomatic selection of methods for solving stiffand nonstiff systems of ordinary differential equationsrdquo SIAMJournal of Scientific and Statistical Computing vol 4 no 1 pp136ndash148 1983
[23] Y Cao D T Gillespie and L R Petzold ldquoEfficient step sizeselection for the tau-leaping simulation methodrdquo The Journalof Chemical Physics vol 124 no 4 2006
[24] M S Nobile D Besozzi P Cazzaniga GMauri andD PescinildquocupSODA a CUDA-powered simulator of mass-action kinet-icsrdquo in Proceedings of the 12th International Conference onParallel Computing Technologies (PaCT rsquo13) V Malyshkin Edvol 7979 of Lecture Notes in Computer Science pp 344ndash357
[25] M S Nobile P Cazzaniga D Besozzi and G Mauri ldquoGPU-accelerated simulations of mass-action kinetics models withcupSODArdquoThe Journal of Supercomputing 2014
[26] M S Nobile P Cazzaniga D Besozzi D Pescini and GMauri ldquocuTauLeaping a GPU-powered tau-leaping stochasticsimulator for massive parallel analyses of biological systemsrdquoPLoS ONE vol 9 no 3 2014
[27] K Rogers Blood Physiology and CirculationTheHuman BodyBritannica Educational Publishing 2010
[28] A D Michelson Platelets Elsevier 2nd edition 2007[29] R W Colman J Hirsh V J Marder A W Clowes and J
N George Hemostasis and Thrombosis Basic Principles andClinical Practice Lippincott Williams ampWilkins 2006
[30] M S Chatterjee W S Denney H Jing and S L DiamondldquoSystems biology of coagulation initiation Kinetics of thrombingeneration in resting and activated human bloodrdquo PLoS Com-putational Biology vol 6 no 9 Article ID e1000950 2010
[31] M Bellini D Besozzi P Cazzaniga GMauri andM S NobileldquoSimulation and analysis of the blood coagulation cascadeaccelerated on GPUrdquo in Proceedings of the 22nd EuromicroInternational Conference on Parallel Distributed and Network-Based Processing (PDP) pp 590ndash593 IEEE Computer SocietyConference Publishing Services 2014
[32] T Wajima G K Isbister and S B Duffull ldquoA comprehensivemodel for the humoral coagulation network in humansrdquo Clini-cal Pharmacology and Therapeutics vol 86 no 3 pp 290ndash2982009
[33] K Lo W S Denney and S L Diamond ldquoStochastic modelingof blood coagulation initiationrdquo Pathophysiology of Haemostasisand Thrombosis vol 34 no 2-3 pp 80ndash90 2006
[34] C M Danforth T Orfeo K G Mann K E Brummel-Ziedins and S J Everse ldquoThe impact of uncertainty in a bloodcoagulation modelrdquo Mathematical Medicine and Biology vol26 no 4 Article ID dqp011 pp 323ndash336 2009
[35] S D Lewis P P Shields and J A Shafer ldquoCharacterizationof the kinetic pathway for liberation of fibrinopeptides duringassembly of fibrinrdquoThe Journal of Biological Chemistry vol 260no 18 pp 10192ndash10199 1985
[36] M F Hockin K C Jones S J Everse and K G Mann ldquoAmodel for the stoichiometric regulation of blood coagulationrdquoThe Journal of Biological Chemistry vol 277 no 21 pp 18322ndash18333 2002
[37] K Brummel-Ziedins C YVossen F R Rosendaal KUmezakiand K G Mann ldquoThe plasma hemostatic proteome thrombin
20 BioMed Research International
generation in healthy individualsrdquo Journal of Thrombosis andHaemostasis vol 3 no 7 pp 1472ndash1481 2005
[38] K E Brummel-Ziedins C Y Vossen S Butenas K G Mannand F R Rosendaal ldquoThrombin generation profiles in deepvenous thrombosisrdquo Journal of Thrombosis and Haemostasisvol 3 no 11 pp 2497ndash2505 2005
[39] S D Bungay P A Gentry and R D Gentry ldquoA mathematicalmodel of lipid-mediated thrombin generationrdquo MathematicalMedicine and Biology vol 20 no 1 pp 105ndash129 2003
[40] S Butenas T Orfeo M T Gissel K E Brummel and K GMann ldquoThe significance of circulating factor IXa in bloodrdquoThe Journal of Biological Chemistry vol 279 no 22 pp 22875ndash22882 2004
[41] D Voet and J G Voet Biochemistry Wiley 2011[42] A C Guyton and J E Hall Textbook of Medical Physiology
Saunders 2010[43] M F M James and A M Roche ldquoDose-response relationship
between plasma ionized calcium concentration and thrombe-lastographyrdquo Journal of Cardiothoracic and Vascular Anesthesiavol 18 no 5 pp 581ndash586 2004
[44] D L Nelson and M M Cox Lehninger Principles of Biochem-istry W H Freeman Company 2004
[45] R M Farber ldquoTopical perspective on massive threading andparallelismrdquo Journal of Molecular Graphics and Modelling vol30 pp 82ndash89 2011
[46] OWolkenhauerMUllahWKolch andK-HCho ldquoModelingand simulation of intracellular dynamics choosing an appropri-ate frameworkrdquo IEEETransactions onNanobioscience vol 3 no3 pp 200ndash207 2004
[47] B H Estridge A P Reynolds and N J Walters Basic MedicalLaboratory Techniques Cengage Learning 4th edition 2000
[48] T Renne A H Schmaier K F Nickel M Blomback and CMaas ldquoIn vivo roles of factor XIIrdquo Blood vol 120 no 22 pp4296ndash4303 2012
[49] P V Jenkins O Rawley O P Smith and J S ODonnellldquoElevated factor VIII levels and risk of venous thrombosisrdquoBritish Journal of Haematology vol 157 no 6 pp 653ndash663 2012
[50] C E Walsh and K M Batt ldquoHemophilia clinical gene therapybrief reviewrdquo Translational Research vol 161 no 4 pp 307ndash3122013
[51] A Van Hylckama Vlieg I K Van Der Linden R M Bertinaand F R Rosendaal ldquoHigh levels of factor IX increase the risk ofvenous thrombosisrdquo Blood vol 95 no 12 pp 3678ndash3682 2000
[52] K Brummel-Ziedins ldquoModels for thrombin generation and riskof diseaserdquo Journal ofThrombosis and Haemostasis vol 11 no 1pp 212ndash223 2013
[53] U Seligsohn and A Lubetsky ldquoGenetic susceptibility to venousthrombosisrdquoTheNew England Journal of Medicine vol 344 no16 pp 1222ndash1231 2001
[54] I Martinelli P Bucciarelli and P M Mannucci ldquoThromboticrisk factors basic pathophysiologyrdquo Critical Care Medicine vol38 no 2 pp S3ndashS9 2010
[55] K Brummel-Ziedins A Undas T Orfeo et al ldquoThrombingeneration in acute coronary syndrome and stable coronaryartery disease dependence on plasma factor compositionrdquoJournal ofThrombosis andHaemostasis vol 6 no 1 pp 104ndash1102008
[56] S Hoops R Gauges C Lee et al ldquoCOPASImdasha complexpathway simulatorrdquo Bioinformatics vol 22 no 24 pp 3067ndash3074 2006
[57] Nvidia NVIDIArsquos Next Generation CUDA Compute Architec-ture Kepler GK110 2012
[58] W Stortelder W Hemker and H Hemker ldquoMathematicalmodelling in blood coagulation simulation and parameterestimationrdquoModelling Analysis and Simulation vol 20 pp 1ndash111997
[59] S M C Orfao G Jank and K Mottaghy ldquoA new mathematicalapproach to modelling thrombin generationrdquo InternationalJournal of Artificial Organs vol 29 no 7 pp 703ndash708 2006
[60] Nvidia Nvidia CUDA C Programming Guide v50 2012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Anatomy Research International
PeptidesInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
International Journal of
Volume 2014
Zoology
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Molecular Biology International
GenomicsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
BioinformaticsAdvances in
Marine BiologyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Signal TransductionJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
BioMed Research International
Evolutionary BiologyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Biochemistry Research International
ArchaeaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Genetics Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in
Virolog y
Hindawi Publishing Corporationhttpwwwhindawicom
Nucleic AcidsJournal of
Volume 2014
Stem CellsInternational
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Enzyme Research
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Microbiology
BioMed Research International 17
0
200
400
600
Tissue factor (M) Factor IX (M
)
Clot
ting
time (
s)
0
100
200
300
400
500
600
700
1eminus12
1eminus10
1eminus8
1eminus131eminus11
1eminus9
Figure 14 Clotting time at different initial concentrations of Tissue Factor and factor IXThe sweep ranges are [500 sdot 10minus14 500 sdot 10minus10] and[900 sdot 10
minus12 900 sdot 10
minus8] respectively
5 10 50 100 500 1000 5000 10000
Runn
ing
time (
s)
Number of simulations
CPUGPU
1e2
1e3
1e4
1e5
1e6
1e7
Figure 15 Comparison of the computational time required to execute an increasing number of simulations of the BCCmodel on CPU (IntelCore i5 23 GHz) and GPU (Nvidia Tesla K20c GPU clock 706MHz) The computational time is expressed in seconds The time valuesrelated to 1000 5000 and 10000 simulations on the CPU were estimated by regression (see also Table 3)
Table 3 CPU versus GPU performance comparison
Number ofsimulations CPU time (sec) GPU time (sec) Speedup
5 3808 1278 298times10 10604 2329 456times50 51704 8058 642times100 116528 10977 1061times500 536058 17394 3081times1000 1073582lowast 19982 5373times5000 5363507lowast 30960 1732times10000 10724975lowast 58951 1819timeslowastValues estimated by regression
to efficiently determine the response of the BCC to differenttherapeutical drugs
4 Conclusions
Thanks to their high-performance computing capabilitiesand the very low costs GPUs nowadays represent a suitable
technology for the development and the application of par-allel computational methods for in silico analysis of complexbiological systems However the implementation of efficientcomputational tools able to fully exploit the large potentialityof GPUs is still challenging since good programming skillsare required to implement GPU-based algorithmic methodsand to handle specific features as an optimal usage ofmemoryor the communication bandwidth between GPU and CPUMoreover algorithms cannot be directly ported on the GPUbecause of the limited programming capabilities allowed byGPUkernels as amatter of fact they need to be redesigned tomeet the requirements of the underlying SIMD architecture
In this work we presented coagSODA a GPU-poweredsimulator specifically developed to carry out fast parallelsimulations of the BCC model coagSODA was designed tooffer a black-box solution usable by any final user in aneasy way It relies on cupSODA [24] a numerical integratorfor ODEs that we previously implemented for the GPUarchitecture based on the LSODA algorithm and capableof automatically translating a reaction-based model into aset of coupled ODEs In addition to mass-action kinet-ics coagSODA implements specific functions to compute
18 BioMed Research International
the kinetics of Hill function based reactions such as thoseinvolved in the platelets activity of the BCC model coag-SODA exploits the massive parallel capabilities of modernGPUs and our results demonstrated that it can achieve arelevant reduction of the computational time required toexecutemany concurrent and independent simulations of theBCC dynamics The mutual independence of the simulationsallows fully exploiting the underlying SIMD architecturemoreover coagSODA benefits from an additional speedupthanks to our choice of storing the state of the systeminto the low-latency shared memory (a solution that wasalready implemented in cupSODA) Since the BCC modelis large (96 reactions 71 chemical species) a large amountof shared memory was assigned to each thread stronglyreducing the theoretical occupancy of the GPU that is theratio of active warps with respect to the maximum numberof warps supported by each streaming multiprocessor of theGPU However the results of the analysis presented in thiswork performed on the BCC model show that coagSODAachieves a relevant boost with respect to a reference CPUimplementation For instance in the case of 10000 simula-tions we achieved a noticeable 181times speedup Interestinglythe performances of coagSODA are better than COPASI evenfor small batches of simulations These results indicate thatfor biological models consisting in many reactions and manyspecies our GPU implementation of LSODA becomes moreprofitable than the CPU counterpart as the number of con-current simulations increases making it suitable especiallywhen performing demanding computational analysis suchas for example parameter sweep parameter estimation orsensitivity analysis
As a test case in this paper we presented several parame-ter sweep analyses over reaction constants and initial concen-trations of factors involved in the BCC model Other com-putational analyses on mathematical models of this pathwaywere previously presented For instance in [58] a parameterestimation of the reaction constants of different modelsinvolving the activation of factor X factor V prothrombinand the inactivation of factors was performed The aim ofthis analysis was to discriminate between different modelsto unravel the mechanisms on the basis of the BCC Similaranalyses were executed on a model describing thrombingeneration in plasma since the reaction mechanism thereaction constants and initial concentrations were unknown[59] In [34] a sensitivity analysis of a model consisting of44 species over 34 chemical species was presented reactionconstants were varied between 10 and 1000 of theirreference value to the aim of identifying the most influentialfactors of the BCC
All these analyses were performed bymeans of sequentialsimulations of the models under investigation thereforein general only small batches of simulations could be run(for instance in [34] only 836 simulations were executed toexecute the sensitivity analysis of the model) On the otherhand in this paper we presented the results of different PSAefficiently executed bymeans of coagSODA overall providinguseful information regarding unknownparameters and inter-esting insights into the functioning of the BCC A thoroughsensitivity analysis of the whole BCC model consisting in
around 5 sdot 105 simulations is currently in progress on GPUsby our group exploiting the great computational efficiency ofcoagSODA
The results of our computational analyses should benow validated by means of specifically designed laboratoryexperiments In particular we identified a plausible valuefor the constant of the reaction describing the autoactivationof factor XII (since no reference values can be found inliterature) moreover the results of the PSA over factor IXsuggest that a deficiency of this factor is not enough to causesevere bleeding disorders as haemophilia B but an alterationof other factors seems to be necessary for the occurrence ofsuch condition (eg the lack of Tissue Factor as suggestedby the PSA-2D over Tissue Factor and factor IX) Finallythe PSA over factors VIII and IX showed in a situationcharacterized by very high concentrations of these factorsa counterintuitive behavior in which the clotting time isincreased with respect to the value obtained in physiologicalconditions Indeed it would be interesting to design ad hoclaboratory experiments to verify if the BCCmodel is actuallypredictive in such extreme situations
The coagSODA software and the SBML version of theBCC model used in this work are available from the authorsupon request
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This work is partially supported by SYSBIO Centre forSystems Biology
References
[1] B B Aldridge JM BurkeDA Lauffenburger andP K SorgerldquoPhysicochemicalmodelling of cell signalling pathwaysrdquoNatureCell Biology vol 8 no 11 pp 1195ndash1203 2006
[2] I-C Chou and E O Voit ldquoRecent developments in parameterestimation and structure identification of biochemical andgenomic systemsrdquoMathematical Biosciences vol 219 no 2 pp57ndash83 2009
[3] D Besozzi P Cazzaniga D Pescini G Mauri S Colomboand EMartegani ldquoThe role of feedback control mechanisms onthe establishment of oscillatory regimes in the RascAMPPKApathway in S cerevisiaerdquo EURASIP Journal on Bioinformaticsand Systems Biology vol 2012 no 10 2012
[4] A Saltelli M Ratto T Andres et al Global Sensitivity AnalysisThe Primer Wiley-Interscience 2008
[5] A Raue C Kreutz T Maiwald et al ldquoStructural and practicalidentifiability analysis of partially observed dynamical modelsby exploiting the profile likelihoodrdquo Bioinformatics vol 25 no15 pp 1923ndash1929 2009
[6] A Drager M Kronfeld M J Ziller et al ldquoModeling metabolicnetworks in C glutamicum a comparison of rate laws incombination with various parameter optimization strategiesrdquoBMC Systems Biology vol 3 article 5 2009
BioMed Research International 19
[7] D Besozzi P Cazzaniga G Mauri D Pescini and L Van-neschi ldquoA comparison of genetic algorithms and particle swarmoptimization for parameter estimation in stochastic biochem-ical systemsrdquo in Proceedings of the 7th European ConferenceEvolutionary Computation Machine Learning and Data Miningin Bioinformatics (EvoBIO rsquo09) C Pizzuti M Ritchie and MGiacobini Eds vol 5483 of Lecture Notes in Computer Sciencepp 116ndash127 Springer 2009
[8] M S Nobile D Besozzi P Cazzaniga GMauri andD PescinildquocupSODA aGPU-basedmulti-swarmPSOmethod for param-eter estimation in stochastic biological systems exploitingdiscrete-time target seriesrdquo in Proceedings of the 10th EuropeanConference Evolutionary Computation Machine Learning andData Mining in Bioinformatics (EvoBIO rsquo12) M Giacobini LVanneschi and W Bush Eds vol 7246 of Lecture Notes inComputer Science pp 74ndash85 Springer
[9] M S Nobile D Besozzi P Cazzaniga D Pescini and GMaurildquoReverse engineering of kinetic reaction networks by meansof Cartesian Genetic Programming and Particle Swarm Opti-mizationrdquo in Proceedings of the IEEE Congress on EvolutionaryComputation (CEC rsquo13) pp 1594ndash1601 June 2013
[10] M S Nobile ldquoEvolutionary inference of biochemical inter-action networks accelerated on Graphics Processing Unitsrdquoin Proceedings of the 11th International Conference on HighPerformance Computingamp Simulation 2013 (HPCS rsquo13) pp 668ndash670 Helsinki Finland 2013
[11] K Kentzoglanakis and M Poole ldquoA swarm intelligence frame-work for reconstructing gene networks searching for bio-logically plausible architecturesrdquo IEEEACM Transactions onComputational Biology and Bioinformatics vol 9 no 2 pp 358ndash371 2012
[12] J R Koza W Mydlowec G Lanza Y Jessen and M A KeaneldquoAutomatic computational discovery of chemical reaction net-works using genetic programmingrdquo inComputationalDiscoveryof Scientific Knowledge S Dzeroski and L Todorovski Edsvol 4660 of Lecture Notes in Computer Science pp 205ndash227Springer 2007
[13] S Ando E Sakamoto and H Iba ldquoEvolutionary modeling andinference of gene networkrdquo Information Sciences vol 145 no3-4 pp 237ndash259 2002
[14] L Dematte and D Prandi ldquoGPU computing for systemsbiologyrdquo Briefings in Bioinformatics vol 11 no 3 Article IDbbq006 pp 323ndash333 2010
[15] J L Payne N A Sinnott-Armstrong and J HMoore ldquoExploit-ing graphics processing units for computational biology andbioinformaticsrdquo Interdisciplinary Sciences Computational LifeSciences vol 2 no 3 pp 213ndash220 2010
[16] M J Harvey and G De Fabritiis ldquoA survey of computationalmolecular science using graphics processing unitsrdquoWiley Inter-disciplinary Reviews Computational Molecular Science vol 2no 5 pp 734ndash742 2012
[17] Y Zhou J Liepe X Sheng M P H Stumpf and C BarnesldquoGPU accelerated biochemical network simulationrdquo Bioinfor-matics vol 27 no 6 Article ID btr015 pp 874ndash876 2011
[18] M Vigelius A Lane and B Meyer ldquoAccelerating reaction-diffusion simulations with general-purpose graphics processingunitsrdquo Bioinformatics vol 27 no 2 Article ID btq622 pp 288ndash290 2011
[19] D J Wilkinson ldquoStochastic modelling for quantitative descrip-tion of heterogeneous biological systemsrdquo Nature ReviewsGenetics vol 10 no 2 pp 122ndash133 2009
[20] J C Butcher ldquoTrees and numerical methods for ordinarydifferential equationsrdquo Numerical Algorithms vol 53 no 2-3pp 153ndash170 2010
[21] D T Gillespie ldquoStochastic simulation of chemical kineticsrdquoAnnual Review of Physical Chemistry vol 58 pp 35ndash55 2007
[22] L Petzold ldquoAutomatic selection of methods for solving stiffand nonstiff systems of ordinary differential equationsrdquo SIAMJournal of Scientific and Statistical Computing vol 4 no 1 pp136ndash148 1983
[23] Y Cao D T Gillespie and L R Petzold ldquoEfficient step sizeselection for the tau-leaping simulation methodrdquo The Journalof Chemical Physics vol 124 no 4 2006
[24] M S Nobile D Besozzi P Cazzaniga GMauri andD PescinildquocupSODA a CUDA-powered simulator of mass-action kinet-icsrdquo in Proceedings of the 12th International Conference onParallel Computing Technologies (PaCT rsquo13) V Malyshkin Edvol 7979 of Lecture Notes in Computer Science pp 344ndash357
[25] M S Nobile P Cazzaniga D Besozzi and G Mauri ldquoGPU-accelerated simulations of mass-action kinetics models withcupSODArdquoThe Journal of Supercomputing 2014
[26] M S Nobile P Cazzaniga D Besozzi D Pescini and GMauri ldquocuTauLeaping a GPU-powered tau-leaping stochasticsimulator for massive parallel analyses of biological systemsrdquoPLoS ONE vol 9 no 3 2014
[27] K Rogers Blood Physiology and CirculationTheHuman BodyBritannica Educational Publishing 2010
[28] A D Michelson Platelets Elsevier 2nd edition 2007[29] R W Colman J Hirsh V J Marder A W Clowes and J
N George Hemostasis and Thrombosis Basic Principles andClinical Practice Lippincott Williams ampWilkins 2006
[30] M S Chatterjee W S Denney H Jing and S L DiamondldquoSystems biology of coagulation initiation Kinetics of thrombingeneration in resting and activated human bloodrdquo PLoS Com-putational Biology vol 6 no 9 Article ID e1000950 2010
[31] M Bellini D Besozzi P Cazzaniga GMauri andM S NobileldquoSimulation and analysis of the blood coagulation cascadeaccelerated on GPUrdquo in Proceedings of the 22nd EuromicroInternational Conference on Parallel Distributed and Network-Based Processing (PDP) pp 590ndash593 IEEE Computer SocietyConference Publishing Services 2014
[32] T Wajima G K Isbister and S B Duffull ldquoA comprehensivemodel for the humoral coagulation network in humansrdquo Clini-cal Pharmacology and Therapeutics vol 86 no 3 pp 290ndash2982009
[33] K Lo W S Denney and S L Diamond ldquoStochastic modelingof blood coagulation initiationrdquo Pathophysiology of Haemostasisand Thrombosis vol 34 no 2-3 pp 80ndash90 2006
[34] C M Danforth T Orfeo K G Mann K E Brummel-Ziedins and S J Everse ldquoThe impact of uncertainty in a bloodcoagulation modelrdquo Mathematical Medicine and Biology vol26 no 4 Article ID dqp011 pp 323ndash336 2009
[35] S D Lewis P P Shields and J A Shafer ldquoCharacterizationof the kinetic pathway for liberation of fibrinopeptides duringassembly of fibrinrdquoThe Journal of Biological Chemistry vol 260no 18 pp 10192ndash10199 1985
[36] M F Hockin K C Jones S J Everse and K G Mann ldquoAmodel for the stoichiometric regulation of blood coagulationrdquoThe Journal of Biological Chemistry vol 277 no 21 pp 18322ndash18333 2002
[37] K Brummel-Ziedins C YVossen F R Rosendaal KUmezakiand K G Mann ldquoThe plasma hemostatic proteome thrombin
20 BioMed Research International
generation in healthy individualsrdquo Journal of Thrombosis andHaemostasis vol 3 no 7 pp 1472ndash1481 2005
[38] K E Brummel-Ziedins C Y Vossen S Butenas K G Mannand F R Rosendaal ldquoThrombin generation profiles in deepvenous thrombosisrdquo Journal of Thrombosis and Haemostasisvol 3 no 11 pp 2497ndash2505 2005
[39] S D Bungay P A Gentry and R D Gentry ldquoA mathematicalmodel of lipid-mediated thrombin generationrdquo MathematicalMedicine and Biology vol 20 no 1 pp 105ndash129 2003
[40] S Butenas T Orfeo M T Gissel K E Brummel and K GMann ldquoThe significance of circulating factor IXa in bloodrdquoThe Journal of Biological Chemistry vol 279 no 22 pp 22875ndash22882 2004
[41] D Voet and J G Voet Biochemistry Wiley 2011[42] A C Guyton and J E Hall Textbook of Medical Physiology
Saunders 2010[43] M F M James and A M Roche ldquoDose-response relationship
between plasma ionized calcium concentration and thrombe-lastographyrdquo Journal of Cardiothoracic and Vascular Anesthesiavol 18 no 5 pp 581ndash586 2004
[44] D L Nelson and M M Cox Lehninger Principles of Biochem-istry W H Freeman Company 2004
[45] R M Farber ldquoTopical perspective on massive threading andparallelismrdquo Journal of Molecular Graphics and Modelling vol30 pp 82ndash89 2011
[46] OWolkenhauerMUllahWKolch andK-HCho ldquoModelingand simulation of intracellular dynamics choosing an appropri-ate frameworkrdquo IEEETransactions onNanobioscience vol 3 no3 pp 200ndash207 2004
[47] B H Estridge A P Reynolds and N J Walters Basic MedicalLaboratory Techniques Cengage Learning 4th edition 2000
[48] T Renne A H Schmaier K F Nickel M Blomback and CMaas ldquoIn vivo roles of factor XIIrdquo Blood vol 120 no 22 pp4296ndash4303 2012
[49] P V Jenkins O Rawley O P Smith and J S ODonnellldquoElevated factor VIII levels and risk of venous thrombosisrdquoBritish Journal of Haematology vol 157 no 6 pp 653ndash663 2012
[50] C E Walsh and K M Batt ldquoHemophilia clinical gene therapybrief reviewrdquo Translational Research vol 161 no 4 pp 307ndash3122013
[51] A Van Hylckama Vlieg I K Van Der Linden R M Bertinaand F R Rosendaal ldquoHigh levels of factor IX increase the risk ofvenous thrombosisrdquo Blood vol 95 no 12 pp 3678ndash3682 2000
[52] K Brummel-Ziedins ldquoModels for thrombin generation and riskof diseaserdquo Journal ofThrombosis and Haemostasis vol 11 no 1pp 212ndash223 2013
[53] U Seligsohn and A Lubetsky ldquoGenetic susceptibility to venousthrombosisrdquoTheNew England Journal of Medicine vol 344 no16 pp 1222ndash1231 2001
[54] I Martinelli P Bucciarelli and P M Mannucci ldquoThromboticrisk factors basic pathophysiologyrdquo Critical Care Medicine vol38 no 2 pp S3ndashS9 2010
[55] K Brummel-Ziedins A Undas T Orfeo et al ldquoThrombingeneration in acute coronary syndrome and stable coronaryartery disease dependence on plasma factor compositionrdquoJournal ofThrombosis andHaemostasis vol 6 no 1 pp 104ndash1102008
[56] S Hoops R Gauges C Lee et al ldquoCOPASImdasha complexpathway simulatorrdquo Bioinformatics vol 22 no 24 pp 3067ndash3074 2006
[57] Nvidia NVIDIArsquos Next Generation CUDA Compute Architec-ture Kepler GK110 2012
[58] W Stortelder W Hemker and H Hemker ldquoMathematicalmodelling in blood coagulation simulation and parameterestimationrdquoModelling Analysis and Simulation vol 20 pp 1ndash111997
[59] S M C Orfao G Jank and K Mottaghy ldquoA new mathematicalapproach to modelling thrombin generationrdquo InternationalJournal of Artificial Organs vol 29 no 7 pp 703ndash708 2006
[60] Nvidia Nvidia CUDA C Programming Guide v50 2012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Anatomy Research International
PeptidesInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
International Journal of
Volume 2014
Zoology
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Molecular Biology International
GenomicsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
BioinformaticsAdvances in
Marine BiologyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Signal TransductionJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
BioMed Research International
Evolutionary BiologyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Biochemistry Research International
ArchaeaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Genetics Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in
Virolog y
Hindawi Publishing Corporationhttpwwwhindawicom
Nucleic AcidsJournal of
Volume 2014
Stem CellsInternational
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Enzyme Research
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Microbiology
18 BioMed Research International
the kinetics of Hill function based reactions such as thoseinvolved in the platelets activity of the BCC model coag-SODA exploits the massive parallel capabilities of modernGPUs and our results demonstrated that it can achieve arelevant reduction of the computational time required toexecutemany concurrent and independent simulations of theBCC dynamics The mutual independence of the simulationsallows fully exploiting the underlying SIMD architecturemoreover coagSODA benefits from an additional speedupthanks to our choice of storing the state of the systeminto the low-latency shared memory (a solution that wasalready implemented in cupSODA) Since the BCC modelis large (96 reactions 71 chemical species) a large amountof shared memory was assigned to each thread stronglyreducing the theoretical occupancy of the GPU that is theratio of active warps with respect to the maximum numberof warps supported by each streaming multiprocessor of theGPU However the results of the analysis presented in thiswork performed on the BCC model show that coagSODAachieves a relevant boost with respect to a reference CPUimplementation For instance in the case of 10000 simula-tions we achieved a noticeable 181times speedup Interestinglythe performances of coagSODA are better than COPASI evenfor small batches of simulations These results indicate thatfor biological models consisting in many reactions and manyspecies our GPU implementation of LSODA becomes moreprofitable than the CPU counterpart as the number of con-current simulations increases making it suitable especiallywhen performing demanding computational analysis suchas for example parameter sweep parameter estimation orsensitivity analysis
As a test case in this paper we presented several parame-ter sweep analyses over reaction constants and initial concen-trations of factors involved in the BCC model Other com-putational analyses on mathematical models of this pathwaywere previously presented For instance in [58] a parameterestimation of the reaction constants of different modelsinvolving the activation of factor X factor V prothrombinand the inactivation of factors was performed The aim ofthis analysis was to discriminate between different modelsto unravel the mechanisms on the basis of the BCC Similaranalyses were executed on a model describing thrombingeneration in plasma since the reaction mechanism thereaction constants and initial concentrations were unknown[59] In [34] a sensitivity analysis of a model consisting of44 species over 34 chemical species was presented reactionconstants were varied between 10 and 1000 of theirreference value to the aim of identifying the most influentialfactors of the BCC
All these analyses were performed bymeans of sequentialsimulations of the models under investigation thereforein general only small batches of simulations could be run(for instance in [34] only 836 simulations were executed toexecute the sensitivity analysis of the model) On the otherhand in this paper we presented the results of different PSAefficiently executed bymeans of coagSODA overall providinguseful information regarding unknownparameters and inter-esting insights into the functioning of the BCC A thoroughsensitivity analysis of the whole BCC model consisting in
around 5 sdot 105 simulations is currently in progress on GPUsby our group exploiting the great computational efficiency ofcoagSODA
The results of our computational analyses should benow validated by means of specifically designed laboratoryexperiments In particular we identified a plausible valuefor the constant of the reaction describing the autoactivationof factor XII (since no reference values can be found inliterature) moreover the results of the PSA over factor IXsuggest that a deficiency of this factor is not enough to causesevere bleeding disorders as haemophilia B but an alterationof other factors seems to be necessary for the occurrence ofsuch condition (eg the lack of Tissue Factor as suggestedby the PSA-2D over Tissue Factor and factor IX) Finallythe PSA over factors VIII and IX showed in a situationcharacterized by very high concentrations of these factorsa counterintuitive behavior in which the clotting time isincreased with respect to the value obtained in physiologicalconditions Indeed it would be interesting to design ad hoclaboratory experiments to verify if the BCCmodel is actuallypredictive in such extreme situations
The coagSODA software and the SBML version of theBCC model used in this work are available from the authorsupon request
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This work is partially supported by SYSBIO Centre forSystems Biology
References
[1] B B Aldridge JM BurkeDA Lauffenburger andP K SorgerldquoPhysicochemicalmodelling of cell signalling pathwaysrdquoNatureCell Biology vol 8 no 11 pp 1195ndash1203 2006
[2] I-C Chou and E O Voit ldquoRecent developments in parameterestimation and structure identification of biochemical andgenomic systemsrdquoMathematical Biosciences vol 219 no 2 pp57ndash83 2009
[3] D Besozzi P Cazzaniga D Pescini G Mauri S Colomboand EMartegani ldquoThe role of feedback control mechanisms onthe establishment of oscillatory regimes in the RascAMPPKApathway in S cerevisiaerdquo EURASIP Journal on Bioinformaticsand Systems Biology vol 2012 no 10 2012
[4] A Saltelli M Ratto T Andres et al Global Sensitivity AnalysisThe Primer Wiley-Interscience 2008
[5] A Raue C Kreutz T Maiwald et al ldquoStructural and practicalidentifiability analysis of partially observed dynamical modelsby exploiting the profile likelihoodrdquo Bioinformatics vol 25 no15 pp 1923ndash1929 2009
[6] A Drager M Kronfeld M J Ziller et al ldquoModeling metabolicnetworks in C glutamicum a comparison of rate laws incombination with various parameter optimization strategiesrdquoBMC Systems Biology vol 3 article 5 2009
BioMed Research International 19
[7] D Besozzi P Cazzaniga G Mauri D Pescini and L Van-neschi ldquoA comparison of genetic algorithms and particle swarmoptimization for parameter estimation in stochastic biochem-ical systemsrdquo in Proceedings of the 7th European ConferenceEvolutionary Computation Machine Learning and Data Miningin Bioinformatics (EvoBIO rsquo09) C Pizzuti M Ritchie and MGiacobini Eds vol 5483 of Lecture Notes in Computer Sciencepp 116ndash127 Springer 2009
[8] M S Nobile D Besozzi P Cazzaniga GMauri andD PescinildquocupSODA aGPU-basedmulti-swarmPSOmethod for param-eter estimation in stochastic biological systems exploitingdiscrete-time target seriesrdquo in Proceedings of the 10th EuropeanConference Evolutionary Computation Machine Learning andData Mining in Bioinformatics (EvoBIO rsquo12) M Giacobini LVanneschi and W Bush Eds vol 7246 of Lecture Notes inComputer Science pp 74ndash85 Springer
[9] M S Nobile D Besozzi P Cazzaniga D Pescini and GMaurildquoReverse engineering of kinetic reaction networks by meansof Cartesian Genetic Programming and Particle Swarm Opti-mizationrdquo in Proceedings of the IEEE Congress on EvolutionaryComputation (CEC rsquo13) pp 1594ndash1601 June 2013
[10] M S Nobile ldquoEvolutionary inference of biochemical inter-action networks accelerated on Graphics Processing Unitsrdquoin Proceedings of the 11th International Conference on HighPerformance Computingamp Simulation 2013 (HPCS rsquo13) pp 668ndash670 Helsinki Finland 2013
[11] K Kentzoglanakis and M Poole ldquoA swarm intelligence frame-work for reconstructing gene networks searching for bio-logically plausible architecturesrdquo IEEEACM Transactions onComputational Biology and Bioinformatics vol 9 no 2 pp 358ndash371 2012
[12] J R Koza W Mydlowec G Lanza Y Jessen and M A KeaneldquoAutomatic computational discovery of chemical reaction net-works using genetic programmingrdquo inComputationalDiscoveryof Scientific Knowledge S Dzeroski and L Todorovski Edsvol 4660 of Lecture Notes in Computer Science pp 205ndash227Springer 2007
[13] S Ando E Sakamoto and H Iba ldquoEvolutionary modeling andinference of gene networkrdquo Information Sciences vol 145 no3-4 pp 237ndash259 2002
[14] L Dematte and D Prandi ldquoGPU computing for systemsbiologyrdquo Briefings in Bioinformatics vol 11 no 3 Article IDbbq006 pp 323ndash333 2010
[15] J L Payne N A Sinnott-Armstrong and J HMoore ldquoExploit-ing graphics processing units for computational biology andbioinformaticsrdquo Interdisciplinary Sciences Computational LifeSciences vol 2 no 3 pp 213ndash220 2010
[16] M J Harvey and G De Fabritiis ldquoA survey of computationalmolecular science using graphics processing unitsrdquoWiley Inter-disciplinary Reviews Computational Molecular Science vol 2no 5 pp 734ndash742 2012
[17] Y Zhou J Liepe X Sheng M P H Stumpf and C BarnesldquoGPU accelerated biochemical network simulationrdquo Bioinfor-matics vol 27 no 6 Article ID btr015 pp 874ndash876 2011
[18] M Vigelius A Lane and B Meyer ldquoAccelerating reaction-diffusion simulations with general-purpose graphics processingunitsrdquo Bioinformatics vol 27 no 2 Article ID btq622 pp 288ndash290 2011
[19] D J Wilkinson ldquoStochastic modelling for quantitative descrip-tion of heterogeneous biological systemsrdquo Nature ReviewsGenetics vol 10 no 2 pp 122ndash133 2009
[20] J C Butcher ldquoTrees and numerical methods for ordinarydifferential equationsrdquo Numerical Algorithms vol 53 no 2-3pp 153ndash170 2010
[21] D T Gillespie ldquoStochastic simulation of chemical kineticsrdquoAnnual Review of Physical Chemistry vol 58 pp 35ndash55 2007
[22] L Petzold ldquoAutomatic selection of methods for solving stiffand nonstiff systems of ordinary differential equationsrdquo SIAMJournal of Scientific and Statistical Computing vol 4 no 1 pp136ndash148 1983
[23] Y Cao D T Gillespie and L R Petzold ldquoEfficient step sizeselection for the tau-leaping simulation methodrdquo The Journalof Chemical Physics vol 124 no 4 2006
[24] M S Nobile D Besozzi P Cazzaniga GMauri andD PescinildquocupSODA a CUDA-powered simulator of mass-action kinet-icsrdquo in Proceedings of the 12th International Conference onParallel Computing Technologies (PaCT rsquo13) V Malyshkin Edvol 7979 of Lecture Notes in Computer Science pp 344ndash357
[25] M S Nobile P Cazzaniga D Besozzi and G Mauri ldquoGPU-accelerated simulations of mass-action kinetics models withcupSODArdquoThe Journal of Supercomputing 2014
[26] M S Nobile P Cazzaniga D Besozzi D Pescini and GMauri ldquocuTauLeaping a GPU-powered tau-leaping stochasticsimulator for massive parallel analyses of biological systemsrdquoPLoS ONE vol 9 no 3 2014
[27] K Rogers Blood Physiology and CirculationTheHuman BodyBritannica Educational Publishing 2010
[28] A D Michelson Platelets Elsevier 2nd edition 2007[29] R W Colman J Hirsh V J Marder A W Clowes and J
N George Hemostasis and Thrombosis Basic Principles andClinical Practice Lippincott Williams ampWilkins 2006
[30] M S Chatterjee W S Denney H Jing and S L DiamondldquoSystems biology of coagulation initiation Kinetics of thrombingeneration in resting and activated human bloodrdquo PLoS Com-putational Biology vol 6 no 9 Article ID e1000950 2010
[31] M Bellini D Besozzi P Cazzaniga GMauri andM S NobileldquoSimulation and analysis of the blood coagulation cascadeaccelerated on GPUrdquo in Proceedings of the 22nd EuromicroInternational Conference on Parallel Distributed and Network-Based Processing (PDP) pp 590ndash593 IEEE Computer SocietyConference Publishing Services 2014
[32] T Wajima G K Isbister and S B Duffull ldquoA comprehensivemodel for the humoral coagulation network in humansrdquo Clini-cal Pharmacology and Therapeutics vol 86 no 3 pp 290ndash2982009
[33] K Lo W S Denney and S L Diamond ldquoStochastic modelingof blood coagulation initiationrdquo Pathophysiology of Haemostasisand Thrombosis vol 34 no 2-3 pp 80ndash90 2006
[34] C M Danforth T Orfeo K G Mann K E Brummel-Ziedins and S J Everse ldquoThe impact of uncertainty in a bloodcoagulation modelrdquo Mathematical Medicine and Biology vol26 no 4 Article ID dqp011 pp 323ndash336 2009
[35] S D Lewis P P Shields and J A Shafer ldquoCharacterizationof the kinetic pathway for liberation of fibrinopeptides duringassembly of fibrinrdquoThe Journal of Biological Chemistry vol 260no 18 pp 10192ndash10199 1985
[36] M F Hockin K C Jones S J Everse and K G Mann ldquoAmodel for the stoichiometric regulation of blood coagulationrdquoThe Journal of Biological Chemistry vol 277 no 21 pp 18322ndash18333 2002
[37] K Brummel-Ziedins C YVossen F R Rosendaal KUmezakiand K G Mann ldquoThe plasma hemostatic proteome thrombin
20 BioMed Research International
generation in healthy individualsrdquo Journal of Thrombosis andHaemostasis vol 3 no 7 pp 1472ndash1481 2005
[38] K E Brummel-Ziedins C Y Vossen S Butenas K G Mannand F R Rosendaal ldquoThrombin generation profiles in deepvenous thrombosisrdquo Journal of Thrombosis and Haemostasisvol 3 no 11 pp 2497ndash2505 2005
[39] S D Bungay P A Gentry and R D Gentry ldquoA mathematicalmodel of lipid-mediated thrombin generationrdquo MathematicalMedicine and Biology vol 20 no 1 pp 105ndash129 2003
[40] S Butenas T Orfeo M T Gissel K E Brummel and K GMann ldquoThe significance of circulating factor IXa in bloodrdquoThe Journal of Biological Chemistry vol 279 no 22 pp 22875ndash22882 2004
[41] D Voet and J G Voet Biochemistry Wiley 2011[42] A C Guyton and J E Hall Textbook of Medical Physiology
Saunders 2010[43] M F M James and A M Roche ldquoDose-response relationship
between plasma ionized calcium concentration and thrombe-lastographyrdquo Journal of Cardiothoracic and Vascular Anesthesiavol 18 no 5 pp 581ndash586 2004
[44] D L Nelson and M M Cox Lehninger Principles of Biochem-istry W H Freeman Company 2004
[45] R M Farber ldquoTopical perspective on massive threading andparallelismrdquo Journal of Molecular Graphics and Modelling vol30 pp 82ndash89 2011
[46] OWolkenhauerMUllahWKolch andK-HCho ldquoModelingand simulation of intracellular dynamics choosing an appropri-ate frameworkrdquo IEEETransactions onNanobioscience vol 3 no3 pp 200ndash207 2004
[47] B H Estridge A P Reynolds and N J Walters Basic MedicalLaboratory Techniques Cengage Learning 4th edition 2000
[48] T Renne A H Schmaier K F Nickel M Blomback and CMaas ldquoIn vivo roles of factor XIIrdquo Blood vol 120 no 22 pp4296ndash4303 2012
[49] P V Jenkins O Rawley O P Smith and J S ODonnellldquoElevated factor VIII levels and risk of venous thrombosisrdquoBritish Journal of Haematology vol 157 no 6 pp 653ndash663 2012
[50] C E Walsh and K M Batt ldquoHemophilia clinical gene therapybrief reviewrdquo Translational Research vol 161 no 4 pp 307ndash3122013
[51] A Van Hylckama Vlieg I K Van Der Linden R M Bertinaand F R Rosendaal ldquoHigh levels of factor IX increase the risk ofvenous thrombosisrdquo Blood vol 95 no 12 pp 3678ndash3682 2000
[52] K Brummel-Ziedins ldquoModels for thrombin generation and riskof diseaserdquo Journal ofThrombosis and Haemostasis vol 11 no 1pp 212ndash223 2013
[53] U Seligsohn and A Lubetsky ldquoGenetic susceptibility to venousthrombosisrdquoTheNew England Journal of Medicine vol 344 no16 pp 1222ndash1231 2001
[54] I Martinelli P Bucciarelli and P M Mannucci ldquoThromboticrisk factors basic pathophysiologyrdquo Critical Care Medicine vol38 no 2 pp S3ndashS9 2010
[55] K Brummel-Ziedins A Undas T Orfeo et al ldquoThrombingeneration in acute coronary syndrome and stable coronaryartery disease dependence on plasma factor compositionrdquoJournal ofThrombosis andHaemostasis vol 6 no 1 pp 104ndash1102008
[56] S Hoops R Gauges C Lee et al ldquoCOPASImdasha complexpathway simulatorrdquo Bioinformatics vol 22 no 24 pp 3067ndash3074 2006
[57] Nvidia NVIDIArsquos Next Generation CUDA Compute Architec-ture Kepler GK110 2012
[58] W Stortelder W Hemker and H Hemker ldquoMathematicalmodelling in blood coagulation simulation and parameterestimationrdquoModelling Analysis and Simulation vol 20 pp 1ndash111997
[59] S M C Orfao G Jank and K Mottaghy ldquoA new mathematicalapproach to modelling thrombin generationrdquo InternationalJournal of Artificial Organs vol 29 no 7 pp 703ndash708 2006
[60] Nvidia Nvidia CUDA C Programming Guide v50 2012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Anatomy Research International
PeptidesInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
International Journal of
Volume 2014
Zoology
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Molecular Biology International
GenomicsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
BioinformaticsAdvances in
Marine BiologyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Signal TransductionJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
BioMed Research International
Evolutionary BiologyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Biochemistry Research International
ArchaeaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Genetics Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in
Virolog y
Hindawi Publishing Corporationhttpwwwhindawicom
Nucleic AcidsJournal of
Volume 2014
Stem CellsInternational
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Enzyme Research
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Microbiology
BioMed Research International 19
[7] D Besozzi P Cazzaniga G Mauri D Pescini and L Van-neschi ldquoA comparison of genetic algorithms and particle swarmoptimization for parameter estimation in stochastic biochem-ical systemsrdquo in Proceedings of the 7th European ConferenceEvolutionary Computation Machine Learning and Data Miningin Bioinformatics (EvoBIO rsquo09) C Pizzuti M Ritchie and MGiacobini Eds vol 5483 of Lecture Notes in Computer Sciencepp 116ndash127 Springer 2009
[8] M S Nobile D Besozzi P Cazzaniga GMauri andD PescinildquocupSODA aGPU-basedmulti-swarmPSOmethod for param-eter estimation in stochastic biological systems exploitingdiscrete-time target seriesrdquo in Proceedings of the 10th EuropeanConference Evolutionary Computation Machine Learning andData Mining in Bioinformatics (EvoBIO rsquo12) M Giacobini LVanneschi and W Bush Eds vol 7246 of Lecture Notes inComputer Science pp 74ndash85 Springer
[9] M S Nobile D Besozzi P Cazzaniga D Pescini and GMaurildquoReverse engineering of kinetic reaction networks by meansof Cartesian Genetic Programming and Particle Swarm Opti-mizationrdquo in Proceedings of the IEEE Congress on EvolutionaryComputation (CEC rsquo13) pp 1594ndash1601 June 2013
[10] M S Nobile ldquoEvolutionary inference of biochemical inter-action networks accelerated on Graphics Processing Unitsrdquoin Proceedings of the 11th International Conference on HighPerformance Computingamp Simulation 2013 (HPCS rsquo13) pp 668ndash670 Helsinki Finland 2013
[11] K Kentzoglanakis and M Poole ldquoA swarm intelligence frame-work for reconstructing gene networks searching for bio-logically plausible architecturesrdquo IEEEACM Transactions onComputational Biology and Bioinformatics vol 9 no 2 pp 358ndash371 2012
[12] J R Koza W Mydlowec G Lanza Y Jessen and M A KeaneldquoAutomatic computational discovery of chemical reaction net-works using genetic programmingrdquo inComputationalDiscoveryof Scientific Knowledge S Dzeroski and L Todorovski Edsvol 4660 of Lecture Notes in Computer Science pp 205ndash227Springer 2007
[13] S Ando E Sakamoto and H Iba ldquoEvolutionary modeling andinference of gene networkrdquo Information Sciences vol 145 no3-4 pp 237ndash259 2002
[14] L Dematte and D Prandi ldquoGPU computing for systemsbiologyrdquo Briefings in Bioinformatics vol 11 no 3 Article IDbbq006 pp 323ndash333 2010
[15] J L Payne N A Sinnott-Armstrong and J HMoore ldquoExploit-ing graphics processing units for computational biology andbioinformaticsrdquo Interdisciplinary Sciences Computational LifeSciences vol 2 no 3 pp 213ndash220 2010
[16] M J Harvey and G De Fabritiis ldquoA survey of computationalmolecular science using graphics processing unitsrdquoWiley Inter-disciplinary Reviews Computational Molecular Science vol 2no 5 pp 734ndash742 2012
[17] Y Zhou J Liepe X Sheng M P H Stumpf and C BarnesldquoGPU accelerated biochemical network simulationrdquo Bioinfor-matics vol 27 no 6 Article ID btr015 pp 874ndash876 2011
[18] M Vigelius A Lane and B Meyer ldquoAccelerating reaction-diffusion simulations with general-purpose graphics processingunitsrdquo Bioinformatics vol 27 no 2 Article ID btq622 pp 288ndash290 2011
[19] D J Wilkinson ldquoStochastic modelling for quantitative descrip-tion of heterogeneous biological systemsrdquo Nature ReviewsGenetics vol 10 no 2 pp 122ndash133 2009
[20] J C Butcher ldquoTrees and numerical methods for ordinarydifferential equationsrdquo Numerical Algorithms vol 53 no 2-3pp 153ndash170 2010
[21] D T Gillespie ldquoStochastic simulation of chemical kineticsrdquoAnnual Review of Physical Chemistry vol 58 pp 35ndash55 2007
[22] L Petzold ldquoAutomatic selection of methods for solving stiffand nonstiff systems of ordinary differential equationsrdquo SIAMJournal of Scientific and Statistical Computing vol 4 no 1 pp136ndash148 1983
[23] Y Cao D T Gillespie and L R Petzold ldquoEfficient step sizeselection for the tau-leaping simulation methodrdquo The Journalof Chemical Physics vol 124 no 4 2006
[24] M S Nobile D Besozzi P Cazzaniga GMauri andD PescinildquocupSODA a CUDA-powered simulator of mass-action kinet-icsrdquo in Proceedings of the 12th International Conference onParallel Computing Technologies (PaCT rsquo13) V Malyshkin Edvol 7979 of Lecture Notes in Computer Science pp 344ndash357
[25] M S Nobile P Cazzaniga D Besozzi and G Mauri ldquoGPU-accelerated simulations of mass-action kinetics models withcupSODArdquoThe Journal of Supercomputing 2014
[26] M S Nobile P Cazzaniga D Besozzi D Pescini and GMauri ldquocuTauLeaping a GPU-powered tau-leaping stochasticsimulator for massive parallel analyses of biological systemsrdquoPLoS ONE vol 9 no 3 2014
[27] K Rogers Blood Physiology and CirculationTheHuman BodyBritannica Educational Publishing 2010
[28] A D Michelson Platelets Elsevier 2nd edition 2007[29] R W Colman J Hirsh V J Marder A W Clowes and J
N George Hemostasis and Thrombosis Basic Principles andClinical Practice Lippincott Williams ampWilkins 2006
[30] M S Chatterjee W S Denney H Jing and S L DiamondldquoSystems biology of coagulation initiation Kinetics of thrombingeneration in resting and activated human bloodrdquo PLoS Com-putational Biology vol 6 no 9 Article ID e1000950 2010
[31] M Bellini D Besozzi P Cazzaniga GMauri andM S NobileldquoSimulation and analysis of the blood coagulation cascadeaccelerated on GPUrdquo in Proceedings of the 22nd EuromicroInternational Conference on Parallel Distributed and Network-Based Processing (PDP) pp 590ndash593 IEEE Computer SocietyConference Publishing Services 2014
[32] T Wajima G K Isbister and S B Duffull ldquoA comprehensivemodel for the humoral coagulation network in humansrdquo Clini-cal Pharmacology and Therapeutics vol 86 no 3 pp 290ndash2982009
[33] K Lo W S Denney and S L Diamond ldquoStochastic modelingof blood coagulation initiationrdquo Pathophysiology of Haemostasisand Thrombosis vol 34 no 2-3 pp 80ndash90 2006
[34] C M Danforth T Orfeo K G Mann K E Brummel-Ziedins and S J Everse ldquoThe impact of uncertainty in a bloodcoagulation modelrdquo Mathematical Medicine and Biology vol26 no 4 Article ID dqp011 pp 323ndash336 2009
[35] S D Lewis P P Shields and J A Shafer ldquoCharacterizationof the kinetic pathway for liberation of fibrinopeptides duringassembly of fibrinrdquoThe Journal of Biological Chemistry vol 260no 18 pp 10192ndash10199 1985
[36] M F Hockin K C Jones S J Everse and K G Mann ldquoAmodel for the stoichiometric regulation of blood coagulationrdquoThe Journal of Biological Chemistry vol 277 no 21 pp 18322ndash18333 2002
[37] K Brummel-Ziedins C YVossen F R Rosendaal KUmezakiand K G Mann ldquoThe plasma hemostatic proteome thrombin
20 BioMed Research International
generation in healthy individualsrdquo Journal of Thrombosis andHaemostasis vol 3 no 7 pp 1472ndash1481 2005
[38] K E Brummel-Ziedins C Y Vossen S Butenas K G Mannand F R Rosendaal ldquoThrombin generation profiles in deepvenous thrombosisrdquo Journal of Thrombosis and Haemostasisvol 3 no 11 pp 2497ndash2505 2005
[39] S D Bungay P A Gentry and R D Gentry ldquoA mathematicalmodel of lipid-mediated thrombin generationrdquo MathematicalMedicine and Biology vol 20 no 1 pp 105ndash129 2003
[40] S Butenas T Orfeo M T Gissel K E Brummel and K GMann ldquoThe significance of circulating factor IXa in bloodrdquoThe Journal of Biological Chemistry vol 279 no 22 pp 22875ndash22882 2004
[41] D Voet and J G Voet Biochemistry Wiley 2011[42] A C Guyton and J E Hall Textbook of Medical Physiology
Saunders 2010[43] M F M James and A M Roche ldquoDose-response relationship
between plasma ionized calcium concentration and thrombe-lastographyrdquo Journal of Cardiothoracic and Vascular Anesthesiavol 18 no 5 pp 581ndash586 2004
[44] D L Nelson and M M Cox Lehninger Principles of Biochem-istry W H Freeman Company 2004
[45] R M Farber ldquoTopical perspective on massive threading andparallelismrdquo Journal of Molecular Graphics and Modelling vol30 pp 82ndash89 2011
[46] OWolkenhauerMUllahWKolch andK-HCho ldquoModelingand simulation of intracellular dynamics choosing an appropri-ate frameworkrdquo IEEETransactions onNanobioscience vol 3 no3 pp 200ndash207 2004
[47] B H Estridge A P Reynolds and N J Walters Basic MedicalLaboratory Techniques Cengage Learning 4th edition 2000
[48] T Renne A H Schmaier K F Nickel M Blomback and CMaas ldquoIn vivo roles of factor XIIrdquo Blood vol 120 no 22 pp4296ndash4303 2012
[49] P V Jenkins O Rawley O P Smith and J S ODonnellldquoElevated factor VIII levels and risk of venous thrombosisrdquoBritish Journal of Haematology vol 157 no 6 pp 653ndash663 2012
[50] C E Walsh and K M Batt ldquoHemophilia clinical gene therapybrief reviewrdquo Translational Research vol 161 no 4 pp 307ndash3122013
[51] A Van Hylckama Vlieg I K Van Der Linden R M Bertinaand F R Rosendaal ldquoHigh levels of factor IX increase the risk ofvenous thrombosisrdquo Blood vol 95 no 12 pp 3678ndash3682 2000
[52] K Brummel-Ziedins ldquoModels for thrombin generation and riskof diseaserdquo Journal ofThrombosis and Haemostasis vol 11 no 1pp 212ndash223 2013
[53] U Seligsohn and A Lubetsky ldquoGenetic susceptibility to venousthrombosisrdquoTheNew England Journal of Medicine vol 344 no16 pp 1222ndash1231 2001
[54] I Martinelli P Bucciarelli and P M Mannucci ldquoThromboticrisk factors basic pathophysiologyrdquo Critical Care Medicine vol38 no 2 pp S3ndashS9 2010
[55] K Brummel-Ziedins A Undas T Orfeo et al ldquoThrombingeneration in acute coronary syndrome and stable coronaryartery disease dependence on plasma factor compositionrdquoJournal ofThrombosis andHaemostasis vol 6 no 1 pp 104ndash1102008
[56] S Hoops R Gauges C Lee et al ldquoCOPASImdasha complexpathway simulatorrdquo Bioinformatics vol 22 no 24 pp 3067ndash3074 2006
[57] Nvidia NVIDIArsquos Next Generation CUDA Compute Architec-ture Kepler GK110 2012
[58] W Stortelder W Hemker and H Hemker ldquoMathematicalmodelling in blood coagulation simulation and parameterestimationrdquoModelling Analysis and Simulation vol 20 pp 1ndash111997
[59] S M C Orfao G Jank and K Mottaghy ldquoA new mathematicalapproach to modelling thrombin generationrdquo InternationalJournal of Artificial Organs vol 29 no 7 pp 703ndash708 2006
[60] Nvidia Nvidia CUDA C Programming Guide v50 2012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Anatomy Research International
PeptidesInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
International Journal of
Volume 2014
Zoology
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Molecular Biology International
GenomicsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
BioinformaticsAdvances in
Marine BiologyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Signal TransductionJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
BioMed Research International
Evolutionary BiologyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Biochemistry Research International
ArchaeaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Genetics Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in
Virolog y
Hindawi Publishing Corporationhttpwwwhindawicom
Nucleic AcidsJournal of
Volume 2014
Stem CellsInternational
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Enzyme Research
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Microbiology
20 BioMed Research International
generation in healthy individualsrdquo Journal of Thrombosis andHaemostasis vol 3 no 7 pp 1472ndash1481 2005
[38] K E Brummel-Ziedins C Y Vossen S Butenas K G Mannand F R Rosendaal ldquoThrombin generation profiles in deepvenous thrombosisrdquo Journal of Thrombosis and Haemostasisvol 3 no 11 pp 2497ndash2505 2005
[39] S D Bungay P A Gentry and R D Gentry ldquoA mathematicalmodel of lipid-mediated thrombin generationrdquo MathematicalMedicine and Biology vol 20 no 1 pp 105ndash129 2003
[40] S Butenas T Orfeo M T Gissel K E Brummel and K GMann ldquoThe significance of circulating factor IXa in bloodrdquoThe Journal of Biological Chemistry vol 279 no 22 pp 22875ndash22882 2004
[41] D Voet and J G Voet Biochemistry Wiley 2011[42] A C Guyton and J E Hall Textbook of Medical Physiology
Saunders 2010[43] M F M James and A M Roche ldquoDose-response relationship
between plasma ionized calcium concentration and thrombe-lastographyrdquo Journal of Cardiothoracic and Vascular Anesthesiavol 18 no 5 pp 581ndash586 2004
[44] D L Nelson and M M Cox Lehninger Principles of Biochem-istry W H Freeman Company 2004
[45] R M Farber ldquoTopical perspective on massive threading andparallelismrdquo Journal of Molecular Graphics and Modelling vol30 pp 82ndash89 2011
[46] OWolkenhauerMUllahWKolch andK-HCho ldquoModelingand simulation of intracellular dynamics choosing an appropri-ate frameworkrdquo IEEETransactions onNanobioscience vol 3 no3 pp 200ndash207 2004
[47] B H Estridge A P Reynolds and N J Walters Basic MedicalLaboratory Techniques Cengage Learning 4th edition 2000
[48] T Renne A H Schmaier K F Nickel M Blomback and CMaas ldquoIn vivo roles of factor XIIrdquo Blood vol 120 no 22 pp4296ndash4303 2012
[49] P V Jenkins O Rawley O P Smith and J S ODonnellldquoElevated factor VIII levels and risk of venous thrombosisrdquoBritish Journal of Haematology vol 157 no 6 pp 653ndash663 2012
[50] C E Walsh and K M Batt ldquoHemophilia clinical gene therapybrief reviewrdquo Translational Research vol 161 no 4 pp 307ndash3122013
[51] A Van Hylckama Vlieg I K Van Der Linden R M Bertinaand F R Rosendaal ldquoHigh levels of factor IX increase the risk ofvenous thrombosisrdquo Blood vol 95 no 12 pp 3678ndash3682 2000
[52] K Brummel-Ziedins ldquoModels for thrombin generation and riskof diseaserdquo Journal ofThrombosis and Haemostasis vol 11 no 1pp 212ndash223 2013
[53] U Seligsohn and A Lubetsky ldquoGenetic susceptibility to venousthrombosisrdquoTheNew England Journal of Medicine vol 344 no16 pp 1222ndash1231 2001
[54] I Martinelli P Bucciarelli and P M Mannucci ldquoThromboticrisk factors basic pathophysiologyrdquo Critical Care Medicine vol38 no 2 pp S3ndashS9 2010
[55] K Brummel-Ziedins A Undas T Orfeo et al ldquoThrombingeneration in acute coronary syndrome and stable coronaryartery disease dependence on plasma factor compositionrdquoJournal ofThrombosis andHaemostasis vol 6 no 1 pp 104ndash1102008
[56] S Hoops R Gauges C Lee et al ldquoCOPASImdasha complexpathway simulatorrdquo Bioinformatics vol 22 no 24 pp 3067ndash3074 2006
[57] Nvidia NVIDIArsquos Next Generation CUDA Compute Architec-ture Kepler GK110 2012
[58] W Stortelder W Hemker and H Hemker ldquoMathematicalmodelling in blood coagulation simulation and parameterestimationrdquoModelling Analysis and Simulation vol 20 pp 1ndash111997
[59] S M C Orfao G Jank and K Mottaghy ldquoA new mathematicalapproach to modelling thrombin generationrdquo InternationalJournal of Artificial Organs vol 29 no 7 pp 703ndash708 2006
[60] Nvidia Nvidia CUDA C Programming Guide v50 2012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Anatomy Research International
PeptidesInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
International Journal of
Volume 2014
Zoology
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Molecular Biology International
GenomicsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
BioinformaticsAdvances in
Marine BiologyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Signal TransductionJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
BioMed Research International
Evolutionary BiologyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Biochemistry Research International
ArchaeaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Genetics Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in
Virolog y
Hindawi Publishing Corporationhttpwwwhindawicom
Nucleic AcidsJournal of
Volume 2014
Stem CellsInternational
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Enzyme Research
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Microbiology
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Anatomy Research International
PeptidesInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
International Journal of
Volume 2014
Zoology
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Molecular Biology International
GenomicsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
BioinformaticsAdvances in
Marine BiologyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Signal TransductionJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
BioMed Research International
Evolutionary BiologyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Biochemistry Research International
ArchaeaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Genetics Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in
Virolog y
Hindawi Publishing Corporationhttpwwwhindawicom
Nucleic AcidsJournal of
Volume 2014
Stem CellsInternational
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Enzyme Research
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Microbiology