Sagar et al. BMC Systems Biology (2018) 12:87 https://doi.org/10.1186/s12918-018-0610-x

METHODOLOGY ARTICLE Open Access

Dynamic Optimization with ParticleSwarms (DOPS): a meta-heuristic for parameterestimation in biochemical modelsAdithya Sagar1†, Rachel LeCover1†, Christine Shoemaker2 and Jeffrey Varner1*

Abstract

Background: Mathematical modeling is a powerful tool to analyze, and ultimately design biochemical networks.However, the estimation of the parameters that appear in biochemical models is a significant challenge. Parameterestimation typically involves expensive function evaluations and noisy data, making it difficult to quickly obtainoptimal solutions. Further, biochemical models often have many local extrema which further complicates parameterestimation. Toward these challenges, we developed Dynamic Optimization with Particle Swarms (DOPS), a novelhybrid meta-heuristic that combined multi-swarm particle swarm optimization with dynamically dimensioned search(DDS). DOPS uses a multi-swarm particle swarm optimization technique to generate candidate solution vectors, thebest of which is then greedily updated using dynamically dimensioned search.

Results: We tested DOPS using classic optimization test functions, biochemical benchmark problems and real-worldbiochemical models. We performed T = 25 trials withN = 4000 function evaluations per trial, and compared theperformance of DOPS with other commonly used meta-heuristics such as differential evolution (DE), simulatedannealing (SA) and dynamically dimensioned search (DDS). On average, DOPS outperformed other commonmeta-heuristics on the optimization test functions, benchmark problems and a real-world model of the humancoagulation cascade.

Conclusions: DOPS is a promising meta-heuristic approach for the estimation of biochemical model parameters inrelatively few function evaluations. DOPS source code is available for download under a MIT license at http://www.varnerlab.org.

Keywords: Biochemical engineering, Systems biology, Modeling, Optimization

BackgroundMathematical modeling has evolved as a powerfulparadigm to analyze, and ultimately design complex bio-chemical networks [1–5]. Mathematical modeling of bio-chemical networks is often an iterative process. First,models are formulated from existing biochemical knowl-edge, and then model parameters are estimated usingexperimental data [6–8]. Parameter estimation is typicallyframed as a non-linear optimization problem whereinthe residual (or objective function) between experimental

*Correspondence: jdv27@cornell.edu†Adithya Sagar and Rachel LeCover contributed equally to this work.1Robert Fredrick Smith School of Chemical and Biomolecular Engineering,Cornell University, 244 Olin Hall, Ithaca, NY, USAFull list of author information is available at the end of the article

measurements and model simulations is minimized usingan optimization strategy [9]. Optimal parameter estimatesare then used to predict unseen experimental data. If thevalidation studies fail, model construction and calibra-tion are repeated iteratively until satisfactory results areobtained. As our biological knowledge increases, modelformulation may not be as significant a challenge, butparameter estimation will likely remain difficult.Parameter estimation is a major challenge to the devel-

opment of biochemical models. Parameter estimation hasbeen a well studied engineering problem for decades[10–13]. However, the complex dynamics of large bio-logical systems and noisy, often incomplete experimen-tal data sets pose a unique estimation challenge. Oftenoptimization problems involving biological systems are

© The Author(s). 2018 Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, andreproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to theCreative Commons license, and indicate if changes were made. The Creative Commons Public Domain Dedication waiver(http://creativecommons.org/publicdomain/zero/1.0/) applies to the data made available in this article, unless otherwise stated.

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non-linear and multi-modal i.e., typical models havemultiple local minima or maxima [7, 9]. Non-linearitycoupled with multi-modality renders local optimizationtechniques such as pattern search [14], Nelder-Meadsimplex methods [15], steepest descent or Levenberg-Marquardt [16] incapable of reliably obtaining globallyoptimal solutions as these methods often terminate atlocal minimum. Though deterministic global optimiza-tion techniques (for example algorithms based on branchand bound) can handle non-linearity and multi-modality[17, 18], the absence of derivative information, discontin-uous objective functions, non-smooth regions or the lackof knowledge about the objective function hampers thesetechniques.Meta-heuristics like Genetic Algorithms (GAs) [19],

Simulated Annealing (SA) [20], Evolutionary Program-ming [21] and Differential Evolution (DE) [22–25] haveall shown promise on non-linear multi-modal prob-lems [26]. These techniques do not make any assump-tions nor do they require, a priori information aboutthe structure of the objective function. Meta-heuristicsare often very effective at finding globally optimal ornear optimal solutions. For example, Mendes et al.used SA to estimate rate constants for the inhibitionof HIV proteinase [27], while Modchang et al. used aGA to estimate parameters for a model of G-protein-coupled receptor (GPCR) activity [28]. Parameter esti-mates obtained using the GA stratified the effectiveness oftwo G-protein agonists, N6-cyclopentyladenosine (CPA)and 5’-N-ethylcarboxamidoadenosine (NECA). Tashkovaet al. compared different meta-heuristics for parameterestimation on a dynamic model of endocytosis; DE wasthemost effective of the approaches tested [29]. Banga andco-workers have also successfully applied scatter-search toestimate model parameters [30–32]. Hybrid approaches,which combine meta-heuristics with local optimizationtechniques, have also become popular. For example, Egeaet al. developed the enhanced scatter search (eSS) method[32], which combined scatter and local search methods,for parameter estimation in biological models [33]. How-ever, despite these successes, a major drawback of mostmeta-heuristics remains the large number of functionevaluations required to explore parameter space. Perform-ing numerous potentially expensive function evaluationsis not desirable (and perhaps not feasible) for many typesof biochemical models. Alternatively, Tolson and Shoe-maker found, using high-dimensional watershed mod-els, that perturbing only a subset of parameters was aneffective strategy for estimating parameters in expensivemodels [34]. Their approach, called Dynamically Dimen-sioned Search (DDS), is a simple stochastic single-solutionheuristic that estimates nearly optimal solutions within aspecified maximum number of function (or model) eval-uations. Thus, while meta-heuristics are often effective at

estimating globally optimal or nearly optimal solutions,they require a large number of function evaluations toconverge to a solution.In this study, we developed Dynamic Optimization with

Particle Swarms (DOPS), a novel hybrid meta-heuristicthat combines the global search capability of multi-swarmparticle swarm optimization with the greedy refinementof dynamically dimensioned search (DDS). The objectiveof DOPS is to obtain near optimal parameter estimatesfor large biochemical models within a relatively few func-tion evaluations. DOPS uses multi-swarm particle swarmoptimization to generate nearly optimal candidate solu-tions, which are then greedily updated using dynamicallydimensioned search. While particle swarm techniques areeffective, they have the tendency to become stuck in smalllocal regions and lose swarm diversity, so we combinedmulti-swarm particle optimization with DDS to escapethese local regions and continue towards better solutions[35]. We tested DOPS using a combination of classic opti-mization test functions, biochemical benchmark prob-lems and real-world biochemical models. First, we testedthe performance of DOPS on the Ackley and Rosenbrockfunctions, and published biochemical benchmark prob-lems. Next, we used DOPS to estimate the parametersof a model of the human coagulation cascade. On aver-age, DOPS outperformed other common meta-heuristicslike differential evolution, a genetic algorithm, CMA-ES (Covariance Matrix Adaptation Evolution Strategy),simulated annealing, single-swarm particle swarm opti-mization, and dynamically dimensioned search on theoptimization test functions, benchmark problems and thecoagulation model. For example, DOPS recovered thenominal parameters for the benchmark problems using anorder of magnitude fewer function evaluations than eSSin all cases. It also produced parameter estimates for thecoagulation model that predicted unseen coagulation datasets. Thus, DOPS is a promising hybrid meta-heuristicfor the estimation of biochemical model parameters inrelatively few function evaluations. However, the relativeperformance of DOPS should be evaluated cautiously;only naive implementations of the other approaches weretested. Thus, it is possible that other optimized meta-heuristics could outperform DOPS on both test and real-world problems.

ResultsDOPS explores parameter space using a combination ofglobal methods.DOPS combines a multi-swarm particle swarm methodwith the dynamically dimensioned search approach ofShoemaker and colleagues (Fig. 1). The goal of DOPS isto estimate optimal or near optimal parameter vectorsfor high-dimensional biological models within a specifiednumber of function evaluations. Toward this objective,

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Fig. 1 Schematic of the dynamic optimization with particle swarms (DOPS) approach. Top: Each particle represents an N dimensional parametervector. Particles are given randomly generated initial solutions and grouped into different sub-swarms. Within each swarm the magnitude anddirection of the movement a particle is influenced by the position of the best particle and also by its own experience. After every g number offunction evaluations the particles are mixed and randomly assigned to different swarms. When the error due to the global best particle (best particleamongst all the sub-swarms) does not drop over a certain number of function evaluations, the swarm search is stopped and the search switches toa Dynamically Dimensioned Search with global best particle as the initial solution vector or candidate vector. Bottom: The candidate vectorperforms a greedy global search for the remaining number of function evaluations. The search neighborhood is dynamically adjusted by varying thenumber of dimensions that are perturbed (in black) in each evaluation step. The probability that a dimension is perturbed decreases as the numberof function evaluations increase

DOPS begins by using a multi-swarm particle swarmsearch and then dynamically switches, using an adap-tive switching criteria, to the DDS approach. The parti-cle swarm search uses multiple sub-swarms wherein theupdate to each particle (corresponding to a parameter vec-tor estimate) is influenced by the best particle amongstthe sub-swarm, and the current globally best particle.Particle updates occur within sub-swarms for a certainnumber of function evaluations, after which the sub-swarms are reorganized. This sub-swarmmixing is similar

to the regrouping strategy described by Zhao et al. [36].DOPS switches out of the particle swarm phase basedupon an adaptive switching criteria that is a function ofthe rate of error convergence. If the error represented bythe best particle does not decrease for a threshold num-ber of function evaluations, DOPS switches automaticallyto the DDS search phase. The DDS search is initializedwith the globally best particle from the particle swarmphase, thereafter, the particle is greedily updated by per-turbing a subset of dimensions for the remaining number

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of function evaluations. The identity of the parametersperturbed is chosen randomly, with fewer parametersperturbed the higher the number of function evaluations.

DOPSminimized benchmark problems using fewerfunction evaluations.On average, DOPS performed similarly or outperformedfour other meta-heuristics for the Ackley and Rastrigintest functions (Fig. 2). The Ackley and Rastrigin functionsboth have multiple local extrema and attain a global mini-mum value of zero. In each case, the maximum number offunction evaluations was fixed at N = 4000, and T = 25independent experiments were run with different initialparameter vectors. DOPS found optimal or near optimalsolutions for both the 10-dimensional Ackley (Fig. 2a) andRastrigin (Fig. 2b) functions within the budget of func-tion evaluations. In each of the 10-dimensional cases,other meta-heurtistics such as DDS and DE also per-formed well. However, DOPS consistently outperformedall other approaches tested. This performance differencewas more pronounced as the dimension of the searchproblem increased; for a 300-dimensional Rastrigin func-tion, DOPS was the only approach to find an optimal ornear optimal solution within the function evaluation bud-get (Fig. 2b). Taken together, DOPS performed at leastas well as other meta-heuristics on small dimensionaltest problems, but was especially suited to large dimen-sional search spaces. Next, we tested DOPS on benchmarkbiochemical models of varying complexity.Villaverde and co-workers published a set of benchmark

biochemical problems to evaluate parameter estimationmethods [33]. They ranked the example problems by com-putational cost from most to least expensive. We evalu-ated the performance of DOPS on problems from the leastand most expensive categories. The least expensive prob-lem was a metabolic model of Chinese Hamster Ovary(CHO) with 35 metabolites, 32 reactions and 117 param-eters [37]. The biochemical reactions were modeled usingmodular rate laws and generalized Michaelis–Mentenkinetics. On the other hand, the expensive problem wasa genome scale kinetic model of Saccharomyces cerevisiaewith 261 reactions, 262 variables and 1759 parameters[38]. In both cases, synthetic time series data gener-ated with known parameter values, was used as trainingdata to estimate the model parameters. For the Saccha-romyces cerevisiaemodel, the time series data consisted of44 observables, while for the CHO metabolism problemthe data corresponded to 13 different metabolite mea-surement sets. The number of function evaluations wasfixed at N = 4000, and we trained both models againstthe synthetic experimental data. DOPS produced goodfits to the synthetic data (Additional file 1: Figure S1 andAdditional file 2: Figure S2), and recapitulated the nom-inal parameter values using only N ≤ 4000 function

Fig. 2 Performance of DOPS and other meta-heuristics for the Ackleyand Rastrigin functions. a: Mean scaled error versus the number offunction evaluations for the 10-dimensional Ackley function. DOPS,DDS and ESS find optimal or near optimal solutions within thespecified number of function evaluations. b: Mean scaled error versusthe number of function evaluations for the 10-dimensional Rastriginfunction. Nearly all the techniques find optimal or near optimalsolutions within the specified number of function evaluations. c:Mean scaled error versus the number of function evaluations for the300-dimensional Rastrigin function. DOPS is the only algorithm thatfinds an optimal or near optimal solution within the specified numberof function evaluations. In all cases, the maximum number of functionevaluations wasN = 4000. Mean and standard deviation werecalculated over T = 25 trials. A star denotes that the average valuewas less than 1E-6

evaluations (Additional file 3: Figure S3). On the otherhand, the enhanced scatter search (eSS) with a localoptimizer method, took on order 105 function evalua-tions for the same problems. DOPS required a comprableamount of time (Additional file 4: Figure S4), faster con-vergence (Additional file 5: Figure S5 and Additional file 6:Figure S6), and also had lower variability in the best

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value obtained (Additional file 7: Figure S7) across multi-ple runs when compared to other meta-heuristics. Thus,DOPS estimated the parameters in benchmark biochem-ical models, and recovered the original parameters fromthe synthetic data, using fewer function evaluations. Next,we compared the performance of DOPS with four othermeta-heuristics for a model of the human coagulationcascade.

DOPS estimated the parameters of a human coagulationmodel.Coagulation is an archetype biochemical network that ishighly interconnected, containing both negative and pos-itive feedback (Fig. 3). The biochemistry of coagulation,

though complex, has been well studied [39–45], and reli-able experimental protocols have been developed to inter-rogate the system [46–49]. Coagulation is mediated bya family proteases in the circulation, called factors anda key group of blood cells, called platelets. The centralprocess in coagulation is the conversion of prothrom-bin (fII), an inactive coagulation factor, to the masterprotease thrombin (FIIa). Thrombin generation involvesthree phases, initiation, amplification and termination.Initiation requires a trigger event, for example a ves-sel injury which exposes tissue factor (TF), which leadsto the activation of factor VII (FVIIa) and the forma-tion of the TF/FVIIa complex. Two converging pathways,the extrinsic and intrinsic cascades, then process and

Fig. 3 Schematic of the extrinsic and intrinsic coagulation cascade. Inactive zymogens upstream (grey) are activated by exposure to tissue factor(TF) following vessel injury. Tissue factor and activated factor VIIa (FVIIa) form a complex that activates factor X (fX) and IX (fIX). FXa activatesdownstream factors including factor VIII (fVIII) and fIX. Factor V (fV) is primarily activated by thrombin (FIIa). In addition, we included a secondary fVactivation route involving FXa. FXa and FVa form a complex (prothrombinase) on activated platelets that converts prothrombin (fII) to FIIa. FIXa andFVIIIa can also form a complex (tenase) on activated platelets which catalyzes FXa formation. Thrombin also activates upstream coagulation factors,forming a strong positive feedback ensuring rapid activation. Tissue factor pathway inhibitor (TFPI) downregulates FXa formation and activity bysequestering free FXa and TF-FVIIa in a FXa-dependent manner. Antithrombin III (ATIII) inhibits all proteases. Thrombin inhibits itself binding thesurface protein thrombomodulin (TM). The IIa-TM complex catalyzes the conversion of protein C (PC) to activated protein C (APC), which attenuatesthe coagulation response by the proteolytic cleavage of fV/FVa and fVIII/FVIIIa

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amplify this initial coagulation signal. There are severalcontrol points in the cascade that inhibit thrombin for-mation, and eventually terminate thrombin generation.Tissue Factor Pathway Inhibitor (TFPI) inhibits upstreamactivation events, while antithrombin III (ATIII) neutral-izes several of the proteases generated during coagulation,including thrombin. Thrombin itself also inadvertentlyplays a role in its own inhibition; thrombin, through inter-action with thrombomodulin, protein C and endothelialcell protein C receptor (EPCR), converts protein C toactivated protein C (APC) which attenuates the coagu-lation response by proteolytic cleavage of amplificationcomplexes. Termination occurs after either prothrombinis consumed, or thrombin formation is neutralized byinhibitors such as APC or ATIII. Thus, the human coag-ulation cascade is an ideal test case; coagulation is chal-lenging because it contains both fast and slow dynamics,but also accessible because of the availability of com-prehensive data sets for model identification and vali-dation. In this study, we used the coagulation model ofLuan et al. [49], which is a coupled system of non-linearordinary differential equations where biochemical inter-actions were modeled using mass action kinetics. TheLuan model contained 148 parameters and 92 speciesand has been validated using 21 published experimentaldatasets.DOPS estimated the parameters of a human coagu-

lation model for TF/VIIa initiated coagulation without

anticoagulants (Fig. 4). The objective function was anunweighted linear combination of two error functions,representing coagulation initiated with different concen-trations of TF/FVIIa (5pM, 5nM) [46]. The number offunction evaluations was restricted toN = 4000 for eachalgorithm we tested, and we performed T = 25 trialsof each experiment to collect average performance data(Table 1). DOPS converged faster and had a lower finalerror compared to the other algorithms (Fig. 5). Withinthe first 25% of function evaluations, DOPS produced arapid drop in error followed by a slower but steady decline(Additional file 8: Figure S8b). Approximately between500-1000 function evaluations DOPS switched to thedynamically dimensioned search phase, however thistransition varied from trial to trial since the switch wasbased upon the local convergence rate. On average, DOPSminimized the coagulation model error to a greater extentthan the other meta-heuristics. However, it was unclear ifthe parameters estimated by DOPS had predictive poweron unseen data. To address this question, we used the finalparameters estimated by DOPS to simulate data that wasnot used for training (coagulation initiated with 500pM,50pM, and 10pM TF/VIIa). The optimal or near optimalparameters obtained by DOPS predicted unseen coagula-tion datasets (Fig. 6). The normalized standard error forthe coagulation predictions was consistent with the train-ing error, with the exception of the 50pM TF/VIIa casewhich was a factor 2.65 worse (Table 2). However, this

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Fig. 4Model fits on experimental data using DOPS. The model parameters were estimated using DOPS. Solid black lines indicate the simulatedmean thrombin concentration using parameter vectors from 25 trials. The grey shaded region represents the 99% confidence estimate of the meansimulated thrombin concentration. The experimental data is reproduced from the synthetic plasma assays of Mann and co-workers. Thrombingeneration is initiated by adding Factor TF/VIIa (5nM (blue) and 5pM (red)) to synthetic plasma containing 200 μmol/L of phospholipid vesicles(PCPS) and a mixture of coagulation factors (II,V,VII,VIII,IX,X and XI) at their mean plasma concentrations

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Table 1 Table with optimization settings and results for thecoagulation problem, the benchmarks and test functions usingDOPS

Coagulation B1 B4 Ackley Rastrigin

Evaluations 4000 4000 4000 4000 4000

Lower Bound 0.001.pnom 0.2.pnom 0.2.pnom -15 -5.12

Upper Bound 1000.pnom 5.pnom 5.pnom 30 5.12

CPU Time 10.1 hrs 38.3 hrs 6.2 min 2.8 s 2.6 s

Scaled initial error 1.0 1.0 1.0 1.0 1.0

Scaled final error < 0.01 < 0.01 < 0.01 < 0.01 < 0.01

Scaled nominal error 0.42 0.1 < 0.01 0 0

For each problem the bounds on the parameter vector, the total number offunction evaluations, the best initial objective value and the best final objectivevalue are specified. Here pnom indicates the nominal or true parameter vector ofthe model. Nominal objective value represents the objective value using the trueparameter vector or the nominal parameter vector. The CPU time is the time takenfor the problem on a 2.4GHz Intel Xeon Architecture running Matlab 2014b

might be expected as coagulation initiation with 50pMTF/FVIIa was the farthest away from the training con-ditions. Taken together, DOPS estimated parameter setswith predictive power on unseen coagulation data usingfewer function iterations than other meta-heuristics.Next, we explored how the number of sub-swarms and

the switch to DDS influenced the performance of theapproach.

Phase switching was critical to DOPS performance.A differentiating feature of DOPS is the switch todynamically dimensioned search following stagnationof the initial particle swarm phase. We quantifiedthe influence of the number of sub-swarms and theswitch to DDS on error convergence by comparingDOPS with and without DDS for different numbers ofsub-swarms (Fig. 7). We considered multi swarm particleswarm optimization with and without the DDS phase forN = 4000 function evaluations and T = 25 trials onthe coagulation model. We used one, two, four, five andeight sub-swarms, with a total of 40 particles dividedevenly amongst the swarms. Hence, we did not considerswarm numbers of three and seven. All other algorithmparameters remained the same for all cases. Generally, thehigher sub-swarm numbers converged in fewer functionevaluations, where the optimum particle partitioning wasin the neighborhood of five sub-swarms. However, thedifference in convergence rate was qualitatively sim-ilar for four, five and eight sub-swarms, suggestingthere was an optimal number of particles per swarmbeyond which there was no significant advantage. The

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Fig. 5 Error convergence rates of the nine different algorithms on the coagulation model. The objective error is the mean over T = 25 trials. DOPS,SA, PSO and DOPS-PSO have the steepest drop in error during first 300 function evaluations. Thereafter the error drop in DDS and SA remains nearlyconstant whereas DOPS continues to drops further. In the alloted budget of function evaluations ESS produces a modest reduction in error. At theend of 4000 function evaluations DOPS attains the lowest error

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Fig. 6Model predictions on unseen experimental data using parameters obtained from DOPS. The parameter estimates that were obtained usingDOPS were tested against data that was not used in the model training. Solid black lines indicate the simulated mean thrombin concentration usingparameter vectors from T = 25 trials. The grey shaded region represents the 99% confidence estimate of the mean simulated thrombinconcentration. The experimental data is reproduced from the synthetic plasma assays of Mann and co-workers. Thrombin generation is initiated byadding Factor VIIa-TF (500pM - blue, 50pM - pink and 10pM - purple, respectively) to synthetic plasma containing 200 μmol/L of phospholipidvesicles (PCPS) and a mixture of coagulation factors (II,V,VII,VIII,IX,X and XI) at their mean plasma concentrations

multi-swarm particle swarm optimization stagnated after25% of the available function evaluations irrespectiveof the number of sub-swarms. However, DOPS (withfive sub-swarms) switched to DDS after detecting thestagnation. The DDS phase refined the globally bestparticle to produce significantly lower error on aver-age when compared to multi-swarm particle swarmoptimization alone. Thus, the automated switching strat-egy was critical to the overall performance of DOPS.However, it was unclear if multiple strategy switches couldfurther improve performance.

Table 2 Error analysis for the human coagulation model

TF/FVIIa concentration Normalized S.E. Category

5 nM 0.1336 Training

500 pM 0.2242 Prediction

50 pM 0.3109 Prediction

10 pM 0.2023 Prediction

5 pM 0.1170 Training

The coagulation model was trained on coagulation initiated with TF/FVIIa at 5 nMand the 5 pM to obtain the optimal parameters. Using these optimal parameters,coagulation dynamics were predicted for varying initiator concentrations (500 pM,50 pM and 10 pM). Model agreement with measurements was quantified usingnormalized squared error. The normalized squared error is defined asN.S.E. = (1/max(X)) ∗ (‖(Y,X)‖/sqrt(N)) where X is the experimental data, Y is themodel simulation data interpolated onto the experimental time scale and N is thetotal number of experimental time points

We explored the performance of DOPS if it waspermitted to switch between the PSO (Particle SwarmOptimization) and DDSmodes multiple times. This mode(msDOPS) had comparable performance to DOPS on10-d Ackley and Rastrigin functions, as well as on the300-dimensional Rastrigin function. However, msDOPSperformed better than DOPS on the CHO metabolismproblem (Fig. 8a), with the average functional value beingnearly half that of DOPS. To further distinguish DOPSfrom msDOPS, we compared the performance of eachalgorithm on the Eggholder function, a difficult functionto optimize given its multiple minima [50]. msDOPS out-performed DOPS on the Eggholder function, however,neither version reached the true minimum at -959.6407on any trial with a budget of N = 4000 function eval-uations (Fig. 8b). We also explored the performance ofmsDOPS and DOPS on the 100 dimensional Styblinksi-Tang function [51] (Fig. 8c). In this comparison, msDOPSsignificantly outperformed DOPS, finding the true min-imum before exhausting its function evaluation budget,while DOPS does not reach the minimum. Since the per-formance of msDOPS was promising on these problems,wemeasured its performance on the coagulation problem.Surprisingly, DOPS performed similarly to msDOPS onthe coagulation problem (Fig. 8d); the final average objec-tive value for DOPS reached 0.9413% of the initial func-tional value, compared to 0.9428% for msDOPS. Taken

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0 1000 2000 3000 4000Number of Function Evalutions

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Fig. 7 Influence of the switching strategy and sub-swarms on DOPSperformance for the coagulation model. DOPS begins by using aparticle swarm search and then dynamically switches (switch region),using an adaptive switching criteria, to the DDS search phase. Wecompared the performance of DOPS with and without DDS fordifferent sub-swarm searches to quantify the effect of number ofsub-swarms and DDS. We used one, two, four, five and eightsub-swarms, with a total of 40 particles divided evenly amongst theswarms. The results presented are the average of T = 25 trials withN = 4000 function evaluations each. The convergence rates withhigher swarm numbers is typically higher but there is no pronounceddifference amongst four, five and eight. The multi-swarm withwithout DDS saturates while DOPS shows a rapid drop due to aswitch to the DDS phase

together, these results indicate that switching plays a keyrole in DOPS’s performance and that for some classes ofproblems, multiple switching between modes produces afaster drop in objective value. However, the coagulationmodel results suggested the advantage of msDOPS wasproblem specific.

DiscussionIn this study, we developed dynamic optimization withparticle swarms (DOPS), a novel meta-heuristic forparameter estimation. DOPS combined multi-swarm par-ticle swarm optimization, a global search approach, withthe greedy strategy of dynamically dimensioned search toestimate optimal or nearly optimal solutions in a fixednumber of function evaluations. We tested the perfor-mance of DOPS and seven widely used meta-heuristicson the Ackley and Rastrigin test functions, a set ofbiochemical benchmark problems and a model of thehuman coagulation cascade. We also compared the per-formance of DOPS to enhanced Scatter Search (eSS),another widely used meta-heuristic approach. As the

number of parameters increased, DOPS outperformedthe other meta-heuristics, generating optimal or nearlyoptimal solutions using significantly fewer function eval-uations compared with the other methods. We tested thesolutions generated by DOPS by comparing the estimatedand true parameters in the benchmark studies, and byusing the coagulationmodel to predict unseen experimen-tal data. For both benchmark problems, DOPS retrievedthe true parameters in significantly fewer function eval-uations than other meta-heuristics. For the coagulationmodel, we used experimental coagulation measurementsunder two different conditions to estimate optimal ornearly optimal parameters. These parameters were thenused to predict unseen coagulation data; the coagula-tion model parameters estimated by DOPS predicted thecorrect thrombin dynamics following TF/FVIIa inducedcoagulation without anticoagulants. Lastly, we showed theaverage performance of DOPS improved when combinedwith dynamically dimensioned search phase, compared toan identical multi-swarm approach alone, and that multi-ple mode switches could improve performance for someclasses of problems. Taken together, DOPS is a promisingmeta-heuristic for the estimation of parameters in largebiochemical models.Meta-heuristics can be effective tools to estimate opti-

mal or nearly optimal solutions for complex, multi-modalfunctions. However, meta-heuristics typically require alarge number of function evaluations to converge toa solution compared with techniques that use deriva-tive information. DOPS is a combination of particleswarm optimization, which is a global search method,and dynamically dimensioned search, which is a greedyevolutionary technique. Particle swarm optimization usescollective information shared amongst swarms of com-putational particles to search for global extrema. Severalparticle swarm variants have been proposed to improvethe search ability and rate of convergence. These vari-ations involve different neighborhood structures, multi-swarms or adaptive parameters. Multi-swarm particleswarm optimization with small particle neighborhoodshas been shown to be better in searching on complexmulti-modal solutions [36]. Multi-swarm methods gen-erate diverse solutions, and avoid rapid convergence tolocal optima. However, at least for the coagulation prob-lem used in this study, multi-swarm methods stagnatedafter approximately 25% of the available function evalua-tions; only the introduction of dynamically dimensionedsearch improved the rate of error convergence. Dynam-ically dimensioned search, which greedily perturbs onlya subset of parameter dimensions in high dimensionalparameter spaces, refined the globally best particle andproduced significantly lower error on average when com-pared to multi-swarm particle swarm optimization alone.However, dynamically dimensioned search, starting from

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Fig. 8 Comparison of DOPS and Multiswitch DOPS Performance of DOPS and Multiswitch DOPS on the CHO metabolism problem (a), theEggholder function (b), the 100 dimensional Styblinksi-Tang function (c) and the coagulation problem (d). Both methods have the same initialdecrease in error, but as the number of function evaluations increases, multiswitch DOPS produces a larger decrease in error. The results presentedare the average of T = 250 trials with for the CHO metabolism problem and T = 250 trials on the Eggholder and Styblinksi-Tang functions withN = 4000 function evaluations each, and T = 25 trials for the coagulation problem

a initial random parameter guess, was not as effective onaverage as DOPS. The initial solutions generated by themulti swarm search had a higher propensity to producegood parameter estimates when refined by dynamicallydimensioned search. Thus, our hybrid combination oftwo meta-heuristics produced better results than eitherconstituent approach, and better results than other meta-heuristic approaches on average. This was true of notonly the convergence rate on the coagulation problem,but also the biochemical benchmark problems; DOPSrequired two-orders of magnitude fewer function evalu-ations compared with enhanced Scatter Search (eSS) toestimate the biochemical benchmark model parameters.What remains to be explored is the performance of DOPScompared to techniques that utilize derivative informa-tion, either on their own or in combination with othermeta-heuristics, and the performance of DOPS in real-world applications compared with other meta-heuristics

such as hybrid genetic algorithms e.g., see [52]. Gradientmethods perform well on smooth convex problems whichhave either a closed form of the gradient of the functionbeing minimized, or a form that can be inexpensively esti-mated numerically. While the biological problems DOPSis intended for often do not have this form, perhapsthe solutions could be further improved by following(or potentially replacing) the DDS phase with a gradientbased technique when applicable. Taken together, thecombination of particle swarm optimization and dynami-cally dimensioned search performed better than either ofthese constituent approaches alone, and required fewerfunction evaluations compared with other commonmeta-heuristics.

ConclusionsDOPS performed well on many different systems with nopre-optimization of algorithm parameters, however there

Sagar et al. BMC Systems Biology (2018) 12:87 Page 11 of 15

are many research questions that should be pursued fur-ther. DOPS comfortably outperformed existing, widelyused meta-heuristics for high dimensional global opti-mization functions, biochemical benchmark models anda model of the human coagulation system. However, it ispossible that highly optimized versions of common meta-heuristics could surpass DOPS; we should compare theperformance of DOPS with optimized versions of theother common meta-heuristics on both test and real-world problems to determine if a performance advantageexists in practice. Next, DOPS has a hybrid architec-ture, thus the particle swarm phase could be combinedwith other search strategies such as local derivative basedapproaches to improve convergence rates. We could alsoconsider multiple phases beyond particle swarm anddynamically dimensioned search, for example switchingto a gradient based search following the dynamicallydimensioned search phase. Lastly, we should updateDOPS to treat multi-objective problems. The identifi-cation of large biochemical models sometimes requirestraining using qualitative, conflicting or even contradic-tory data sets. One strategy to address this challengeis to estimate experimentally constrained model ensem-bles using multiobjective optimization. Previously, wedeveloped Pareto Optimal Ensemble Techniques (POETs)which integrates simulated annealing with Pareto opti-mality to identify models near the optimal tradeoff surfacebetween competing training objectives [53]. Since DOPSconsistently outperformed simulated annealing on bothtest and real-world problems, we expect a multi-objectiveform of DOPS would more quickly estimate solutionswhich lie along high dimensional trade-off surfaces.

MethodsOptimization problem formulation.Model parameters were estimated by minimizing the dif-ference between model simulations and E experimentalmeasurements. Simulation error is quantified by an objec-tive function K (p) (typically the Euclidean norm of thedifference between simulations and measurements) sub-ject to problem and parameter constraints:

minp

K(p) =E∑

i=1

(gi(ti, x,p,u) − yi

)2

subject to x = f(t, x(t,p),u(t),p)

x(t0) = x0c(t, x,p,u) � 0pL � p � pU

(1)

The term K(p) denotes the objective function (sum ofsquared error), t denotes time, gi(ti, x,p,u) is the modeloutput for experiment i, while yi denotes the measuredvalue for experiment i. The quantity x (t,p) denotes the

state variable vector with an initial state x0, u(t) is amodel input vector, f(t, x(t,p),u(t),p) is the system ofmodel equations (e.g., differential equations or algebraicconstraints) and p denotes the model parameter vec-tor (quantity to be estimated). The parameter search (ormodel simulations) can be subject to c(t, x,p,u) linear ornon-linear constraints, and parameter bound constraintswhere pL and pU denote the lower and upper parameterbounds, respectively. Optimal model parameters are thengiven by:

p∗ = argminp

K (p) (2)

In this study, we considered only parameter boundconstraints, and did not include the c(t, x,p,u) linearor non-linear problem constraints. However, additionalthese constraints can be handled, without changing theapproach, using a penalty function method.

Dynamic optimization with particle swarms (DOPS).DOPS combines multi-swarm particle swarm optimiza-tion with dynamically dimensioned search (Fig. 1) and(Algorithm 1). The goal of DOPS is to estimate optimalor near optimal parameter vectors for high-dimensionalbiological models within a specified number of functionevaluations. Toward this objective, DOPS begins by usinga particle swarm search and then dynamically switches,using an adaptive switching criteria, to a DDS searchphase.

Phase 1: Particle swarm phase.Particle swarm optimization is an evolutionary algorithmthat uses a population of particles (solutions) to findan optimal solution [54, 55]. Each particle is updatedbased on its experience (particle best) and the experi-ence of all other particles within the swarm (global best).The particle swarm phase of DOPS begins by randomlyinitializing a swarm of K-dimensional particles (repre-sented as zi), wherein each particle corresponded to aK-dimensional parameter vector. After initialization, par-ticles were randomly partitioned into k equally sized sub-swarms S1, . . . ,Sk . Particles within each sub-swarm Skwere updated according to the rule:

zi,j = θ1,j−1zi,j−1+θ2r1(Li − zi,j−1

)+θ3r2(Gk − zi,j−1

)

(3)

where (θ1, θ2, θ3) were adjustable parameters, Li denotesthe best solution found by particle i within sub-swarm Skfor function evaluation 1 → j − 1, and Gk denotes thebest solution found over all particles within sub-swarmSk . The quantities r1 and r2 denote uniform random vec-tors with the same dimension as the number of unknownmodel parameters (K × 1). Equation 3 is similar to thegeneral particle swarm update rule, however, it does not

Sagar et al. BMC Systems Biology (2018) 12:87 Page 12 of 15

Algorithm 1: Pseudo code for the dynamic optimiza-tion with particle swarms (DOPS) methodinput : A randomized swarm of particles of size

NP × K and fixed number of functionevaluations N

output: Optimized parameter vector of size 1 × K1 Initialize the particles randomly and assign particlesrandomly to various sub-swarms;

2 while j ≤ N do3 ifmod(j,G)=0 then4 Reassign particles to different sub-swarms;5 end6 for i ← 1 to NS do7 Update particles within sub-swarms

according to Eq. 3;8 end9 Find best particle G amongst all sub-swarms;

10 if besterror(j) ≥ 0.99 ∗ besterror(j + 1) then11 failurecounter ← failurecounter + 1;12 else13 failurecounter ← 0;14 end15 if failurecounter ≥ threshold then16 G ← DDS(G,N − j);17 return G18 else19 j ← j + 1;20 end21 return G22 end

contain velocity terms. In DOPS, the parameter θ1,j−1 issimilar to the inertia weight parameter for the velocityterm described by Shi and Eberhart [56]; Shi and Eberhartproposed a linearly decreasing inertia weight to improveconvergence properties of particle swarm optimization.Our implementation of θ1,j−1 is inspired by this and thedecreasing perturbation probability proposed by Tolsonand Shoemaker [34]. It is an analogous equivalent to iner-tia weight on velocity. However θ1,j−1 places inertia onthe position rather than velocity and uses the same ruledescribed by Shi and Eberhart to adaptively change withthe number of function evaluations:

θ1,j = (N − j) ∗ (wmax − wmin)

(N − 1)+ wmin (4)

where N represents the total number of function eval-uations, wmax and wmin are the maximum and mini-mum inertia weights, respectively. In this study, we usedwmax = 0.9 and wmin = 0.4, however, these values are userconfigurable and could be changed depending upon theproblem being explored. Similarly, θ2 and θ3 were treated

as constants, where θ2 = θ3 = 1.5; the values of θ2and θ3 control how heavily the particle swarm weighs theprevious solutions it has found when generating a newcandidate solution. If θ2 � θ3 the new parameter solu-tion will resemble the best local solution found by particlei (Li), while θ3 � θ2 suggests the new parameter solutionwill resemble the best global solution found so far. Whileupdating the particles, parameter bounds were enforcedusing reflection boundary conditions (Algorithm 2).

Algorithm 2: Pseudo code for the reflective bound-ary conditions used by the dynamic optimization withparticle swarms (DOPS) method1 if zoldi,j < zmin

i then2 znewi,j = zoldi,j + (zmin

i − zoldi,j ) if znewi,j > zmaxi then

3 znewi,j = zmaxi

4 end5 end6 if zoldi,j > zmax

i then7 znewi,j = zoldi,j + (zoldi,j − zmax

i ) if znewi,j < zmini then

8 znewi,j = zmini

9 end10 end

After everyM function evaluations, particles were ran-domly redistributed to a new sub-swarm, and updatedaccording to Eq. (3). This process continued for a maxi-mum of FN functions evaluations, where F denotes thefraction of function evaluations used during the particleswarm phase of DOPS:

F =(NPN

)j (5)

The quantity NP denotes the total number of particles inthe swarm, N denotes the total possible number of func-tion evaluations, while the counter j denotes the numberof successful particle swarm iterations (each costing NPfunction evaluations). If the simulation error stagnatede.g., did not change by more than 1% for a specified num-ber of evaluations (default value of 4), the swarm phasewas terminated and DOPS switched to exploring param-eter space using the DDS approach using the remaining(1 − F)N function evaluations.

Phase 2: DDS phase.Dynamically Dimensioned Search (DDS) is a single solu-tion based search algorithm. DDS is used to obtain goodsolutions to high-dimensional search problems within afixed number of function evaluations. DDS starts as aglobal search algorithm by perturbing all the dimensions.

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Algorithm 3: Pseudo code for the DynamicallyDimensioned Search (DDS) methodinput : Candidate vector G from swarm search and

(1 − F)N evaluationsoutput: Optimized parameter vector of size 1 × K

1 while j ≤ (1 − F)N do2 Assign probability of perturbation to each

dimension Pi according to Eq. 7;3 Select a subset of dimensions based on a threshold

value for perturbation;4 Update candidate solution G(J) according to Eq. 5;5 Ensure updated solution Gnew(J) is within bounds

using Algorithm 2;6 end

Later the number of dimensions that are perturbed isdecreased with a certain probability. The probability thata certain dimension is perturbed reduces (a minimumof one dimension is always perturbed) as the iterationsincrease. This causes the algorithm to behave as a localsearch algorithm as the number of iterations increase.The perturbation magnitude of each dimension is fromnormal distribution with zero mean. The standard devi-ation that was used in the original DDS paper and thecurrent study is 0.2. DDS performs a greedy searchwhere the solution is updated only if it is better thanthe previous solution. The combination of perturbinga subset of dimensions along with greedy search indi-rectly relies on model sensitivity to a specific param-eter combination. The reader is requested to refer tothe original paper by Tolson and Shoemaker for furtherdetail [34].At the conclusion of the swarm phase, the overall best

particle, Gk , over the k sub-swarms was used to initializethe DDS phase. DOPS takes at least (1 − F)N functionevaluations during the DDS phase and then terminates thesearch. For the DDS phase, the best parameter estimatewas updated using the rule:

Gnew(J) ={G(J) + rnormal(J)σ (J), if Gnew(J) < G(J).G(J), otherwise.

(6)

where J is a vector representing the subset of dimensionsthat are being perturbed, rnormal denotes a normal randomvector of the same dimensions as G, and σ denotes theperturbation amplitude:

σ = R(pU − pL

)(7)

where R is the scalar perturbation size parameter, pU andpL are (K × 1) vectors that represent the maximum and

minimum bounds on each dimension. The set J was con-structed using a probability function Pi that represents athreshold for determining whether a specific dimensionj was perturbed or not; Pi is monotonically decreasingfunction of function evaluations:

Pi = 1 − log[

i(1 − F)N

](8)

where i is the current iteration. After Pi is determined, wedrew Pj from a uniform distribution for each dimension j.If Pj < Pi was included in J. Thus, the probability thata dimension j was perturbed was inversely proportionalto the number of function evaluations. DDS updates aregreedy; Gnew becomes the new solution vector only if it isbetter than G.

Multiswitch DOPSWe investigated whether switching search methods morethan once would result in better performance; this DOPSvariant is referred to as multiswitch DOPS or msDOPS.msDOPS begins with the PSO phase and uses the samecriteria as DOPS to switch to the DDS phase. How-ever, msDOPS can switch back to a PSO search whenthe DDS phase has reduced the functional value to90% of its initial value. Should the DDS phase fail toimprove the functional value sufficiently, this versionis identical to DOPS. When the switch from DDS toPSO occurs, we use the best solution from DDS toseed the particle swarm. DOPS and msDOPS sourcecode is available for download under a MIT licenseat http://www.varnerlab.org.

Comparison techniquesThe implementations of particle swarm optimization,simulated annealing, and genetic algorithms are theones given in Matlab R2017A (particleswarm,simulannealbnd and ga). The implementation of DEused was developed by R.Storn and available at http://www1.icsi.berkeley.edu/~storn/code.html. The versionof eSS used was Release 2014B - AMIGO2014benchVERSION WITH eSS MAY-2014-BUGS FIXED - JRB,released by the Process Engineering Group IIM-CSIC.The genetic algorithm, particle swarm, and differentialevolution algorithms were run with a 40 particles to bedirectly comparable to the number of particles used inthe PSO phase of DOPS. For comparison, the versionof CM-AES used was cmaes.m, Version 3.61.beta fromhttp://cma.gforge.inria.fr/cmaes_sourcecode_page.html.The scripts used to run the comparison methods are alsoavailable at http://www.varnerlab.org.

Sagar et al. BMC Systems Biology (2018) 12:87 Page 14 of 15

Additional files

Additional file 1: Figure S1. Data fits for CHO metabolism problem.(PDF 130 kb)

Additional file 2: Figure S2. Data fits for S.cerevisiaemetabolismproblem. (PDF 22 kb)

Additional file 3: Figure S3. Comparison of states and parameters.(PDF 192 kb)

Additional file 4: Figure S4. Time Comparison. (PNG 28 kb)

Additional file 5: Figure S5. Convergence Curves. (PDF 116 kb)

Additional file 6: Figure S6. Comparison of DOPS to ESS. (PDF 73 kb)

Additional file 7: Figure S7. Comparison of functional values.(PDF 123 kb)

Additional file 8: Figure S8. Dispersion Curves. (PDF 89 kb)

FundingThis study was supported by an award from the US Army and Systems Biologyof Trauma Induced Coagulopathy (W911NF-10-1-0376).

Availability of data andmaterialsDOPS is open source, available under an MIT software license from http://www.varnerlab.org. All scripts used in this study are included in theutilities directory. The optimization problems used are located in theCoagulationFiles and testFunctions directories.

Authors’ contributionsAS developed original DOPS algorithm. RL developed msDOPS and generatedfigures in the manuscript. CS developed the dynamic dimensional searchapproach, and assisted in editing the manuscript. JV directed the study,prepared and edited the manuscript. All authors read and approved the finalmanuscript.

Ethics approval and consent to participateNot applicable.

Consent for publicationNot applicable.

Competing interestsThe authors declare that they have no competing interests.

Publisher’s NoteSpringer Nature remains neutral with regard to jurisdictional claims inpublished maps and institutional affiliations.

Author details1Robert Fredrick Smith School of Chemical and Biomolecular Engineering,Cornell University, 244 Olin Hall, Ithaca, NY, USA. 2School of Civil andEnvironmental Engineering, Cornell University, Ithaca, NY, USA.

Received: 29 December 2017 Accepted: 17 September 2018

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