Mathematical Biosciences 233 (2011) 111–125

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Mathematical Biosciences

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Multiobjective H2/H1 synthetic gene network design based on promoter libraries

Chih-Hung Wu a, Weihei Zhang b, Bor-Sen Chen a,⇑a Lab of Systems Biology, Department of Electrical Engineering, National Tsing Hua University, Hsinchu 30013, Taiwanb Collage of Information and Electrical Engineering, Shandong University of Science and Technology, Qingdo 266510, China

a r t i c l e i n f o

Article history:Received 18 April 2011Received in revised form 29 June 2011Accepted 1 July 2011Available online 19 July 2011

Keywords:Synthetic gene networkMultiobjective H2/H1 designLMIsReference trackingNonlinear stochastic system

0025-5564/$ - see front matter Crown Copyright � 2doi:10.1016/j.mbs.2011.07.001

⇑ Corresponding author. Tel.: +886 3 573 1155; faxE-mail address: bschen@ee.nthu.edu.tw (B.-S. Che

a b s t r a c t

Some current promoter libraries have been developed for synthetic gene networks. But an efficientmethod to engineer a synthetic gene network with some desired behaviors by selecting adequate pro-moters from these promoter libraries has not been presented. Thus developing a systematic method toefficiently employ promoter libraries to improve the engineering of synthetic gene networks with desiredbehaviors is appealing for synthetic biologists.

In this study, a synthetic gene network with intrinsic parameter fluctuations and environmental distur-bances in vivo is modeled by a nonlinear stochastic system. In order to engineer a synthetic gene networkwith a desired behavior despite intrinsic parameter fluctuations and environmental disturbances in vivo,a multiobjective H2/H1 reference tracking (H2 optimal tracking and H1 noise filtering) design is intro-duced. The H2 optimal tracking can make the tracking errors between the behaviors of a synthetic genenetwork and the desired behaviors as small as possible from the minimum mean square error point ofview, and the H1 noise filtering can attenuate all possible noises, from the worst-case noise effect pointof view, to achieve a desired noise filtering ability. If the multiobjective H2/H1 reference tracking designis satisfied, the synthetic gene network can robustly and optimally track the desired behaviors, simulta-neously.

First, based on the dynamic gene regulation, the existing promoter libraries are redefined by their pro-moter activities so that they can be efficiently selected in the design procedure. Then a systematicmethod is developed to select an adequate promoter set from the redefined promoter libraries to synthe-size a gene network satisfying these two design objectives. But the multiobjective H2/H1 reference track-ing design problem needs to solve a difficult Hamilton–Jacobi Inequality (HJI)-constrained optimizationproblem. Therefore, the fuzzy approximation method is employed to simplify the HJI-constrained optimi-zation problem to an equivalent linear matrix inequality (LMI)-constrained optimization problem, whichcan be easily solved by selecting an adequate promoter set from the redefined promoter libraries usingthe LMI toolbox in Matlab.

Based on the confirmation of in silico design examples, we can select an adequate promoter set from theredefined promoter libraries to achieve the multiobjective H2/H1 reference tracking design. The proposedmethod can reduce the number of trial-and-error experiments in selecting an adequate promoter set for asynthetic gene network with desired behaviors. With the rapid increase of promoter libraries, this sys-tematic method will accelerate progress of synthetic biology design.

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1. Introduction

Synthetic biology is hybrid discipline, combining elements ofboth engineering and science to engineer synthetic organisms.Through detailed understanding of cellular mechanisms and im-proved experimental techniques for manipulating cell genotypes,it has become possible to engineer a cell for the rational designof genetic and protein circuits [1]. In the past decade, syntheticbiologists have built several gene networks such as toggle switches

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[2–5], transcriptional cascades [6], pulse generators [7], time-delayed circuits [8,9], oscillators [2,10–12] and digital logicevaluators [13,14]. The field has also yielded several technologicalapplications and provided new avenues for drug manufacture[15,16], bio-fabrication [17], therapeutics [18,19] and biofuelproduction [20–22].

From the bottom-up approach, simple and well-characterizedbiological parts can be coupled together into more complexnetworks with predicted behaviors. Synthetic biologists alsoprovide a concept that programmable cells can be constructed bydesigning appropriate interfaces, which couple engineered genenetworks to the regulatory circuitry of the cell [23,24]. Hence,

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112 C.-H. Wu et al. / Mathematical Biosciences 233 (2011) 111–125

the well-characterized biological parts as well as a mathematicalmodel can help us engineer a synthetic gene network. Recently,many studies have discussed computation design for syntheticgene networks. For transcription networks, their qualitative depic-tion and an estimation of produced proteins can be both obtainedby employing mathematical modeling frameworks such as an or-dinary differential equation (ODE). Kuepfer et al. developed an ap-proach for synthetic biology based on semidefinite programmingfor partitioning the parameter spaces of polynomial differentialequation models into the so-called feasible and infeasible regions[25]. In this approach, a feasible region refers simply to the exis-tence of a steady state of the synthetic system. Rosenfeld et al. usedtime-lapse fluorescence microscopy to quantitatively analyze aut-oregulatory negative feedback circuits in order to test the assump-tion that quantitative characterization of regulatory elements canpredict the behavior of gene circuits [26]. More recently, somegene circuit designs have been proposed to embed gene circuitsinto an existing gene network to improve its robust stability[27,28]. In the biological implementation, synthetic biologists havebuilt the combined gene circuits to produce a push-on/push-offcircuit [29] and a counter [30]. These synthetic gene networksare implemented by some basic biological parts for industrial bio-technology. Hence, the synthetic gene networks can be imple-mented by these basic biological parts and rational designprinciple [31]. From the engineering viewpoint, the design purposein synthetic biology is to engineer a completely new gene networkand then insert this gene network into a host cell to perform newtasks in spite of intrinsic parameter fluctuations and environmen-tal disturbances on the host cell. Hence, despite intrinsic parameterfluctuations and environmental disturbances, the robust syntheticgene network is designed to achieve a desired steady state [32,33]or an H2 optimal tracking of a desired oscillatory behavior [34]. Theabove robust design method is to tune some parameters of biolog-ical devices for a synthetic gene network to achieve the desiredsteady or oscillatory state. But, in fact, tuning the parameters ofbiological devices to fit the designed parameters for a real suitablebiological network is currently quite a difficult or even unfeasibletask for biotechnology. With this concept, we think that selectingadequate promoters from the existing promoter libraries for syn-thetic gene networks is more convenient than tuning the parame-ters of some components to achieve the designed values ofsynthetic gene networks.

In general, gene circuits can be constructed from a handful ofbasic biological parts. For example, in order to completely imple-ment a transcription unit, one needs the promoter, ribosome bind-ing site (RBS), protein coding region and terminator, which can beobtained from the BioBrick. However, BioBrick parts are only stan-dardized in terms of how individual parts are physically assembledinto a multi-component system, and most parts remain uncharac-terized [35]. Accurate prediction and computational design for syn-thetic gene networks with adequate component properties havegot to be developed for system-level circuitry to integrate basicparts and modules [36]. Quantitative characterization of parts isneeded to be measured to improve faster construction of circuits.Kelly et al. developed standard measurement kits for characteriz-ing BioBrick promoters and ribosome binding sites [35]. However,incomplete knowledge about components on modules makes itdifficult predict the precise behaviors of synthetic gene networks.The complexity of reactions, variety of molecules, intrinsic andextrinsic disturbances all makes it hard to accurately predict thebehaviors even of simpler biological devices.

More recently, Ellis et al. presented an approach that couples li-braries of diversified components with in silico modeling to guidepredictable gene network construction without the need for posthoc tweaking [37]. Cantone et al. created a relatively sophisticatedsynthetic gene network of five genes that serves as an ‘oracle’ that

is queried by different perturbations. Finally, they tested their syn-thetic gene network by some methods based on ordinary differen-tial equations, Bayesian inference and information theory touncover the connectivity of the network [38]. Ellis et al. andCantone et al. provided some guidance through the introductionof benchmark networks.

Next, an efficient approach for selecting suitable promoter setsfrom the existing promoter libraries is important for the progressof synthetic biology. Synthetic biologists can use the standard bio-logical parts that have a clear definition of the function and inter-face of the device, and the operating context of the device, tomeasure characteristics and describe the quantitative behavior.However, at present, there is still no accepted method for syntheticbiologists to efficiently select biological parts from promoter li-braries for engineering a synthetic gene network with some pre-scribed behaviors.

In an actual real biological environment, the physical intercon-nections cannot be extended as in electrical and mechanical sys-tem because the interoperability has to be derived from chemicalspecificity of parts and their desired targets [39]. Thus rational con-struction in synthetic biology is always hampered by chemicalspecificity of biological parts. Fortunately, promoter libraries havebeen widely constructed and their input–output properties can beeasily measured via systematic experiments. Since different pro-moters in promoter libraries have different biochemical kinetics,to engineer a synthetic gene network with desired behaviors byselecting some adequate promoters from promoter libraries hasbecome possible. However, it is still difficult to directly select pro-moters from these promoter libraries for engineering a syntheticgene network to achieve desired behaviors because these promot-ers still lack promoter activities which can be easily selected forsynthetic biologists. Therefore, for the convenience of design, weshould redefine the promoter activities for promoters in these con-ventional promoter libraries so that they can be more easilyselected.

In this study, we also develop a systematic method to efficientlyselect adequate promoters from these redefined promoter librariesso that the synthetic gene network with desired behaviors can beimplemented. Based on the proposed design procedure, syntheticbiologists can first create a gene circuit topology as a guide for con-structing a synthetic gene network. With a reference model to gen-erate a desired behavior to be tracked by a synthetic gene network,the proposed multiobjective H2/H1 reference tracking design canefficiently select an adequate promoter set from the redefined pro-moter libraries. Then the synthetic gene network with a desiredbehavior can be achieved by employing the adequate promoterset, which can simultaneously guarantee a desired H1 noise filter-ing and H2 optimal reference tracking. In order to meet these twodesign objectives, we need to solve an HJI-constrained optimiza-tion problem for the multiobjective H2/H1 reference tracking de-sign of a synthetic gene network, which is not easy solvedanalytically or numerically at present. We employ Takagi–Sugeno(T–S) fuzzy system to interpolate several linear systems to approx-imate the nonlinear gene network system. Based on the T–S fuzzysystem, the HJI-constrained optimization problem for multiobjec-tive H2/H1 reference tracking design of synthetic gene networksis transformed to an equivalent LMIs-constrained optimizationproblem, which can be easily solved with the help of the LMItoolbox in Matlab [40]. Finally, two in silico design examples ofselecting an adequate promoter set from the redefined promoterlibraries to achieve the multiobjective H2/H1 reference trackingdesign are given to illustrate the design procedure. Although mul-tiobjective H2/H1 control designs have been widely proposed forrobust stabilization of linear and nonlinear systems [41–44], thisis the first proposed method for selecting an adequate promoterset from refined promoter libraries to achieve a multiobjective

C.-H. Wu et al. / Mathematical Biosciences 233 (2011) 111–125 113

H2/H1 reference tracking design of nonlinear stochastic gene net-works. In the future, synthetic biologists can use the proposedmethod to select an adequate set of promoters from the redefinedpromoter libraries to develop a predictable synthetic gene networkand facilitate the development of gene network engineering.

2. Methods

2.1. Redefinition of promoter libraries based on the promoter activityof dynamic gene regulation

Quantitative fluorescence measurements of promoters havebeen used to build promoter libraries over the past decade [45–48]. The prospect of engineering a synthetic gene network with de-sired behaviors by selecting adequate promoters from these exist-ing promoter libraries is appealing for synthetic biologists.However, the current promoter libraries provide only the experi-mental output of fluorescence as an indirect measure of promotersand still lack of information on the promoter activities, which areuseful for engineering synthetic gene networks. At present, thereis still not an efficient method to quickly select adequate promot-ers from the current promoter libraries for synthetic genenetworks.

In general, a synthetic gene network always consists of a set ofpromoter-regulation circuits as shown in Fig. 1. Therefore, the dy-namic model of promoter-regulation gene circuit in Fig. 1 shouldbe studied based on promoter activities in promoter libraries be-fore the design of synthetic gene network. Since directly measuringpromoter activities is still challenging, measuring fluorescence asan indirect measure of promoter activities is widely used[26,35,37,48,49]. Nowadays the current promoter libraries onlycontain the fluorescence of promoter. However, the promoteractivities rather than fluorescence are directly employed as designparameters to select adequate promoters in engineering a syn-thetic gene network. For the convenience of selection, we redefineprevious promoter libraries in the following.

A dynamic model can be used to indirectly evaluate the pro-moter activities [35,37,50] to help redefine existing promoter li-braries shown in Fig. 1. Suppose the promoter c is selected fromthe TetR-regulated promoter library. In general, the promoter c al-ways has the minimum expression with the repressor TetR bindingand the maximum expression without it. Furthermore, the repres-sor TetR is also regulated by the inducer adhydrotetracycline (ATc).Hence we denote a repressor activity r, which is regulated by theinducer ATc, of the repressor TetR with the following form [51]:

r ¼ TetR1þ ðATc=KATcÞnATc

; ð1Þ

where KATc denotes the ATc–TetR dissociation rate and nATc denotesthe binding cooperativity between ATc and TetR. Then the promoter

Fig. 1. Single schematic diagram of the synthetic promoter-regulation gene circuit. The exof fluorescence {ymin,ymax} with and without the repressor TetR binding, respectively. Theto build the redefined promoter library. By a similar process, the promoter activities ofpromoter libraries. The detailed redefined procedure is illustrated in Appendix A.

regulation function pTetR(c, r), which is dependent on the repressoractivity r and the choice of the promoter c, has the following form:

pTetRðc; rÞ ¼ cr þ cs�cr1þðr=KTetRÞnTetR ¼ cr þ ðcs � crÞHTetRðrÞ;

c ¼ fcr ; csg 2 LibTetR;ð2Þ

where the promoter c has two promoter activities cr and cs for theTetR-regulated promoter library LibTetR, i.e., c = {cr,cs} 2 LibTetR; andKTetR and nTetR denote the TetR–DNA binding affinity and bindingcooperativity between regulatory protein TetR and DNA, respec-tively; HTetRðrÞ ¼ 1=ð1þ ðr=KTetRÞnTetR Þ is a Hill function which cap-tures the regulatory effect of a regulatory protein [51]. Accordingto the above definition, the promoter regulation function pTetR(c, r)in (2) has the minimum value cr and maximum value cs with andwithout the saturating concentration of the repressor activity in(1). These are related to the minimum fluorescence ymin and maxi-mum fluorescence ymax of fluorescent protein. By the simple esti-mation from fluorescence of fluorescent protein, cr and cs cansimply transformed through ymin and ymax from (A3) and (A2) inAppendix A, respectively. The detailed discussion is listed inAppendix A.

Assume that x(c, t) and X(c, t) denote the concentrations of themRNA and protein of gene yegfp, respectively. Then the dynamicmodel of promoter-regulation gene circuit in Fig. 1 is constructedwith

_xðc; tÞ ¼ pTetRðc; rÞ � bxðc; tÞ_Xðc; tÞ ¼ axðc; tÞ � cyEGFPXðc; tÞ

; ð3Þ

where b and cyEGFP denote the degradation rates of mRNA and pro-tein yEGFP, respectively; a denotes the translation rate. Accordingto the dynamic model (3), it is desirable for the promoter libraryto contain the promoter activity c = {cr,cs} for each promoter so thatadequate promoters could be easily selected from the correspond-ing promoter libraries to achieve desired behaviors or gene circuits.Therefore, the promoter libraries should be redefined based on thepromoter activity c = {cr,cs}, which needs to be estimated from theexperimental data in conventional promoter libraries, i.e., the fluo-rescence of indirect measure of a promoter in the conventional pro-moter libraries should be used to estimate its promoter activity cr

and cs and then promoter libraries should be redefined accordingto these estimated promoter activities. For example, we use Ellis’experimental data [37] to reconstruct the TetR- and LacI-regulatedpromoter libraries shown in Table 1. The detailed procedure ofhow to use Ellis’ experimental data to redefine the TetR- andLacI-regulated promoter libraries is illustrated in Appendix A. Theother promoters from the corresponding promoter libraries couldbe similarly redefined based on the promoter activity. Since acomplex synthetic gene network always consists of a set ofpromoter-regulation gene circuit in Fig. 1, the design of the complexsynthetic gene network becomes how to select a set of promoters

isting TetR-regulated promoter library contains the minimum and maximum valuespromoter activities are estimated by fluorescence from (A2) and (A3) in Appendix Aeach promoter in the redefined promoter libraries can be estimated from existing

114 C.-H. Wu et al. / Mathematical Biosciences 233 (2011) 111–125

from the corresponding promoter libraries with adequate promoteractivities to achieve the design specifications.

Remark. The BioBricks have provided a description of theexchangeable protocol of BioBrick-related data and developed astandardized, extensible, scalable and machine-processable inter-face. However, only few BioBrick parts are well-characterized, sothat at present there is no standard approach to utilize BioBricksfor engineering a synthetic gene network. For the reason, we aregoing to characterize and utilize the biological parts in BioBricks inour next research.

2.2. Nonlinear stochastic gene network model based on gene circuittopology

After providing the redefined promoter libraries for the simplepromoter-regulation gene circuit in Fig. 1, we discuss how to

Table 1Redefined TetR- and LacI-regulated promoter libraries. The redefined TetR- and LacI-regulated promoter libraries (i.e., LibTetR and LibLacI) comprise different promoters (i.e.,Tk and Lk, k = 0, . . ., 20) with their corresponding promoter activities of cs and cr

obtained from experimental data of previous promoter libraries in [37] (see AppendixA).

TetR-regulated promoter library(LibTetR)

LacI-regulated promoter library(LibLacI)

Promoter Promoter activity Promoter Promoter activity

cs cr cs cr

T0 2121 0.1724 L0 1657.5 0.3018T1 1604 0.7576 L1 923.97 0.2567T2 1376.6 0.1936 L2 860.87 0.2244T3 1169.8 0.4672 L3 674.92 1.9189T4 974.52 0.0753 L4 651.58 1.1680T5 942.77 0.2281 L5 570.07 3.5062T6 967.17 0.1493 L6 527.83 0.5497T7 738.57 0.0702 L7 323.45 0.1248T8 641.74 0.7135 L8 327.77 0.1772T9 564.24 0.2620 L9 309.74 0.5439T10 501.35 0.0756 L10 298.35 0.1146T11 469.35 0.0788 L11 250.16 0.1326T12 466.16 0.1636 L12 248.03 0.1171T13 356.88 0.0927 L13 239.32 0.1010T14 348.95 0.1483 L14 190.2 0.0959T15 274.79 0.1067 L15 163.84 0.4813T16 250.04 0.0857 L16 166.42 0.0989T17 188.77 0.1366 L17 131.63 0.1190T18 119.57 0.0753 L18 108.96 0.0903T19 111.57 0.1185 L19 101.89 0.0982T20 70.909 0.1606 L20 85.673 0.2174

Fig. 2. Synthetic gene circuit topology: simple toggle switch [3]. The regulatory proteinthe inducer ATc. The protein LacI inhibits the transcription of tetR by binding promoter cyegfp through binding promoter c3 to repress the protein yEGFP expression. The gene cirstates by changing the inducers ATc and IPTG. If an adequate promoter set c = {c1,c2,c3} isfluorescent protein yEGFP will track the desired behaviors generated by a reference modeselected from TetR-regulated promoter library in Table 1, i.e., c1 2 LibLacI, c2,c3 2 LibTetR.

engineer a more complex gene network by selecting adequate pro-moters from the redefined promoter libraries and then provide asystematic design method.

In fact, a complex synthetic gene network always contains alarge number of the simple gene circuit in Fig. 1. Thus, the selectionof adequate promoters from the redefined promoter libraries canbe easily extended to the selection of an adequate promoter setin the complex gene network. For the convenience of illustration,a well-known gene circuit topology, the simple toggle switch, isshown in Fig. 2. The toggle switch has two distinct stable statesand can be reversibly switched between the two states by chang-ing the inducers ATc and IPTG. The proteins TetR and LacI inhibittranscription through the promoters c2 and c1 and are induced bythe inducers ATc and IPTG, respectively. The fluorescent proteinyEGFP is repressed through the promoter c3 by the repressor TetR.Assume that x1(c1, t),x2 (c2, t) and x3(c3, t) denote the concentrationsof mRNAs tetR, lacI and yegfp, respectively. In addition, X1(c1, t),X2(c2, t) and X3(c3, t) denote the concentrations of proteins TetR,LacI and yEGFP, respectively. Then the dynamic model of the toggleswitch gene network in Fig. 2 is modeled as follows [3,52]

_x1ðc1; tÞ ¼ pLacIðc1; r1Þ � bx1ðc1; tÞ;_X1ðc1; tÞ ¼ ax1ðc1; tÞ � c1X1ðc1; tÞ;_x2ðc2; tÞ ¼ pTetRðc2; r2Þ � bx2ðc2; tÞ;_X2ðc2; tÞ ¼ ax2ðc2; tÞ � c2X2ðc2; tÞ;_x3ðc3; tÞ ¼ pTetRðc3; r3Þ � bx3ðc3; tÞ;_X3ðc3; tÞ ¼ ax3ðc3; tÞ � c3X3ðc3; tÞ;

8>>>>>>>>><>>>>>>>>>:yðc; tÞ ¼ X3ðc3; tÞ;c ¼ fc1; c2; c3g; c1 2 LibLacI; c2; c3 2 LibTetR

ð4Þ

with r1 ¼ X2ðc2; tÞ=ð1þ ðIPTG=KIPTGÞnIPTG Þ and r2 ¼ r3 ¼ X1ðc1; tÞ=ð1þðATc=KATcÞnATc Þ for the gene circuit topology in Fig. 2, where a de-notes the translation rate; and b and ci, i = 1,2,3 denote the degra-dation rates for mRNA and proteins, respectively. The promoterregulation functions pLacI(c1, r1), pTetR(c2,r2) and pTetR(c3,r3) aredependent on the selection of promoters c1, c2 and c3 from the cor-responding promoter libraries. The output y(c, t) denotes the outputof interest and is dependent on the selected promoter setc = {c1,c2,c3} with adequate promoter activities from the corre-sponding promoter libraries in Table 1. Obviously, the dynamicmodel of toggle switch gene network (4) consists of three interac-tive dynamic models of promoter-regulation gene network in (3).

In general, biological parts are inherently uncertain in thisnano-scale biochemical system [51]. We assume that the promoteractivities of promoters, degradation rates of mRNAs and proteins,translation rates are stochastically uncertain in vivodue to gene

TetR inhibits the transcription of lacI by binding the promoters c2 and is induced by1 and is induced by the inducer IPTG. The protein TetR also inhibits transcription ofcuit has two distinct stable states, and can be reversibly switched between the twoselected from corresponding promoter libraries by the systematic method, then thel, where c1 is selected from LacI-regulated promoter library in Table 1; c2 and c3 are

C.-H. Wu et al. / Mathematical Biosciences 233 (2011) 111–125 115

expression noises in transcriptional and translational processes,thermal fluctuations, DNA mutations, parameter estimation errorsand evolutions [51] as follows:

crk! crk

þ DcrknkðtÞ; csk

! cskþ Dcsk

nkðtÞ; ck ! ck þ DcknkðtÞ;b! bþ DbnkðtÞ; a! aþ DankðtÞ; k ¼ 1;2;3;

ð5Þ

where Dcrk; Dcsk

; Dck; Db and Da denote the standard deviations ofstochastic parameter, and nk(t) is a Gaussian noise with zero meanand unit variance. Thus Dcrk

; Dcsk; Dck; Db and Da denote the

deterministic parts of parameter variations and nk(t) denotes thekth random fluctuation source.

Suppose the synthetic gene network also suffers from environ-mental disturbances on the host cell due to various extracellularenvironments in the cellular context. Then the synthetic gene net-work in (4) with a promoter set c = {c1,c2,c3} selected from theredefined promoter libraries, with intrinsic parameter fluctuationsas (5) and environmental disturbances on the host cell could be de-scribed as

_Xðc; tÞ ¼ f ðX; c; r; tÞ þP3k¼1

MkgkðX; ck; rk; tÞnkðtÞ þ vðtÞ;

Yðc; tÞ ¼ HXðc; tÞ;c ¼ fc1; c2; c3g; c1 2 LibLacI; c2; c3 2 LibTetR;

ð6Þ

where

Xðc; tÞ ¼

x1ðc1; tÞX1ðc1; tÞx2ðc2; tÞX2ðc2; tÞx3ðc3; tÞX3ðc3; tÞ

2666666664

3777777775; f ðX; c; r; tÞ ¼

pLibLacIðc1; r1Þ � bx1ðc1; tÞ

ax1ðc1; tÞ � c1X1ðc1; tÞpLibTetR

ðc2; r2Þ � bx2ðc2; tÞax2ðc2; tÞ � c2X2ðc2; tÞ

pLibTetRðc3; r3Þ � bx3ðc3; tÞ

ax3ðc3; tÞ � c3X3ðc3; tÞ

2666666664

3777777775;

vðtÞ ¼

v1ðtÞv2ðtÞv3ðtÞv4ðtÞv5ðtÞv6ðtÞ

2666666664

3777777775; H ¼

000001

2666666664

3777777775

T

;

M1 ¼

Dcs1 Dcr1 �Db 0

0 0 Da �Dc1

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

26666666664

37777777775;

M2 ¼

0 0 0 0

0 0 0 0

Dcs2 Dcr2 �Db 0

0 0 Da �Dc2

0 0 0 0

0 0 0 0

26666666664

37777777775;

M3 ¼

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

Dcs3 Dcr3 �Db 0

0 0 Da �Dc3

26666666664

37777777775;

g1ðX; c1; r1; tÞ ¼

HLacIðr1Þ1� HLacIðr1Þ

x1ðc1; tÞX1ðc1; tÞ

26664

37775;

g2ðX; c2; r2; tÞ ¼

HTetRðr2Þ1� HTetRðr2Þ

x2ðc2; tÞX2ðc2; tÞ

26664

37775;

g3ðX; c3; r3; tÞ ¼

HTetRðr3Þ1� HTetRðr3Þ

x3ðc3; tÞX3ðc3; tÞ

26664

37775:

For the convenience of analysis and design, the stochastic syn-thetic gene network in (6) can be represented by the followingIto stochastic differential equation [53,54]

dXðc; tÞ ¼ ðf ðX; c; r; tÞ þ vðtÞÞdt þP3k¼1

MkgkðX; ck; rk; tÞdxkðtÞ;

Yðc; tÞ ¼ HXðc; tÞ;c ¼ fc1; c2; c3g; c1 2 LibLacI; c2; c3 2 LibTetR:

ð7Þ

The stochastic part of intrinsic parameter fluctuations is ab-sorbed to nk(t) with dxk(t) = nk(t)dt, where xk(t) denotes a stan-dard Wiener process or Brownian motion [53,54]. Since thestochastic differential equation in (7) is dependent on the selectionof the promoter set c from the redefined promoter libraries, thesynthetic gene network with desired behaviors can be achievedby selecting adequate promoter set c.

Now, consider a more general design case of the synthetic genenetwork with n genes as follows

dXðc; tÞ ¼ ðf ðX; c; r; tÞ þ vðtÞÞdt þPnk¼1

MkgkðX; ck; rk; tÞdxk;

Yðc; tÞ ¼ HXðc; tÞ;c ¼ fc1; c2; . . . ; cng 2 Libj; j ¼ 1;2; . . . ;m;

ð8Þ

where the state vector X(c, t) = [x1(c1, t),X1(c1, t), . . .,xn(cn, t),Xn(cn, t)]T

denotes the concentrations of mRNAs and proteins in the syntheticgene network, and f(X,c,r, t) denotes the nonlinear gene regulationfunction consisting of promoter set c from the redefined promoterlibraries Libj, j = 1,2, . . .,m and other parameters (i.e., translationrates and degradation rates of mRNA and proteins). The output vec-tor Y(c, t) = [y1(c, t), . . .,yl(c, t)]T denotes the concentrations of theobserved proteins of interest. H is a l � 2n matrix, where l is thenumber of observed proteins of interest. The elements of the pertur-bative matrix Mk denote the corresponding perturbation amplitudesor the standard deviations of the corresponding stochastic parame-ter fluctuations due to n random noises. v(t) denotes the environ-mental disturbances. For the stochastic differential Eq. (8), weassume the equilibrium point (phenotype) of interest is at origin.If the equilibrium point of interest is not at origin, it can be shiftedto origin.

In order to engineer a stochastic synthetic gene network torobustly and optimally track a desired trajectory by selecting anadequate promoter set from the redefined promoter libraries,two design objectives are provided for multiobjective H2/H1reference tracking design from a systematic point of view.

2.3. Multiobjective H2/H1 reference tracking design: the desired H1noise attenuation level and H2 optimal reference tracking

Multiobjective H2/H1 reference tracking design can be achievedby selecting an adequate promoter set from the redefined pro-moter libraries to engineer a synthetic gene network to track the

116 C.-H. Wu et al. / Mathematical Biosciences 233 (2011) 111–125

desired behavior. Initially, in order to attenuate the effects ofintrinsic parameter fluctuations and environmental disturbancesin vivo, we consider a desired noise attenuation level as the first de-sign objective. Then, in order to minimize the tracking error be-tween the synthetic gene network and reference model, theoptimal reference tracking is considered as the second designobjective. Therefore, our design purpose is to engineer a syntheticgene network with desired behavior by selecting an adequate pro-moter set from the redefined promoter libraries to satisfy two de-sign objectives, i.e., the desired noise attenuation level and optimalreference tracking. Based on the real biological circuit topology andits stochastic differential Eq. (8), how to engineer a synthetic genenetwork with some desired behaviors becomes how to select anadequate promoter set from the redefined promoter libraries totrack the reference trajectories generated by the following refer-ence model

_XrðtÞ ¼ ArXrðtÞ þ rðtÞ;YrðtÞ ¼ HrXrðtÞ;

ð9Þ

where Ar and r(t) are specified beforehand by the designer to gener-ate a desired behavior, Yr(t), to be tracked by Y(c, t) in (8); and Hr is al � n matrix, where l is the number of observed proteins. Based onthese two design objectives, the stochastic differential Eq. (8) couldbe designed so that Y(c, t) could track the desired Yr(t) generated bythe reference model in (9) despite intrinsic parameter fluctuationsand environmental disturbances. From the engineering point ofview, the multiobjective H2/H1 reference tracking design is speci-fied as follows: a desired noise attenuation level qd and the optimalreference tracking, i.e., the following two design objectives need tobe achieved simultaneously [41–43,55].

(i) H1 noise attenuation level:

ER tf

0 ðYðc; tÞ � YrðtÞÞT QðYðc; tÞ � YrðtÞÞdt

ER tf

0 vTðtÞvðtÞdt6 q2

d

for c 2 Libj; j ¼ 1; . . . ;m: ð10Þ

(ii) H2 optimal reference tracking:

minc2Libj

j¼1;...;m

EZ tf

0ðYðc; tÞ � YrðtÞÞT QðYðc; tÞ � YrðtÞÞdt; ð11Þ

where Y(c, t) and Yr(t) are the system outputs in (8) and (9),respectively; and Q is a symmetric weighting matrix. Thephysical meaning of the H1 noise attenuation level qd in (i)is that the effect of all possible environmental disturbanceson the tracking error Y(c, t) � Yr(t) should be less than the de-sired noise attenuation level qd from the average energypoint of view. If the desired qd in (10) holds, the noise filter-ing ability of synthetic gene networks will be better than adesired noise attenuation level qd, despite the intrinsicparameter fluctuations and environmental disturbances fromthe average energy perspective. The optimal reference track-ing performance in (ii) is to select a promoter set from theredefined promoter libraries to achieve the minimum meansquare tracking error. Hence these two design objectivesare that the synthetic gene network can simultaneouslyachieve both robustness and optimal reference trackingobjectives. In summary, the multiobjective H2/H1 referencetracking design is to select an adequate promoter setc = {c1,c2, . . .,cn} from the redefined promoter libraries Libj,j = 1,2, . . .,m so that the desired noise attenuation level qd

in (10) and the optimal reference tracking in (11) for the syn-thetic gene network are achieved simultaneously.

3. Results

3.1. Design procedure of multiobjective H2/H1 reference trackingdesign

Based on the analysis above, the design steps of multiobjectiveH2/H1 reference tracking for the synthetic gene network are out-lined as follows. (i) Based on the prescribed gene circuit topologyas a guide, the dynamic models of the synthetic gene networkand reference model should be constructed at first. (ii) The desiredstandard deviations of parameter fluctuations Mk in (8) to be toler-ated by the stochastic gene network in vivo are thus specified. (iii)An adequate promoter set c = {c1,c2 , . . .,cn} is selected from pro-moter libraries Libj, j = 1,2, . . .,m for a synthetic gene network tosimultaneously satisfy the desired noise attenuation level qd in(10) and optimal reference tracking in (11). Therefore, followingthe above design steps, the synthetic gene network with some de-sired behaviors can be achieved without the large number of trial-and-error experiments in conventional methods.

Our design purpose is to select an adequate promoter setc = {c1,c2 ,. . .,cn} for the synthetic gene network from promoter li-braries Libj, j = 1,2, . . .,m to satisfy the multiobjective H2 /H1 designobjectives in (10) and (11). To illustrate the design procedure, wecombine the stochastic gene network in (8) with the referencemodel in (9) as an augmented system

dXðc; tÞdXrðtÞ

� �¼

f ðX; c; r; tÞArXrðtÞ

� �þ

vðtÞrðtÞ

� �� �dt þ

Pnk¼1

MkgkðX; ck; rk; tÞ0

� �dxk

Yðc; tÞYrðtÞ

� �¼

H 00 Hr

� �Xðc; tÞXrðtÞ

� �ð12Þ

or equivalently,

dXðc; tÞ ¼ ðf ðX; c; r; tÞ þ vðtÞÞdt þPnk¼1

MkgkðX; ck; rk; tÞdxk;

Yðc; tÞ ¼ HXðc; tÞ;ð13Þ

where Xðc; tÞ ¼ Xðc; tÞXrðtÞ

� �; Yðc; tÞ ¼ Yðc; tÞ

YrðtÞ

� �; v ¼ vðtÞ

rðtÞ

� �;

f ðX; c; r; tÞ ¼ f ðX; c; r; tÞArXrðtÞ

� �; Mk ¼ ½Mk 0�T ; gkðX; ck; rk; tÞ ¼ gkðX; ck; rk;

tÞ and H ¼ diagðH;HrÞ. Then the two design objectives in (10) and(11) are equivalent to the following:

ER tf

0 YTðc; tÞQ Yðc; tÞdt

ER tf

0 vTðtÞvðtÞdt6 q2

d or EZ tf

0YTðc; tÞQ Yðc; tÞdt

6 q2dEZ tf

0vTðtÞvðtÞdt ð14Þ

and

minc2Libj

j¼1;...;m

EZ tf

0YTðc; tÞQYðc; tÞdt; ð15Þ

respectively, where Q ¼ Q �Q�Q Q

� �.

Remark. If the initial value Xðc;0Þ – 0, then the inequalityin (14) should be modified as E

R tf

0 Yðc; tÞQ Yðc; tÞdt 6 VðXðc;0ÞÞþq2

dER tf

0 vTðtÞvðtÞdt for some Lyapunov functions VðXðc; tÞÞ > 0, i.e.,the effect of initial values of system states should be taken intoconsideration.

Based on the augmented system in (12) or (13), our designpurpose is to select some adequate promoter sets c = {c1,c2, . . .,cn}from the redefined promoter libraries Libj, j = 1,2, . . .,m such thatthe desired noise attenuation level qd in (14) can be achieved. Thenfrom these adequate promoter sets, we select one promoter set toachieve the optimal reference tracking in (15). In order to achieve

C.-H. Wu et al. / Mathematical Biosciences 233 (2011) 111–125 117

the above two design objectives by a systematic method, designstep (iii) can be divided into two steps. In the first step, we selectall possible promoter sets from the redefined promoter libraries tosatisfy the desired noise attenuation level qd in (14), though theremay exist several promoter sets satisfying the desired noiseattenuation level qd in (14). Then our second step is to select apromoter set from these promoter sets, which satisfies the desirednoise attenuation level, to achieve the optimal reference trackingin (15). In this case, we choose a Lyapunov (energy) functionVðXÞ > 0 with V(0) = 0 for the augmented stochastic gene networkin (13). Based on the Lyapunov function, we obtain the followingresult.

Proposition 1. For the stochastic gene network in (8). If some pro-moter sets c = {c1, c2, . . ., cn} are selected from the redefined promoterlibraries Libj, j = 1,2, . . .,m so that the following HJI has a positive solu-tion VðXÞ > 0 for each promoter set

@VðXðc; tÞÞ@X

!T

f ðX; c; r; tÞ þ 12

Xn

k¼1

gTkðX; ck; rk; tÞMT

k

� @2VðXðc; tÞÞ@X2

MkgkðX; ck; rk; tÞ

þ 14q2

d

@VðXðc; tÞÞ@X

!T@VðXðc; tÞÞ

@X

!

þ XTðc; tÞHT QHXðc; tÞ 6 0: ð16Þ

Then the synthetic gene network with these promoter sets havea desired noise attenuation level qd.

Proof. See Appendix B. h

There may exist several promoter sets selected from the rede-fined promoter libraries Libj, j = 1,2, . . .,m, which could solve HJIin (16) for the synthetic gene network with a desired noise atten-uation level qd. Our second step is to select a promoter set that hasan H1 noise attenuation level from these candidate promoter setsto achieve the H2 optimal reference tracking in (15). Based on thesuboptimal tracking design, we can obtain the following result.

Proposition 2. The suboptimal tracking control design problem in(15) for synthetic gene networks in (8) becomes how to select onepromoter set to solve the following HJI-constrained optimizationproblem

minc2Libj

j¼1;...;m

EVðXðc;0ÞÞ

subjectto @VðXðc;tÞÞ@X

� �Tf ðX; c; r; tÞ þ 1

2

Pnk¼1

gTkðX; ck; rk; tÞMT

k

� @2VðXðc;tÞÞ@X2 MkgkðX; ck; rk; tÞ

þ 14

@VðXðc;tÞÞ@X

� �T@VðXðc;tÞÞ

@X

� �þ XTðc; tÞHT Q HXðc; tÞ 6 0;

ð17Þ

where VðXðc; tÞÞ > 0 is the Lyapunov function of nonlinear stochasticgene networks in (8).

Proof. See Appendix C. h

In order to simultaneously satisfy the two design objectives in(14) and (15), a promoter set should be selected so that (16) and(17) simultaneously hold, i.e., we need to select a promoter set tosolve the following HJI-constrained optimization problem

minc2Libj

j¼1;...;m

EVðXðc;0ÞÞ

subject to ð16Þ and ð17Þ:ð18Þ

From the analysis above, engineering a synthetic gene networkto achieve the multiobjective H2/H1 reference tracking design be-comes how to select an adequate promoter set c = {c1,c2, . . .,cn}from the redefined promoter libraries Libj, j = 1,2, . . .,m to solvethe HJI-constrained optimization in (18), i.e., the two-step designprocedures in Proposition 1 for H1 noise filtering and Proposition2 for H2 optimal tracking are merged in a one-step design proce-dure in (18) for solving the H2 optimal tracking and H1 noise filter-ing problems simultaneously by only selecting one promoter setfrom promoter libraries. But it is still generally very difficult tosolve the above HJI-constrained optimization due to the highlynonlinear properties of synthetic gene networks since there is stillno efficient analytic or numerical method to solve the HJI-constrained optimization problem in (18).

Recently, however, the T–S fuzzy model has been widely ap-plied to approximate the nonlinear system via interpolating sev-eral linearized systems at different operation points. Suppose thenonlinear gene network in (8) can be represented by the T–S fuzzymodel [56], which is a piecewise interpolation of several local lin-earized models through the membership functions. The fuzzymodel is described by several If-then rules and can be employedto approximate nonlinear gene network. The ith rule of the fuzzymodel for the nonlinear stochastic system in (8) can be expressedas the following form [27,56]

Rule i: if X1(c1, t) is F1i and . . . and Xn(cn,t) is Fni, then

dXðc; tÞ ¼ ðAiðcÞXðc; tÞ þ vðtÞÞdt þXn

k¼1

MkBikðcÞXðck; tÞdxk;

i ¼ 1; . . . ; L; ð19Þ

where Fji is the fuzzy set, and L is the number of If-then rules. Thenthe fuzzy system in (19) can be inferred as follows:

dXðc; tÞ ¼PL

i¼1liðXÞ ðAiðcÞXðc; tÞ þ vðtÞÞdtþPn

k¼1MkBikðcÞXðck; tÞdxk�

PLi¼1liðXÞ

¼XL

i¼1

hiðXÞ ðAiðcÞXðc; tÞ þ vðtÞÞdtþXn

k¼1

MkBikðcÞXðck; tÞdxk

" #;

ð20Þ

where liðXÞ ¼Qn

j¼1FjiðXjðcj; tÞÞ;hiðXÞ ¼ liðXÞ=PL

i¼1liðXÞ, and Fji(Xj(cj,t)) is the grade of the membership function of Xj in Fij. We assume li

(X) > 0 andPL

i¼1liðXÞ > 0; 8t. Therefore, we obtain the fuzzy basesas hi(X) P 0 and

PLi¼1hiðXÞ ¼ 1; 8t.

The T–S fuzzy model in (20) is to interpolate L linear local sto-chastic dynamic systems to approximate the stochastic nonlineargene network in (8) via the fuzzy basis function hi(X(c, t)). Thematrices Ai(c) and Bik(c), i = 1,2, . . .,L, are specified so thatPL

i¼1hiðXÞAiðcÞXðc; tÞ andPL

i¼1hiðXÞBikðcÞXðc; tÞ in (20) can approxi-mate f(X,c,r, t) and gk(X,c,r, t) in (13) by the fuzzy identificationmethod [56], respectively. Since this paper focuses on the topicof multiobjective H2/H1 synthetic gene network design, for sim-plicity, the fuzzy approximation error is neglected in (20) and ismerged in the disturbances v(t). Then the augmented system canbe approximated by

dXðc; tÞ ¼PLi¼1

hiðXÞ ðAiðcÞXðc; tÞ þ vðtÞÞdtþPnk¼1

MkBikðcÞXðc; tÞdxðtÞ� �

;

Yðc; tÞ ¼ HXðc; tÞ;ð21Þ

where AiðcÞ ¼ diagðAiðcÞ;ArÞ; Mk ¼ ½Mk 0�T and BikðcÞ ¼ ½BikðcÞ 0�.After approximating the stochastic gene network in (8) by using

118 C.-H. Wu et al. / Mathematical Biosciences 233 (2011) 111–125

the T–S fuzzy system in (20), the stochastic nonlinear multiobjec-tive H2 /H1 reference tracking problem can be replaced by solvingthe fuzzy stochastic multiobjective H2/H1 reference tracking prob-lem in (14) and (15) for the fuzzy system in (21). We could choose aLyapunov (energy) function VðXðc; tÞÞ ¼ Xðc; tÞT PXðc; tÞ > 0 for thefuzzy system in (21). Then the multiobjective H2 /H1 referencetracking design problem for synthetic gene networks becomeshow to select a promoter set from promoter libraries Libj,j = 1,2, . . .,m to solve the following constrained optimizationproblem.

Proposition 3. Based on T–S fuzzy approximation model in (21), ifwe could select a promoter set from the redefined promoter librariesLibj, j = 1,2, . . .,m to solve the following LMI-constrained optimization

minc2Libj

j¼1;2;...;m

PR0 ð22Þ

subject to P > 0;

ATi ðcÞP þ PAiðcÞ þ HT Q H þ

Pnk¼1

BTikðcÞMT

k PMkBikðcÞ P

P �q2d

264

375 < 0;

ð23Þ

ATi ðcÞP þ PAiðcÞ þ HT Q H þ

Pnk¼1

BTikðcÞMT

k PMkBikðcÞ P

P �I

24

35 < 0;

ð24Þ

where R0 ¼ EfXð0ÞXTð0Þg denotes the covariance matrix of initial con-dition Xð0Þ, then the H2/H1 reference tracking in (14) and (15) can beachieved.

Fig. 3. Synthetic gene circuit topology: repressilator [57]. The gene circuit topology is brepressor protein, LacI, inhibits the transcription of the second repressor gene, cI, from proTetR, from promoter c3. Finally, TetR inhibits LacI and yEGFP expressions, completing thregulated promoter library, and the promoters c2 and c3 are selected from LacI- and CI-

Fig. 4. Flowchart of design procedure. After a gene circuit topology as a guide is establishconventional promoter libraries, the synthetic gene network with desired behaviors can breference model with a fuzzy synthetic gene network. Then the synthetic biologists canoptimal reference tracking, to achieve the multiobjective H2/H1 reference tracking viaMatlab to select an adequate promoter set c = {c1,c2, . . .,cn} from the redefined promoteradequate promoter set is selected for the multiobjective H2/H1 reference tracking desreference model in (9).

Proof. See Appendix D. h

The LMIs-constrained optimization problem for multiobjectiveH2/H1 reference tracking design of the synthetic gene network in(22)–(24) could be efficiently solved by selecting an adequate pro-moter set from the redefined promoter libraries with the help ofthe LMI toolbox in Matlab [40].

Based on the analysis above, the design procedure proposed inthis study is summarized as follows.

Design Procedure

(1) Create a synthetic gene circuit topology (e.g. Fig. 2 or 3) as aguide with promoters to be selected from the refined pro-moter libraries in Table 1, and then the stochastic differen-tial equation in (8) is constructed for the synthetic genenetwork.

(2) Provide a desired reference model in (9) to be tracked.(3) Provide the standard deviations of parameter fluctuations

Mk in (8) to be tolerated by the synthetic gene networkin vivo.

(4) Construct a T–S fuzzy model in (20) to approximate the syn-thetic gene network in (8); and then the augmented systemin (21), combining the T–S fuzzy model in (20) with the ref-erence model in (9), is constructed.

(5) Provide the desired H1 noise attenuation level qd.(6) Solve the LMI-constrained optimization problem in (22)–

(24) by selecting a promoter set via searching algorithmfrom the redefined promoter libraries.

A simple flowchart is shown in Fig. 4.

uilt by three repressors LacI, CI, TetR and one fluorescent protein yEGFP. The firstmoter c2. Then the protein product, CI, in turn inhibits the expression of a third gene,

e cycle. In this circuit topology, the promoter c1 and c4 are both selected from TetR-regulated promoter libraries, respectively.

ed and the redefined promoter libraries are built from the experimental data of thee engineered. The augmented T–S fuzzy model could be constructed via combining aprovide two design objectives, i.e., the desired H1 noise attenuation level qd and H2

solving LMI-constrained optimization in (22)–(24) with the help of LMI toolbox inlibraries. If the LMI-constrained optimization in (22)–(24) has been solved, then anign of the synthetic gene network to track the desired behaviors generated by a

Table 2Mathematical characteristics of promoters. Mathematical characteristics of promoters and their regulation parameters. Parameter values are obtained from empirical studies inthe literature or estimated via experimental data from Ellis et al. [37]. The parameters KTetR, KLacI, KATc and KIPTG are identified from Ellis et al. [37] (see Appendix A).

Parameter Description Value Units Ref.

KTetR TetR binding affinity 7.3093 M +KLacI LacI binding affinity 60.1405 M +KATc ATc–TetR dissociation rate 26.3236 ng/ml +KIPTG IPTG–LacI dissociation rate 0.0598 mM +nTetR Binding cooperativity between TetR and DNA 2 – [69]nLacI Binding cooperativity between LacI and DNA 2 – [70]nATc Binding cooperativity between ATc and TetR 4 – [69]nIPTG Binding cooperativity between IPTG and LacI 1 – [70]cyEGFP yEGFP degradation rate 1.925 � 10�3 min�1 [37]cTetR and cLacI TetR and LacI degradation rates 0.1386 min�1 [71]cCI CI degradation rate 0.042 min�1 [72]b mRNA degradation rate 0.288 min�1 [35,49]a Translation rate 24 min�1 [35,49]

C.-H. Wu et al. / Mathematical Biosciences 233 (2011) 111–125 119

3.2. Design of in silicoexamples based on promoter libraries

In this section, we provide two in silico examples to illustratethe design procedure of synthetic gene networks, and then an ade-quate promoter set is selected from the redefined promoter li-braries LibTetR and LibLacI in Table 1 to achieve the multiobjectiveH2/H1 reference tracking design. The redefined promoter librariesLibTetR and LibLacI are listed in Table 1, and all design parameters forsimulation are listed in Table 2.

Example 1. Consider the synthetic toggle switch shown in Fig. 2.The gene network will employ adequate promoters from TetR- andLacI-regulated promoter libraries (i.e., LibTetR and LibLacI), whichhave both 21 promoters T0–T20 and L0–L20 in Table 1. Underintrinsic parameter fluctuations and environmental disturbances,the dynamic system in Fig. 2 has been constructed in (6),where c1 ¼ fcr1 ; cs1g 2 Lk; c2 ¼ fcr2 ; cs2g 2 Tk; c3 ¼ fcr3 ; cs3g 2 Tk;

k ¼ 0; . . . ;20. The elements of parameter fluctuation matrices Mk,k = 1,2,3 in (6) are given as follows

Dcrk¼ 0:1� crk

; Dcsk¼ 0:1� csk

; k ¼ 1;2;3;Db ¼ 0:1� b; Da ¼ 0:1� a; Dc ¼ 0:1� c;

ð25Þ

i.e., the standard deviations of parameter fluctuations are allowed tobe 10% of their nominal values. Finally, the reference model to betracked by the stochastic synthetic gene network in Fig. 2 is given as

_XrðtÞ ¼ �0:0019XrðtÞ þ r;

YrðtÞ ¼ XrðtÞ;ð26Þ

where r = 60,000 if ATc induces or r = 2000 if IPTG induces (seeFig. 5). For simplicity, the stochastic synthetic gene network inFig. 2 is approximated by the T–S fuzzy model. We take 5 trian-gle-type membership functions for Rule 1 to Rule 5 (i.e., 25 local lin-earized dynamic systems) and the operating points for all states areall distributed from 0 to 1000. The parameters for simulation arelisted in Table 2. Then the augmented fuzzy system in (21) is ob-tained as follows:

dXðc; tÞ ¼X25

i¼1

hiðXÞ ðAiðcÞXðc; tÞþ vðtÞÞdtþX3

k¼1

MkBikðcÞXðc; tÞdxk

" #;

ð27Þ

where AiðcÞ and BikðcÞ are obtained by the fuzzy approximatingmethod and Mk is shown in (25). Assume the weighing matrixQ = 1, the covariance of initial value R0 ¼ Xð0ÞXTð0Þ withXð0Þ ¼ 103 � ½1 1 1 1 1 103 103�T and the desired H1 noise attenu-ation level qd is 0.1, then the LMIs-constrained optimization prob-lem in (22)–(24) is solved by selecting a promoter set from theredefined promoter libraries. With the help of the LMI toolbox in

Matlab [40], the adequate promoter set c = {c1,c2,c3} = {L9,T2,L8}from the corresponding promoter libraries can be selected so thatthe synthetic gene network in Fig. 2 can achieve multiobjectiveH2/H1 reference tracking in (22)–(24). For the convenience of sim-ulation, we consider the environmental disturbances v(t) = 10 � [n1,n2, . . .,n6]T, where ni, i = 1, . . .,6 are independent Gaussian whitenoises with zero mean and unit variance. In order to confirm themultiobjective reference tracking, the inducer ATc is added to in-duce the gene network from 80 h and is removed at 160 h, and thenthe inducer IPTG is added to the synthetic gene network from 160 hand is removed at 240 h. The simulation results for reference track-ing of the synthetic gene networks are shown in Fig. 5. Clearly, theoutput y(c, t) of the synthetic gene network can robustly and opti-mally track the desired reference yr(t) despite the intrinsic parame-ter fluctuations and environmental disturbances.

Example 2. Consider the repressilator shown in Fig. 3 [57–61]. Inthis gene circuit topology, the repressor protein LacI inhibits thetranscription of the repressor gene cI, whose protein product inturn inhibits the expression of the repressor gene tetR. Finally,TetR, the protein product of repressor gene tetR, inhibits lacI andyegfp expression. The negative feedback loop leads to temporaloscillations in the concentration of each component which can beseen from a simple model of transcriptional regulation. The con-centrations of mRNAs lacI, cI, tetR and yegfp are denoted byx1(c1, t), x2(c2, t), x3(c3, t) and x4(c4, t), respectively. In addition,X1(c1, t), X2(c2, t), X3(c3, t) and X4(c4, t) are concentrations of theircorresponding protein products. Therefore, the nonlinear stochas-tic model with the four promoters c1, c2, c3 and c4 to be specifiedunder intrinsic parameter fluctuations and environmental distur-bances is given as

dXðc; tÞ ¼ ðf ðX; c; r; tÞ þ vðtÞÞdt þP4k¼1

MkgkðX; ck; rk; tÞdxk;

yðC; tÞ ¼ X4ðc4; tÞ;ð28Þ

where

Xðc; tÞ ¼

x1ðc1; tÞX1ðc1; tÞx2ðc2; tÞX2ðc2; tÞx3ðc3; tÞX3ðc3; tÞx4ðc4; tÞX4ðc4; tÞ

266666666666664

377777777777775;

120 C.-H. Wu et al. / Mathematical Biosciences 233 (2011) 111–125

f ðX; c; r; tÞ ¼

pLibTetRðc1; r1Þ � bx1ðc1; r1Þ

ax1ðc1; tÞ � c1X1ðc1; tÞpLibLacI

ðc2; r2Þ � bx2ðc2; r2Þax2ðc2; tÞ � c2X2ðc2; tÞ

pLibCIðc3; r3Þ � bx3ðc3; r3Þ

ax3ðc3; tÞ � c3X3ðc3; tÞpLibTetR

ðc4; r4Þ � bx4ðc4; r4Þax4ðc4; tÞ � c4X4ðc4; tÞ

266666666666664

377777777777775;

vðtÞ ¼

v1ðtÞv2ðtÞv3ðtÞv4ðtÞv5ðtÞv6ðtÞv7ðtÞv8ðtÞ

266666666666664

377777777777775;

M1 ¼

Dcs1 Dcr1 �Db 00 0 Da �Dc1

0 0 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 0 0

266666666666664

377777777777775;

Fig. 5. Simulation of toggle switch. By solving the LMIs-constrained optimization problec = {c1,c2,c3} = {L9,T2,T8} is selected from the corresponding promoter libraries. The synthe160 h, and then is added with inducer IPTG from 160 h to 240 h. Obviously, the output

M2 ¼

0 0 0 0

0 0 0 0

Dcs2 Dcr2 �Db 0

0 0 Da �Dc2

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

26666666666666664

37777777777777775

;

M3 ¼

0 0 0 00 0 0 00 0 0 00 0 0 0

Dcs3 Dcr3 �Db 00 0 Da �Dc3

0 0 0 00 0 0 0

266666666666664

377777777777775;

M4 ¼

0 0 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 0 0

Dcs4 Dcr4 �Db 00 0 Da �Dc4

266666666666664

377777777777775;

m in (22)–(24) for the synthetic gene network in Fig. 2, an adequate promoter settic gene network is added with inducer ATc to induce the gene network from 80 h toY(c, t) can robustly track the desired reference output Yr(t).

C.-H. Wu et al. / Mathematical Biosciences 233 (2011) 111–125 121

g1ðX; c1; r1; tÞ ¼

HTetRðr1Þ1� HTetRðr1Þ

x1ðc1; tÞX1ðc1; tÞ

26664

37775; g2ðX; c2; r2; tÞ ¼

HLacIðr2Þ1� HLacIðr2Þ

x2ðc2; tÞX2ðc2; tÞ

26664

37775;

g3ðX; c3; r3; tÞ ¼

HCIðr3Þ1� HCIðr3Þ

x3ðc3; tÞX3ðc3; tÞ

26664

37775; g4ðX; c4; r4; tÞ ¼

HTetRðr4Þ1� HTetRðr4Þ

x4ðc4; tÞX4ðc4; tÞ

26664

37775

with r1 = X3(c3, t), r2 = X1(c1, t), r3 = X2(c2, t) and r4 = X3(c3, t).Suppose the perturbation elements Db; Da; Dc; Dcsk and

Dcrk ; k ¼ 1; . . . ;4 of parameter fluctuation matrices Mk in (28) tobe tolerated are listed as follows

Dcrk¼ 0:1� crk

; Dcsk¼ 0:1� csk

; k ¼ 1; . . . ;4;Db ¼ 0:1� b; Da ¼ 0:1� a; Dc ¼ 0:1� c:

ð29Þ

Since there is currently no CI-regulated promoter library, thepromoter regulation function of c3 takes the following form [62]:

pCIðc3; r3Þ ¼ 150=ð1þ ðr3=20Þ2Þ: ð30Þ

The promoters c1, c2 and c4 are to be selected from promotersTk,Lk and Tk, k = 0,1, . . .,20 in Table 1 in promoter libraries LibTetR,LibLacI and LibTetR, respectively.

Our design purpose is to engineer a synthetic gene networkwhose behavior can achieve the multiobjective H2/H1 referencetracking for the following reference oscillation system with thedesired amplitude 400,000 (M) and frequency 8 � 10�3 (min�1) as

0 2.5 5 7.5 10 1

1.5

2

2.5

3

x 106

Tim

Con

cent

ratio

n

The adequate promoter set

Fig. 6. Simulation of repressilator. The adequate promoter set c = {c1,c2,c4} = {T11,L15,T8}in (22)–(24) for the synthetic gene network in Fig. 3. The desired output Y(c, t) of the gendesired behavior Yr(t) generated by the reference model. Our proposed design methodselecting an adequate promoter set from the existing promoter libraries.

_Xr1ðtÞ_Xr2ðtÞ_Xr3ðtÞ

264

375 ¼

0 0:029 �0:029�0:029 0 0:0290:029 �0:029 0

264

375

Xr1ðtÞXr2ðtÞXr3ðtÞ

264

375: ð31Þ

Take triangle-type membership functions for Rules 1–5 and theoperating points for all states are distributed from 0 to 1000. Thenthe augmented system combining the T–S fuzzy model with thereference model in (31) is obtained as

dXðc; tÞ ¼X125

i¼1

hiðXÞ ðAiðcÞXðc; tÞ þ vðtÞÞdtþX3

k¼1

MkBikðcÞXðc; tÞdxk

" #;

ð32Þwhere AiðcÞ and BikðcÞ are obtained by the fuzzy approximatingmethod and Mk is shown in (28). Assume the weighting matrixQ = 1, the covariance of initial value R0 ¼ Xð0ÞXTð0Þ withXð0Þ ¼ 104 � ½1 1 1 1 1 1 1 1 260 200 200�T and the desired H1noise attenuation level qd is 0.1. Then the LMIs-constrained optimi-zation problem in (22)–(24) is solved by selecting a promoter setfrom the redefined promoter libraries in Table 1. The adequate pro-moter set c = {c1,c2,c4} = {T11,L15,T8} can be obtained by solving theLMIs-constrained optimization problem in (22)–(24) with the helpof the LMI toolbox in Matlab, so that the synthetic gene networkin Fig. 3 can achieve multiobjective H2/H1 reference tracking de-sign. For convenience of simulation, we consider the environmentaldisturbances v(t) = 10 � [n1,n2 , . . .,n8]T, in which ni(t), i = 1, . . .,8 areindependent Gaussian white noises with zero mean and unit vari-ance. The simulation results in Fig. 6 show that the gene networkcan robustly track the desired behavior generated by the referencemodel in (31) despite intrinsic parameter fluctuations and environ-mental disturbances.

2.5 15 17.5 20 22.5 25 e (h)

c=(c1, c2, c4)=(T11, L15, T8)

y(c, t)yr(t)

is obtained via solving the multiobjective H2/H1 reference tracking design probleme network employs the adequate promoter set c to robustly and optimally track the

could provide a genetic oscillator with the prescribed amplitude and period via

122 C.-H. Wu et al. / Mathematical Biosciences 233 (2011) 111–125

4. Discussions and conclusions

Nowadays, promoter library studies have significant progress inquantitative measurements. Based on the knowledge about the ef-fect of the promoter architecture on transcriptional activity[45,47,63,64] and mutation-selection techniques [46,65,66,45],the promoter libraries can be widely built. Hence the proposedmethod can be employed to redefine these promoter libraries to ex-tend a more complex synthetic gene network. For example, thenext-generation gene networks, such as tunable filters, analog-to-digital and digital-to-analog converters, adaptive learning networksand protein-based computational circuits, have been proposed toenable the construction of more complex biological systems basedon diverse biological parts [31]. In the future, a much larger andmore complex synthetic gene network can be represented by acomplex nonlinear stochastic system in (8) and the desired behav-ior of a synthetic gene network can be described by the referencemodel in (9). Then, based on the fuzzy interpolation method in(21), the multiobjective H2/H1 reference tracking design can be effi-ciently solved by Proposition3. By the conventional selecting algo-rithm, an adequate promoter set can be efficiently searched frompromoter libraries to achieve the minimum value in (22). Whenthe number and content of promoter libraries become larger inthe future, the library-based search method can be efficiently em-ployed in the systematic design of synthetic biology [67]. Therefore,one of the future challenges in synthetic biology is to engineer someapplication-orientated systems. After providing a novel and func-tional network, the proposed systematic method can be employedto implement the synthetic gene network with the desired functionby selecting an adequate promoter set. By the proposed method, theengineering efforts are needed to focus on creating a systems-levelnetwork to promote the second wave of synthetic biology [1]. If theBioBrick parts can be characterized by their promoter activities bythe proposed method of redefined promoter library in AppendixA, then the BioBricks can be utilized as possible resources of pro-moter libraries for synthetic gene networks. In this situation, theproposed systematic method based on well-characterized BioBrickparts will surely shape the future design in synthetic biology.

In this study, we propose a multiobjective H2/H1 tracking de-sign for a stochastic synthetic gene network to achieve H1robustnoise filtering and H2 optimal reference tracking simultaneouslyby selecting an adequate promoter set from the existing promoterlibraries. In order to efficiently select promoters from the corre-sponding promoter libraries, the conventional promoter librariesare redefined according to their estimated promoter activitiesfirstly so that the promoters could be suitable for the dynamic reg-ulation of gene circuits. The multiobjective H2/H1 tracking designproblem is also transformed into an HJI-constrained optimizationproblem. Then the T–S fuzzy method is employed to interpolateseveral local linear stochastic systems for approximating the non-linear stochastic gene network in vivo. Thus the HJI-constrainedoptimization problem is transformed to that of selecting a pro-moter set to solve an LMIs-constrained optimization problem,which can be efficiently solved using the LMI toolbox in Matlab.For the two design examples in silico, robust and optimal referencetracking of synthetic gene networks can be guaranteed simulta-neously despite the intrinsic parameter fluctuations and environ-mental disturbances in vivo. Hence, the proposed systematicmethod will help synthetic biologists to simplify the design proce-dure of a synthetic gene network with a desired behavior. A largenumber of trail-and-error experimental processes in selecting anadequate promoter set from promoter libraries can be avoided.Therefore, the proposed systematic method will accelerate the pro-gress of synthetic biology, especially as the available biologicalpromoter libraries increase rapidly.

Appendix A. The redefined promoter libraries based onestimated promoter activities

Here, we provide a method to redefine promoter libraries basedon the estimated promoter activities from experimental data on thecurrent promoter libraries in Ellis [37]. Although the experimentaldata of TetR- and LacI-regulated promoter libraries have alreadybeen reported [37], it is necessary to have an efficient method ofselecting promoters from promoter libraries to engineer a dynamicsynthetic gene network in (2) and (3). Hence the promoter librariesneed to be reconstructed by the dynamic model of promoter-regulation gene circuit in (2) and (3). Since TetR- and LacI-regulatedpromoter libraries are characterized similarly, only the redefinitionof TetR-regulated promoter library needs to be described.

A simple scheme of Ellis’ experimental model is shown in Fig. 1.The constitutive expression of TetR represses yegfp by inhibitingthe promoter c (TetR-regulated promoter) and then ensures lowbasal levels of yEGFP. On the other hand, TetR is repressed firstby the saturating concentration of the inducer ATc and then bythe promoter c activities so the gene yegfp transcripts and trans-lates to ensure the high levels of yEGFP. Therefore, the maximumand minimum values of yEGFP are measured with and withoutthe saturating concentrations of the inducer ATc, respectively.Based on these experimental data, we estimate the correspondingpromoter activities and redefine these two promoter libraries bytheir promoter activities. The experimental model for estimatingpromoter activity in Fig. 1 can be described as the following dy-namic model [35,50]

_xðc; tÞ ¼ pTetRðc; rÞ � bxðc; tÞ;_Xðc; tÞ ¼ axðc; tÞ � cyEGFPXðc; tÞ;

(

yðc; tÞ ¼ gXðc; tÞ;; ðA1Þ

where y(c, t) denotes the fluorescence of the protein yEGFP. g is theratio of fluorescence to the concentration of yEGFP [50]. The pro-moter regulation function pTetR(c,r) in (2) and (A1) has the maxi-mum value cs and minimum value cr with and without saturatingconcentrations of the inducer ATc, respectively. This is because,with the saturating concentration of inducer ATc (i.e., the value ofATc is much larger than KATc), the repressor activity r in (1) becomesmuch smaller than KTetR in (2) and the maximum ofpTetRðc; rÞ;pmax

TetRðc; rÞ � cr þ cs � cr ¼ cs in (2). Furthermore, in the casewithout saturation of the inducer ATc, r in (1) becomes much largerthan KTetR in (2) so that the minimum of pTetRðc; rÞ;pmin

TetRðc; rÞ � cr in(2). Therefore, in the case of saturation of inducer ATc, the steadystate of y(c, t) will have a maximum value, i.e.,

ymaxðc; tÞ �ab

gcyEGFP

pmaxTetRðc; rÞ ¼

ab

gcyEGFP

cs: ðA2Þ

Similarly, in the case without the inducer ATc, the steady stateof y(c, t) will have a minimum value, i.e.,

yminðc; tÞ �ab

gcyEGFP

pminTetRðc; rÞ ¼

ab

gcyEGFP

cr : ðA3Þ

Since the maximum and minimum values of ymax(c, t) andymin(c, t) are both at steady states of experimental data in theTetR-regulated promoter library [37] and the parameters a,b, g,cy-

EGFP can also be obtained from the experimental data of TetR-regu-lated promoter library [37], the promoter activities cr and cs ofevery promoter in the promoter libraries can be estimated by theseexperimental data from (A2) and (A3), as listed in Table 1. Otherparameters to be needed for the above promoter activity estima-tion process are identified by least square parameter identificationmethod in the following.

The repressor activity r in (1) becomes much larger or smallerthan KTetR with or without the inducer ATc and will lead to the

C.-H. Wu et al. / Mathematical Biosciences 233 (2011) 111–125 123

promoter regulation functions pminTetRðc; rÞ and pmax

TetRðc; rÞ, respectively.We need to identify KTetR and KATc to determine the promoter activ-ity of promoter regulation function pTetR(c,r). By employing Ellis’experimental data, in which the inducer ATc is the input and thenormalized fluorescence is the output, concentrations of the indu-cer ATc in (1) and the promoter regulation function in (2) are usedto identify two parameters KTetR and KATc by the least squareparameter identification method. The identified results are listedin Table 2. By a similar process, the promoter activity c = {cr,cs} ofevery promoter in the LacI-regulated promoter library can be ob-tained in Table 1 and two parameters KLacI and KIPTG can be identi-fied by the least square parameter estimated method in Table 2.More detailed experimental data are presented in Ellis [37].

Appendix B. Proof of Proposition 1

Let us choose a Lyapunov function VðXðc; tÞÞ > 0 for Xðc; tÞ – 0with V(0) = 0 for the synthetic gene network in (13). Then weobtain

EZ tf

0YTðc; tÞQ Yðc; tÞdt ¼ EVðXðc;0ÞÞ � EVðXðc; tf ÞÞ

þ EZ tf

0YTðc; tÞQ Yðc; tÞ þ dVðXðc; tÞÞ

dt

" #dt:

ðB1ÞBy the Ito formula [53] and E(dxk/dt) = 0,k = 1,. . .,n, we obtain

[43]

EdVðXðc; tÞÞ

dt¼ E

@VðXðc; tÞÞ@Xðc; tÞ

!T

f ðX; c; r; tÞ

þ E@VðXðc; tÞÞ@Xðc; tÞ

!T

vðtÞ

þ 12

EXn

k¼1

gTkðX; ck; rk; tÞMT

k

� @2VðXðc; tÞÞ@X2ðc; tÞ

MkgkðX; ck; rk; tÞ: ðB2Þ

Substituting (B2) into (B1), we get

EZ tf

0YTðc; tÞQ Yðc; tÞdt

¼ EVðXðc;0ÞÞ � EVðXðc; tf ÞÞ þ EZ tf

0XTðc; tÞHT Q HXðc; tÞh

þ @VðXðc; tÞÞ@Xðc; tÞ

!T

f ðX; c; r; tÞ þ @VðXðc; tÞÞ@Xðc; tÞ

!T

vðtÞ

þ12

Xn

k¼1

gTkðX; ck; rk; tÞMT

k@2VðXðc; tÞÞ@X2ðc; tÞ

MkgkðX; ck; rk; tÞ#

dt: ðB3Þ

By the fact [53] that for any vector with the desired noise atten-uation level qd aT bþ bT a 6 q�2

d aT aþ q2dbT b, (B3) is derived as

EZ tf

0YTðc; tÞQ Yðc; tÞdt

6 EVðXðc; 0ÞÞ þ EZ tf

0XTðc; tÞHT Q HXðc; tÞh

þ @VðXðc; tÞÞ@X

!T

f ðX; c; r; tÞ þ 14q2

d

@VðXðc; tÞÞ@X

!T@VðXðc; tÞÞ

@X

!

þ12

Xn

k¼1

gTkðX; ck; rk; tÞMT

k@2VðXðc; tÞÞ

@x2 MkgkðX; ck; rk; tÞ

þ q2dvTðtÞvðtÞ

�dt: ðB4Þ

If the inequality in (16) holds, then we have

EZ tf

0YTðc; tÞQ Yðc; tÞdt 6 EVðXðc;0ÞÞ þ E

Z tf

0q2

dvðtÞvðtÞdt; ðB5Þ

where qd is the desired noise attenuation level. If the initialXðc; 0Þ ¼ 0, then (B5) will be reduced to the H1 noise attenuation le-vel qd in (14). Hence the H1 noise attenuation level qd is guaranteedif the inequality in (16) holds.

Appendix C. Proof of Proposition 2

Let us choose a Lyapunov function VðXðc; tÞÞ > 0 for Xðc;0Þ– 0with V(0) = 0 for the synthetic gene network in (13). Following(B1) and (B2) and by the fact

@VðXðc; tÞÞ@Xðc; tÞ

!T

vðtÞ 6 14

@VðXðc; tÞÞ@Xðc; tÞ

!T@VðXðc; tÞÞ@Xðc; tÞ

!þ vTðtÞvðtÞ;

we get

EZ tf

0YTðc; tÞQ Yðc; tÞdt

6 EVðXðc;0ÞÞ þ EZ tf

0XTðc; tÞHT QHXðc; tÞh

þ @VðXðc; tÞÞ@X

!T

f ðX; c; r; tÞ þ 14

@VðXðc; tÞÞ@X

!T@VðXðc; tÞÞ

@X

!

þ12

Xn

k¼1

gTkðX; ck; rk; tÞMT

k@2VðXðc; tÞÞ

@x2

�MkgkðX; ck; rk; tÞ þ vTðtÞvðtÞ�

dt: ðC1Þ

Therefore, if the HJI in (17) holds, we have

EZ tf

0YTðc; tÞQ Yðc; tÞdt 6 EVðXðc;0ÞÞ þ E

Z tf

0vTðtÞvðtÞdt: ðC2Þ

In other words, EVðXðc;0ÞÞ þ ER tf

0 vTðtÞvðtÞdt is the upper boundof the tracking error energy E

R tf0 Yðc; tÞQ Yðc; tÞdt. Since the subop-

timal tracking design is to minimize its upper bound instead of thetracking error energy, the suboptimal tracking design for a syn-thetic gene network is to select a promoter set to minimizeEVðXðc;0ÞÞ þ E

R tf0 vTðtÞvðtÞdt. Since the environmental disturbance

term ER tf

0 vTðtÞvðtÞdt is independent on the choice of promoter set.Therefore the suboptimal tracking design of a synthetic gene net-work is reduced to min c2Libj

j¼1;...;m

EVðXðc;0ÞÞ subject to HJI in (17).

Appendix D. Proof of Proposition 3

Let us denote a Lyapunov energy function VðXðc; tÞÞ ¼ XTðc; tÞPXðc; tÞ > 0 for Xðc;0Þ – 0 with V(0) = 0. By using the T–S fuzzyapproximation method, we have

f ðX; c; r; tÞ ¼XL

i¼1

hiðXiðc; tÞÞAiðcÞXðc; tÞ ðD1Þ

and

gkðX; ck; rk; tÞ ¼XL

i¼1

hiðXiðc; tÞÞBiðcÞXðc; tÞ: ðD2Þ

Substituting (D1) and (D2) into HJIs in (16) and (17), we get

XL

i¼1

hiðXiðc; tÞÞ XTðc; tÞATi ðcÞPXðc; tÞ þ XTðc; tÞPAiðcÞXðc; tÞ

n

þXn

k¼1

XTðc; tÞBTi ðcÞMT

k PMkBiðcÞXðc; tÞ

þ 1q2

d

XTðc; tÞPPXðc; tÞþXTðc; tÞHT Q HXðc; tÞo< 0 ðD3Þ

124 C.-H. Wu et al. / Mathematical Biosciences 233 (2011) 111–125

and

XL

i¼1

hiðXiðc; tÞÞ XTðc; tÞATi ðcÞPXðc; tÞ þ XTðc; tÞPAiðcÞXðc; tÞ

n

þXn

k¼1

XTðc; tÞBTi ðcÞMT

k PMkBiðcÞXðc; tÞ þ XTðc; tÞPPXðc; tÞ

þ XTðc; tÞHT Q HXðc; tÞo< 0 ðD4Þ

or equivalently [68]

ATi ðcÞP þ PAiðcÞ þ

Xn

k¼1

BTi ðcÞMT

k PMkBiðcÞ þ1q2

d

PP þ HT Q H < 0 ðD5Þ

and

ATi ðcÞP þ PAiðcÞ þ

Xn

k¼1

BTi ðcÞMT

k PMkBiðcÞ þ PP þ HT Q H < 0: ðD6Þ

By Schur complements [68], (D5) and (D6) are equivalent to theLMIs in (24) and (23), respectively. Finally,

EfVðXðc;0ÞÞg ¼ EfXTðc;0ÞPXðc;0Þg ¼ TrðEfPXðc;0ÞXTðc;0ÞgÞ ¼ TrPR0;

where R0 ¼ EfXðc;0ÞXTðc; 0Þg.Therefore, the HJI-constrained optimization in (18) is equiva-

lent to the LMIs-constrained optimization problem in (22)–(24).

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