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Mathematical Biosciences 230 (2011) 23–36
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Mathematical Biosciences
journal homepage: www.elsevier .com/locate /mbs
Robust model matching design methodology for a stochastic synthetic gene network
Bor-Sen Chen a,⇑, Chia-Hung Chang a, Yu-Chao Wang a, Chih-Hung Wu a, Hsiao-Ching Lee b
a Laboratory of Control and Systems Biology, Department of Electrical Engineering, National Tsing Hua University, Hsinchu 30013, Taiwanb Department of Biological Science and Technology, National Chiao Tung University, Hsinchu 30068, Taiwan
a r t i c l e i n f o a b s t r a c t
Article history:Received 23 November 2009Received in revised form 22 December 2010Accepted 29 December 2010Available online 6 January 2011
Keywords:Robust model matching designStochastic synthetic gene networkIntrinsic parameter fluctuationsExternal disturbancesGlobal linearizationLMI
0025-5564/$ - see front matter � 2010 Elsevier Inc. Adoi:10.1016/j.mbs.2010.12.007
⇑ Corresponding author. Tel.: +886 3 5731155; fax:E-mail address: [email protected] (B.-S. Che
Synthetic biology has shown its potential and promising applications in the last decade. However, manysynthetic gene networks cannot work properly and maintain their desired behaviors due to intrinsicparameter variations and extrinsic disturbances. In this study, the intrinsic parameter uncertaintiesand external disturbances are modeled in a non-linear stochastic gene network to mimic the real envi-ronment in the host cell. Then a non-linear stochastic robust matching design methodology is introducedto withstand the intrinsic parameter fluctuations and to attenuate the extrinsic disturbances in order toachieve a desired reference matching purpose. To avoid solving the Hamilton–Jacobi inequality (HJI) inthe non-linear stochastic robust matching design, global linearization technique is used to simplify thedesign procedure by solving a set of linear matrix inequalities (LMIs). As a result, the proposed matchingdesign methodology of the robust synthetic gene network can be efficiently designed with the help of LMItoolbox in Matlab. Finally, two in silico design examples of the robust synthetic gene network are given toillustrate the design procedure and to confirm the robust model matching performance to achieve thedesired behavior in spite of stochastic parameter fluctuations and environmental disturbances in the hostcell.
� 2010 Elsevier Inc. All rights reserved.
1. Introduction
Synthetic biology is essentially an engineering discipline todesign and to construct simplified biological systems that providea desired behavior [1–6]. The ability to design and construct bio-logical systems can also offer scientists a useful way to test theirunderstanding of the complex functional networks of genes andbiomolecules that mediate life processes [7]. Therefore, syntheticbiology is foreseen to have important applications in biotechnol-ogy and medicine. Other practical applications such as biosensing,the production of biofuels and novel biomaterials are also thoughtto be accomplished by synthetic biology [8]. At present, the con-struction of biological networks of inter-regulating genes, so-calledgene networks, has demonstrated the feasibility of syntheticbiology [9]. A bottom–up approach, which couples simple, well-characterized modules into more complex networks with behav-iors that can be predicted from that of the individual components,has also shown to be achievable in synthetic gene network con-struction [10]. However, the development of gene networks is stilldifficult, and most newly created networks are non-functioningand need tuning [11]. One important reason for this difficulty isthat a synthetic gene network suffers from intrinsic parameter
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+886 3 5715971.n).
variations due to thermal fluctuations, transcription factor (TF)bindings, gene expression noises, mutations, and extrinsic distur-bances caused by changing extracellular environments [2,11,12].These uncertainties and disturbances come from current biotech-nological limitations and also from the fluctuations of intra- andextra-cellular environments in the host cell [11,13]. Furthermore,current approaches of gene network construction typically use asmall set of components taken from nature systems, which arethen assembled and tested in vivo, often without guidance from apriori mathematical modeling for design and assembly [12,14]. Inthis case, the networks rarely behave as intended the first time.They usually need to be resolved over months of iterative retrofit-ting, often by fine-tuning imperfect parts by mutation, identifyingalternative parts or adding extra features to counterbalance theproblem [12,14]. Because of these limitations that hamper thedesign of synthetic gene networks, the stability robustness againstparameter fluctuation and the filtering ability to attenuate theenvironmental noises, which have been widely discussed fromthe systems biology perspective [13,15–21], are taken into consid-eration for synthetic gene network design.
Recently, Batt et al. proposed an approach to analyze a class ofuncertain piecewise-multiaffine differential equation model of asynthetic gene network [11]. This modeling framework is adaptedto the experimental data and allows the development for solvingrobustness analysis and tuning problems under parameter varia-tions. This approach can find parameter sets for which the system
24 B.-S. Chen et al. / Mathematical Biosciences 230 (2011) 23–36
presents a particular behavior. However, the external disturbancesare not included in their design procedure. Kuepfer et al. developedan approach based on semi-finite programming for partitioning theparameter space of polynomial differential equation models intothe so-called feasible and infeasible regions [22]. In this approach,‘feasible’ simply refers to the existence of a steady state of the sys-tem. Chen et al. developed a circuit design scheme to implementsome gene circuits into an existing gene network in an organismto improve the stability robustness to tolerate parameter varia-tions [23,24]. However, this gene circuit design is different fromthe synthetic gene network design. The conventional gene circuitdesign implements some gene circuits to an existing gene networkto improve its function, but the synthetic gene network designs anew artificial gene network, which has to be inserted into a hostcell (for example, E. coli) to perform its function. In the host cell,the synthetic gene network will suffer from intrinsic parameterfluctuations and external disturbances, which will destroy thefunction of the synthetic gene network if it is not robustly de-signed. More recently, a robust synthetic biology design withmolecular noises was developed to achieve a desired steady state[25]. A robust synthetic biology design was also proposed toachieve a desired steady state despite parameter variations andexternal disturbances via engineering design specifications [26].These design schemes are of the robust stabilization scheme to adesired steady state (or an equilibrium point). There are, however,many other behaviors of interest in synthetic gene networkdesigns to which the stabilization scheme behavior cannot beapplied. In fact, other behaviors of interest such as oscillations ortransient behaviors are more complex than the steady state behav-iors. How to engineer a synthetic gene network with desiredoscillations or transient behaviors is a model matching (tracking)design problem. In this tracking design case, the desired behaviorshould be generated by a reference dynamic model _xr ¼ frðxrÞþhrðxrÞur and then the parameters of the synthetic gene networkshould be designed so that the time responses of the syntheticgene network can match the desired behavior xr of reference modeldespite the parameter fluctuations and external disturbances inthe host cell. More effort is needed for the robust model matchingdesign than the conventional robust stabilization design of syn-thetic gene network with a desired steady state. Based on a fuzzyapproximation method, a suboptimal H2 tracking design methodis proposed for synthetic gene network [27].
In this study, the intrinsic parameter variations and externaldisturbances are modeled into the non-linear stochastic equationof synthetic gene networks to mimic the real environments inthe host cell [13]. The desired behavior is simulated by the input/output response of a reference model that has been widely usedto generate a desired output response for model reference controldesign [28]. In this situation, the design problem of a syntheticgene network becomes how to specify the kinetic parametersand degradation rates within allowable ranges for the syntheticgene network to robustly track the desired response of the refer-ence model in spite of intrinsic parameter variations and externaldisturbances in the host cell (see Fig. 1). A stochastic H1 robustmodel matching design methodology is introduced for the syn-thetic gene network to tolerate intrinsic parameter perturbationsand to attenuate the extrinsic disturbances on the matching errorto the reference model response to achieve robust tracking of thedesired behavior (see Fig. 1).
In general, it is still very difficult to solve the so-called non-linear stochastic model matching design problems. It is necessaryto solve a non-linear Hamilton–Jacobi inequality (HJI) for the robustmatching design of non-linear stochastic synthetic gene network,and there are currently no good analytic or numerical methodsto solve the HJI [29,30]. In order to avoid solving the HJI for thesynthetic gene network design, the global linearization technique
[31] is employed to simplify the model matching design procedureof the robust synthetic gene regulatory network by solving a set oflinear matrix inequalities (LMIs) instead [13,32]. In this case, therobust model matching problem for synthetic gene networks canbe efficiently solved with the help of LMI toolbox in Matlab [33].Finally, two in silico design examples of synthetic gene networkare given to illustrate the design procedure and to confirm therobust model matching performance to achieve a desired input/output behavior under stochastic parameter fluctuations andenvironmental disturbances in the host cell.
2. Design specifications and problem description
Before the introduction of a general design methodology of therobust model matching synthetic gene network, an example is pro-vided for the convenience of illustrating the design specificationsand problem formulation. We consider a cascade of transcriptionalinhibition built in E. coli by Hooshangi et al. [34]. The syntheticgene network is represented in Fig. 2. It is made of four genes: tetR,lacI, cI and eyfp that codes respectively for three repressor proteins,TetR, LacI, CI, and the fluorescent protein EYFP (enhanced yellowfluorescent protein). The system can be controlled by the additionor removal of a chemical anhydrotetracycline (aTc), which can in-hibit protein TetR and is considered as the control input u(t).
We denote the concentrations of these proteins as
x1 x2 x3 x4½ �T ¼ xtetR xlacI xcI xeyfp½ �T ; ð1Þ
where xtetR, xlacI, xcI, xeyfp denote the concentrations of proteins TetR,LacI, CI, EYFP, respectively and u denotes the constant input of aTc.Then the dynamic model of synthetic transcriptional cascade inFig. 2 can be represented by the following differential equations[11]
_x1 ¼ k10 � r1x1;
_x2 ¼ k20 þ k2 d2ðx1Þ þ aðuÞ � d2ðx1ÞaðuÞð Þ � r2x2;
_x3 ¼ k30 þ k3d3ðx2Þ � r3x3;
_x4 ¼ k40 þ k4d4ðx3Þ � r4x4;
ð2Þ
where the constants k10, k20, k30, k40 are the basal levels of the cor-responding production rates of proteins due to the regulations otherthan upstream regulatory genes, which are not easy to measure inthe host cell and can be considered as system uncertainties in ourdesign procedure; k2, k3, k4 are the kinetic parameters which repre-sent regulation abilities of the gene network. The magnitudes of ki,i = 2, . . . ,4 are proportional to the lengths or affinities of the tran-scription factor binding sites inserted in the promoter region of tar-get genes by using a high-efficiency phage-based homologousrecombination system, called recombineering [35,36]. The parame-ters r1, r2, r3, r4 are the degradation rates of the corresponding pro-teins in the host cell. These parameters (ki and ri) can be designed bythe synthetic biological designers; d2(x1), d3(x2) and d4(x3) are Hill(or sigmoid) functions (see Fig. 3) of inhibition regulation on thegene expression by the corresponding proteins TetR, LacI, CI anda(u) for activation regulation by chemical aTc.
In the synthetic gene network (2), suppose ki and ri suffer fromparameter perturbations, due to gene expression disturbances inTF bindings, transcription and translation processes, alternativesplicing, mutation, etc. [37]. Based on our previous work [13], theintrinsic fluctuations can be represented as the following additivenoises
r1 ! r1 þ Dr1n1;
k2 ! k2 þ Dk2n2; r2 ! r2 þ Dr2n2;
k3 ! k3 þ Dk3n3; r3 ! r3 þ Dr3n3;
k4 ! k4 þ Dk4n4; r4 ! r4 þ Dr4n4;
ð3Þ
Reference model( ) ( )r r r r r rx f x h x u
Reference input ( )ru t
Synthetic gene networkwith intrinsic parameter fluctuations
1( , , ) ( , , )
m
i ii
x f x k r v dt g x k r dw
External disturbances ( )v t
The desired output ( )rx t
System output ( )x t
Tracking error
( )x t
Fig. 1. System block description of robust model matching design for a synthetic gene network. The desired behavior is simulated by the input/output response of a referencemodel. The synthetic gene network is designed to match the desired output response of the reference model to achieve the desired behavior xr in spite of externaldisturbances and intrinsic parameter fluctuations. Here, ur(t) is the reference input, xr(t) is the desired state vector to be matched, and x(t) is the state vector of the syntheticgene network. v0(t) are the stochastic external disturbances and v = v0 + k0 denote the total external disturbances and uncertain basal values.
tetR lacI cI eyfpTetR LacI CI EYFP
Fig. 2. The 4-gene synthetic transcriptional cascade. The protein TetR represses gene lacI. The protein LacI represses gene cI, and the protein CI represses gene eyfp. aTccontrols the repression of TetR and is considered as system input.
B.-S. Chen et al. / Mathematical Biosciences 230 (2011) 23–36 25
where Dkini and Drini denotes the additive parameter noises, inwhich Dki and Dri denote the amplitudes of the kinetic parametervariations; ni is a white noise with zero mean and unit variance,i.e., Dki and Dri denote the amplitudes of the corresponding para-metric variations and ni absorbs the stochastic property of intrinsicparametric variations. n1, n2, n3 and n4 are independent whitenoises to indicate that there are four independent stochastic sourcesof random parameter fluctuations over time, whose covariances aregiven as
CovðDkiniðtÞ;DkiniðsÞÞ ¼ Dk2i dt;s;
CovðDriniðtÞ;DriniðsÞÞ ¼ Dr2i dt;s;
CovðDkiniðtÞ;DriniðsÞÞ ¼ DkiDridt;s
ð4Þ
0 5 10 15 200
0.2
0.4
0.6
0.8
1
Repressor concentration TetR
Reg
ulat
ion
activ
ity
0 5 10 150
0.2
0.4
0.6
0.8
1
Repressor concentration CI
Reg
ulat
ion
activ
ity
d2(x1 )
d4(x3 )
Fig. 3. Hill functions (sigmoid function) as regulatory functions of the gene expressioregulatory activity. TetR, LacI and CI are repressors, whereas aTc acts as an activator.
in which dt,s denotes the delta function, i.e., dt,s = 1 if t = s and dt,s = 0if t – s, i.e., Dki and Dri denote the corresponding standard deviationsof stochastic kinetic parameter fluctuations Dkini and Drini respec-tively. Further, the synthetic gene network (2) is also affected bythe environmental disturbances from the cellular context and inter-actions with other networks in the host cell. In this situation, the per-turbative synthetic gene network is represented as follows [13]
_x1 ¼ k10 � r1x1 � Dr1n1x1 þ v 01;_x2 ¼ k20 þ k2 d2ðx1Þ þ aðuÞ � d2ðx1ÞaðuÞð Þ � r2x2
þ Dk2ðd2ðx1Þ þ aðuÞ � d2ðx1ÞaðuÞÞ � Dr2x2ð Þn2 þ v 02;_x3 ¼ k30 þ k3d3ðx2Þ � r3x3 þ ðDk3d3ðx2Þ � Dr3x3Þn3 þ v 03;_x4 ¼ k40 þ k4d4ðx3Þ � r4x4 þ ðDk4d4ðx3Þ � Dr4x4Þn4 þ v 04;
ð5Þ
0 5 100
0.2
0.4
0.6
0.8
1
Repressor concentration LacI
Reg
ulat
ion
activ
ity
0 100 200 3000
0.2
0.4
0.6
0.8
1
Activator concentration aTc
Reg
ulat
ion
activ
ity
d3(x2 )
a(u)
n. The Hill functions indicate the relation between the regulatory protein and its
26 B.-S. Chen et al. / Mathematical Biosciences 230 (2011) 23–36
where v 01, v 02, v 03 and v 04 denote the corresponding stochastic exter-nal disturbances, whose statistics are assumed to be unavailable ingeneral, i.e., a real synthetic gene network within the host cell mustsuffer from intrinsic stochastic parameter fluctuations and environ-mental disturbances. If a synthetic gene network wants to functionproperly to achieve its design goal, these parameter fluctuationsand external disturbances should be overcome and the eliminationof their effects should be considered in the design procedure of syn-thetic gene network.
For the convenience of analysis and design, the stochastic syn-thetic gene regulatory network of (5) in the host cell can be repre-sented by the following equivalent Ito stochastic differentialequations [29,30]
dx1 ¼ ð�r1x1 þ v1Þdt þ ð�Dr1x1Þdw1;
dx2 ¼ k2ðd2ðx1Þ þ aðuÞ � d2ðx1ÞaðuÞÞ � r2x2 þ v2ð Þdtþ Dk2ðd2ðx1Þ þ aðuÞ � d2ðx1ÞaðuÞÞ � Dr2x2ð Þdw2
dx3 ¼ ðk3d3ðx2Þ � r3x3 þ v3Þdt þ ðDk3d3ðx2Þ � Dr3x3Þdw3;
dx4 ¼ ðk4d4ðx3Þ � r4x4 þ v4Þdt þ ðDk4d4ðx3Þ � Dr4x4Þdw4;
ð6Þ
where v i ¼ v 0i þ ki0 denote the total external disturbances anduncertain basal value of the ith gene because the basal levels ki0
are always unavailable and maybe fluctuant. wi(t) is a standardWiener process or Brownian motion with dwi(t) = ni(t)dt. The abovestochastic gene network could be represented by
dx1
dx2
dx3
dx4
26664
37775¼
�r1x1
k2 d2ðx1Þþ aðuÞ�d2ðx1ÞaðuÞð Þ� r2x2
k3d3ðx2Þ� r3x3
k4d4ðx3Þ� r4x4
26664
37775þ
v1
v2
v3
v4
26664
37775
0BBB@
1CCCAdt
þ
�Dr1x1
000
26664
37775dw1þ
0Dk2ðd2ðx1ÞþaðuÞ�d2ðx1ÞaðuÞÞ�Dr2x2
00
26664
37775dw2
þ
00
Dk3d3ðx2Þ�Dr3x3
0
26664
37775dw3þ
000
Dk4d4ðx3Þ�Dr4x4
26664
37775dw4:
If a synthetic gene regulatory network consists of n genes, amore general perturbative synthetic network can be representedby the following stochastic differential equation
dx ¼ ðf ðx; k; rÞ þ hðxÞvÞdt þXm
i¼1
giðx;Dki;DriÞdwi; xð0Þ ¼ x0; ð7Þ
where x = [x1, . . . ,xn]T denotes a vector of protein concentrations of ngenes; v = [v1, . . . ,vn]T denotes the total external disturbances of thesynthetic gene network; k = [k1, . . . ,kn] and r = [r1, . . . , rn] denote thekinetic parameters and degradation rates to be specified by the de-signer; h(x) denotes the non-linear coupling between the externaldisturbances and the synthetic gene network. The term
Pmi¼1gi
ðx;Dki;DriÞdwi denotes the stochastic parameter fluctuations dueto m random sources. Obviously, a synthetic gene network is af-fected by a large number of disturbances and parameter fluctua-tions in the host cell. If their effects on the synthetic genenetwork cannot be efficiently attenuated by the robustness ofsynthetic gene network, the synthetic gene network cannot workproperly and its desired behaviors will decay quickly [11]. Afterthe stochastic parameter variations to be tolerated and environ-mental disturbances to be attenuated are both modeled into the dy-namic equation of synthetic gene network in (7) to mimic therealistic behavior in vivo, some design specifications for robust syn-thetic gene network to remedy these uncertainties to achieve refer-ence model matching in the host cell are described as follows:
2.1. Design specifications for robust reference model matching
(i) The kinetic parameters and the degradation rates should bedesigned from the following feasible ranges, respectively
k 2 ½k‘; ku�; r 2 ½r‘; ru�: ð8Þ
These feasible ranges are defined by the designer according tothe engineering ability of biotechnologies and the character-istics of the genes in the network.
(ii) The intrinsic parameter fluctuationsPm
i¼1giðx;Dki;DriÞdwi
should be tolerated by the synthetic gene network, i.e., thedesigned gene network could tolerate the stochastic param-eter fluctuations with standard deviations D ki and Dri pre-scribed by designers according to the statistics of realisticparameter variations in the host cell.
(iii) The following prescribed non-linear reference model shouldbe input/output matched
_xr ¼ frðxrÞ þ hrðxrÞur; xrð0Þ ¼ xr0; ð9Þ
where fr(xr) and hr(xr) are specified beforehand such that xr(t)could represent the desired behavior of the designed stochas-tic synthetic gene network in (7). ur(t) denotes the referenceinput to generate the desired reference behavior xr(t) by (9).This is the non-linear reference model to be tracked by syn-thetic gene network in the robust model matching design. Ifa linear reference model will be tracked, then (9) is modifiedas
_xr ¼ Arxr þ Brur ; xrð0Þ ¼ xr0; ð10Þ
where Ar and Br are specified beforehand so that the desiredreference behavior of synthetic gene network could be gener-ated by the linear reference model in (10). In general, the ref-erence input signal ur(t) is related to the steady state of xr(t)and is always given or changed by users. Therefore, ur(t)may be unavailable at the design stage and can be consideredas a disturbance.
(iv) The following prescribed disturbance attenuation should beachieved by filtering uncertain external disturbances [30]
ER1
0 ðx� xrÞT Qðx� xrÞdtER1
0 vTv þ uTr ur
� �dt
6 q2 or
EZ 1
0ðx� xrÞT Qðx� xrÞdt 6 q2E
Z 1
0vTv þ uT
r ur� �
dt
ð11Þ
for all possible bounded external disturbances v(t), where q isa prescribed attenuation level, i.e., the effect of external dis-turbances v(t) and ur on the matching error x � xr should beless than q from the average energy perspective. Q is a sym-metric weighting matrix on the matching errors. If Q = I, thenall state variables are required to match the desired statevariables. If Q = diag(0, . . . ,0,1), then only the output variablexn is required to match the last variable xrn. If the externaldisturbances are of deterministic molecular signals ratherthan of stochastic noises, then E
R10 vTv þ uT
r ur� �
dt in (11) isreplaced by
R10 vTv þ uT
r ur� �
dt, i.e., the expectation E can beneglected.
Remark.
(1) The physical meaning of (11) is that the effect of externaldisturbances on matching errors should be attenuated tothe level q 2 from the energy point of view. For example, ifq = 0.1, then the effect of v and ur on matching error x � xr
should be attenuated to the level of q2 = 0.01 from theenergy perspective.
B.-S. Chen et al. / Mathematical Biosciences 230 (2011) 23–36 27
(2) If the effect of the uncertain initial condition x0 � xr0 is alsoconsidered in the disturbance attenuation problem, Eq. (11)should be modified as follows [30]
EZ 1
0ðx� xrÞT Qðx� xrÞdt 6 EVðx0 � xr0Þ þ q2E
�Z 1
0vTv þ uT
r ur� �
dt ð12Þ
for some positive function V(x) > 0(3) For the convenience of analysis and the simplicity of design,
the origin of non-linear stochastic gene network should beshifted to the desired steady state of the reference modelin (9).
Based on the above four specifications given by users, our goal isto design a synthetic gene network by selecting kinetic parametersk 2 [k‘,ku] and degradation rates r 2 [r‘,ru] to satisfy the engineer-ing specifications (8)–(11) so that the synthetic gene network canachieve the reference model response under the intrinsic parame-ter fluctuations and external disturbances which appear in thehost cell. In other words, the design specifications (i)–(iv) canguarantee that the synthetic gene network has enough robuststability and filtering ability to tolerate the parameter fluctuationsand to attenuate the external disturbances to track the desiredreference signals.
3. Robust model matching design methodology of syntheticgene network
From the analyses in the above section, the robust modelmatching design of the synthetic gene network is to select the ki-netic parameters k and degradation rates r to engineer a stochasticgene network in (7) to achieve the design specifications (i)–(iv). Ingeneral, it is not easy to design the robust model matching genenetwork directly. Before further discussion of the robust matchingdesign methodology of synthetic gene network, the attenuationlevel of disturbance on the matching error x � xr in (12) can bederived as
EZ 1
0�xT Q�xdt 6 EVð�x0Þ þ q2E
Z 1
0�vT �vdt; ð13Þ
where the augmented state vector �x, the generalized disturbance �vand Q are denoted, respectively, by
�x ¼xr
x
� �; Q ¼
�I
I
� �Q �I I½ � ¼
Q �Q
�Q Q
� �; �v ¼
ur
v
� �:
Since ur can be arbitrarily assigned by users and can not bepredicted by the designer, for the convenience of design, it canbe considered as a disturbance in the design procedure. Let usdenote
Fð�x; k; rÞ ¼frðxrÞ
f ðx; k; rÞ
� �; �gið�x;Dki;DriÞ
0giðx;Dki;DriÞ
� �;
Hð�xÞ ¼hrðxrÞ 0
0 hðxÞ
� �: ð14Þ
Then the non-linear augmented system of (7) and (9) is given by
d�x ¼ ðFð�x; k; rÞ þ Hð�xÞ�vÞdt þXm
i¼1
�gið�x;Dki;DriÞdwi; �xð0Þ ¼ �x0:
ð15Þ
Based on the above analysis, the matching design of robust syn-thetic gene network becomes how to specify the kinetic parametersk 2 [k‘,ku] and degradation rates r 2 [r‘,ru] in the augmented system
(15) such that the disturbance attenuation in (13) is achieved for aprescribed disturbance attenuation level q. Then we get the follow-ing modeling matching design method for robust synthetic genenetwork.
Proposition 1. If we can specify kinetic parameters k 2 [k‘,ku] anddegradation rates r 2 [r‘, ru] such that the following HJI has a positivesolution Vð�xÞ > 0
@Vð�xÞ@�x
� �T
Fð�x; k; rÞ þ �xT Q�xþ 14q2
@Vð�xÞ@�x
� �T
Hð�xÞHð�xÞT @Vð�xÞ@�x
� �
þ 12
Xm
i¼1
gTi ð�x;Dki;DrÞ @
2Vð�xÞ@�x2 gið�x;Dki;DriÞ 6 0 ð16Þ
then the stochastic synthetic gene network in (7) can achieve the robustmodel matching design specifications (i)–(iv) from (8) to (11).
Proof. see Appendix A. h
In general, it is not easy to solve HJI in (16) for Vð�xÞ > 0 analyt-ically or numerically at present. In such a situation, the global lin-earization technique is employed to globally linearize theaugmented system in (15) to simplify the model matching designproblem of robust synthetic gene network. By the global lineariza-tion theory [31], if
@Fð�x;k;rÞ@�x
@Hð�xÞ@�x
@gið�x;Dki ;DriÞ@�x
2664
3775 2 Co
A1ðk; rÞB1
Ci1
264
375; A2ðk; rÞ
B2
Ci2
264
375; . . . ;
AMðk; rÞBM
CiM
264
375
8><>:
9>=>;; 8�xð17Þ
where Co denotes the convex hull of a polytope with M verticesdefined in (17), then the state trajectories of the non-linearaugmented system of (7) and (9) in (15) will belong to the convexcombination of the state trajectories of the following M linearizedaugmented systems at the M vertices of the convex hull
d�x ¼ ðAjðk; rÞ�xþ Bj �vÞdt þXm
i¼1
Cij�xdwi; j ¼ 1; . . . ;M; ð18Þ
where Ajðk; rÞ, Bj and Cij are obtained from M vertices in (17).The physical meaning of (18) is that if all the linearizations of
non-linear augmented system in (15) at all possible points �xðtÞ be-long to the convex hull Co which consists of linear systems at the Mvertices in (17), then the behavior of non-linear augmented systemin (15) can be represented by the convex combinations of M linearsystems in (18) [31]. Then the augmented system’s matrices of theglobally linearized system of (15) at the M vertices are given asfollows
Ajðk; rÞ ¼Arj 00 Ajðk; rÞ
� �; Bj ¼
Brj 00 Bj
� �; Cij ¼
0 00 Cij
� �:
Remark. In the linear reference model case in (10), the abovesystem matrices of global linearization are modified as
Ajðk; rÞ ¼Ar 00 Ajðk; rÞ
� �; Bj ¼
Br 00 Bj
� �:
Based on the global linearization theory [31], the augmentednon-linear stochastic system in (15) can be represented by the fol-lowing interpolation system
d�x ¼XM
j¼1
ajð�xÞ ðAjðk; rÞ�xþ Bj �vÞdt þXm
i¼1
Cij�xdwi
" #; ð19Þ
28 B.-S. Chen et al. / Mathematical Biosciences 230 (2011) 23–36
where the interpolation function ajð�xÞ must satisfy the constraints0 6 ajð�xÞ 6 1 and
PMj¼1ajð�xÞ ¼ 1. Unlike the conventional lineariza-
tion method to approximate the non-linear system by the linearizedsystem at the origin, the global linearization method interpolatesseveral linearized systems at the vertices of convex hull Co in (17)to represent the non-linear system.
Proposition 2. If we can specify the kinetic parameters k 2 [k‘, ku]and degradation rates r 2 [r‘, ru] such that the following inequalitieshas a common solution P = PT > 0
PAjðk;rÞþATj ðk;rÞP
T þQ þ 1q2 PBjBT
j PT þXm
i¼1
CTijPCij 6 0; j¼ 1; ::::;M
ð20Þ
then the robust model matching design of stochastic gene network sat-isfies the design specifications (i)–(iv)
Proof. see Appendix B. h
For convenience, we replace the difficult HJI in (16) by a set ofeasier algebraic inequalities in (20) to simplify the design proce-dure of robust model matching problem of non-linear syntheticgene networks. By Schur complement [31], the inequalities in(20) are equivalent to the following LMIs
PAjðk; rÞ þ ATj ðk; rÞP
T þ Q þPmi¼1
CTijPCij PBj
BTj PT �q2I
0B@
1CA 6 0; j ¼ 1; . . . ;M
ð21Þ
Then the model matching problem for synthetic gene networkdesign becomes how to select the kinetic parameters k 2 [k‘,ku]and degradation rates r 2 [r‘, ru] such that the LMIs in (21) have apositive definite solution P > 0 to satisfy the design specifications(i)–(iv). Because the LMIs in (21) can be easily solved by the LMItoolbox in Matlab, the proposed method has good potential forthe desired model reference matching design of robust syntheticgene network in the future.
Remark.
(1) If we want to design a robust synthetic gene network, whichcan tolerate intrinsic parameter fluctuation and attenuateexternal disturbance to optimally achieve reference modelmatching, then we need to solve the following constrainedoptimization problem
mink2½k‘;ku �r2½r‘ ;ru �
q2;
subject to LMIs in ð21Þ; P > 0:
ð22Þ
This constrained optimization can be obtained by decreasingq until there exists no solution P > 0 for all k 2 [k‘,ku] andr 2 [r‘,ru].
(2) In the oscillation design case, the eigenvalues of Arj inreference model are always on the jx-axis, i.e., one half ofeigenvalues of Aj in (21) are on the jx-axis (i.e., with zeroreal parts). In this situation, it is not easy to specifyk 2 [k‘,ku] and r 2 [r‘,ru] to satisfy the LMIs in (21). In orderto overcome this design difficulty, an eigenvalue-shiftedtechnique is proposed as an expedient scheme to deal withthe oscillation tracking design problem. Let us adjust thesystem variables in (18) by �xsðtÞ ¼ e�kt�xðtÞ and �v sðtÞ ¼e�kt �vðtÞ for some positive value k, then �xsðtÞ can be obtainedby the following [31].
d�xs ¼XM
j¼1
ajð�xsÞ ðAjðk; rÞ � kIÞ�xs þ Bj �v s
� dt þ
Xm
i¼1
Cij�xsdwi
" #:
ð23Þ
For the eigenvalue-shifted system in (23), suppose we wantto specify k 2 [k‘,ku] and r 2 [r‘,ru] to achieve the followingattenuation level of disturbances
EZ 1
0�xT
s Q�xsdt 6 EVð�x0Þ þ q2EZ 1
0�vT
s �v sdt; ð24Þ
which is the same as (13) except �xðtÞ and �vðtÞ are replaced by�xsðtÞ and �vsðtÞ, respectively, i.e., we use the matching perfor-mance in (24) to replace the matching performance in (13).
In this transformation case, the robust matching design
problem is relaxed to how to specify k and r in (23) toachieve the disturbance attenuation performance in (24). ByProposition 2 and (21), we need to specify k 2 [k‘,ku] andr 2 [r‘,ru] to solve P > 0 for the following LMIs.� � 0 1
P Ajðk;rÞ�kI þ ATj ðk;rÞ�kI PT þQþPmi¼1
CTijPCij PBj
BTj PT �q2I
B@ CA60; j¼1; . . .;M:
ð25Þ
In the above expedient method, due to the more negativeeigenvalues of Aðk;rÞj � kI, we have more feasible way to solvethe robust reference matching problem for periodic referencesignals. However, in order to avoid some distortion due tosignal transformation, k should be selected as small aspossible.
Based on the analyses above, a design procedure for robustmodel matching synthetic gene network is proposed as follows:
3.1. Design procedure
(a) Formulate a stochastic dynamic equation to mimic a pertur-bative gene network as (7) in the host cell.
(b) Give design specifications (i)–(iv) to the synthetic gene net-works to robustly match a desired input/output responsemodel _xr ¼ frðxrÞ þ hrðxrÞur in spite of intrinsic parameterfluctuations, external disturbances and any input signal ur.
(c) Augment the stochastic dynamic equation and the desiredreference model as (15).
(d) Perform the global linearization as (17).(e) Solve the LMIs in (21) for design parameters k and r from the
feasible intervals k 2 [k‘,ku] and r 2 [r‘,ru], respectively.
4. Design examples in silico
4.1. Case 1: Synthetic transcriptional cascade design
Based on the above analyses of model matching design method-ology of robust synthetic gene network, two in silico design exam-ples are given to illustrate the proposed design procedure and toconfirm the robust matching performance of the proposed method.The first case is the synthetic transcriptional cascade design. Thesynthetic gene network in Fig. 2 is made of four genes: tetR, lacI,cI and eyfp that code respectively for three repressor proteins, TetR,LacI and CI, and the fluorescent protein EYFP. The additional chem-ical aTc is considered as the system input to be used to control thebehavior of the system output of EYFP. The regulatory dynamicequation of the synthetic gene network is given in (2). In thedynamic Eq. (2), the basal levels k10, k20, k30 and k40 are given asconstants 3, 1, 0.9, and 2.5, respectively. In addition, the inhibitionfunctions d2(x1), d3(x2), d4(x3) are all Hill functions as diðxÞ ¼
B.-S. Chen et al. / Mathematical Biosciences 230 (2011) 23–36 29
bd1þ x=Kið Þn, where bd is maximal expression level of promoter. Ki is therepression coefficient. The Hill coefficient n governs the steepnessof the input function. For the activator a(u), Hill function can be de-scribed in the form aðuÞ ¼ baun
Knaþun. ba is also the maximal expression
level of promoter. Ka is the activation coefficient. n determinesthe steepness of the input function [38]. These Hill functions areshown in Fig. 3. The kinetic parameters k2, k3, k4 and degradationrates r1, r2, r3, r4 also suffer from parameter perturbations in thehost cell, and we want to design the three kinetic parametersand four degradation rates from the biological allowable range tosatisfy the following four design specifications to achieve robustmodel matching design.
(i) The kinetic parameters and degradation rates are to bedesigned in the following allowable ranges.
r1 2 ½0:1;3�;k2 2 ½1;40�; r2 2 ½0:1;5�;k3 2 ½30;600�; r3 2 ½0:1;1:5�;k4 2 ½300;1200�; r4 2 ½0:1;1�:
(ii) The standard deviations of parameter fluctuations to be tol-erated in the host cell are assumed as follows
Dr1 ¼ 0:05;Dk2 ¼ 5; Dr2 ¼ 0:05;Dk3 ¼ 5; Dr3 ¼ 0:05;Dk4 ¼ 5; Dr4 ¼ 0:05:
(iii) The reference model of the synthetic gene network is givenas follows
_xr ¼ Arxr þ Brur ¼
�1 0 0 03 �2 0 00 2 �1:5 00 0 1:5 �1
0BBB@
1CCCAxr
þ
3 0 0 00 5 0 00 0 7 00 0 0 10
0BBB@
1CCCAur ; ð26Þ
where xr is the reference state to be matched, and ur is thereference input.
010
2030
4050
200400
600200
400
600
800
1000
1200
k2k3
k4
Fig. 4. The parameter space of kinetic parameters (k2,k3 and k4). The biologicalfeasible ranges for kinetic parameters are located in the green cubic region. Theinfeasible parameters are spread widely outside the green region i.e., the pinkregion. (For interpretation of the references to colour in this figure legend, thereader is referred to the web version of this article.)
(iv) The disturbance attenuation level is described as q =0.05(q2 = 0.0025).
In order to apply the global linearization technique to simplify-ing the design procedure, the desired steady state xr of the refer-ence model in (26) to be matched by the synthetic gene networkin (2) is shifted to the origin, and the global linearization is em-ployed to obtain Aj, Bj and Cij with 6 vertices (M = 6) (see AppendixC). The non-linear augmented system in (15) can be approximatedvia the six linearized augmented systems in (19) through the fol-lowing suitable interpolation functions [39]
ajð�xÞ ¼1
�xj � �xðtÞ 2
2
,XM
j¼1
1
�xj � �xðtÞ 2
2
; j ¼ 1; . . . ;M; ð27Þ
where �xj are the polytope operating points.The next step is to solve LMIs in (21) by tuning the kinetic
parameters ki and degradation rates ri iteratively within the allow-able ranges to find a common positive definite matrix solution P.The weighting matrix is selected as Q ¼ diagð½0:0005 0:00050:00050:0005�Þ. Following the design procedure above and usingLMI Toolbox in Matlab, we find that a P > 0 will be solved when
the kinetic parameters ki are selected from the following feasibleparameter sets (the green cubic region in Fig. 4).
k2 2 ½12;19�; k3 2 ½174;466�; k4 2 ½510;977�
and degradation rates ri within the following ranges
r1 2 ½1:0106;1:3366�; r2 2 ½1:538;2:4585�;r3 2 ½0:3311;0:77817�; r4 2 ½0:1633;0:2422�:
ð28Þ
In order to verify the disturbance filtering ability and modelmatching performance, a set of kinetic parameters are chosen fromthe feasible parameter sets which lie in the green cubic region inFig. 4 and the degradation rates are selected from the design-spec-ified ranges in (28) as follows
k2 k3 k4½ � ¼ 18 309 713½ �;r1 r2 r3 r4½ � ¼ 1:0691 2:0071 0:39826 0:18203½ �:
Thirty simulation results of the synthetic gene network areshown in Fig. 5 with random initial condition of normal distributionunder intrinsic parameter fluctuations mentioned above and exter-nal disturbances v 0ðtÞ ¼ n1ðtÞ n2ðtÞ n3ðtÞ n4 ðtÞ½ �T where ni(t), i =1, . . . ,4 denote the standard white noises with zero mean and unitvariance. The desired model reference states xr are also shown asred dashed curves in Fig. 5. The desired transient and steady statescan be properly matched under the external disturbances and intrin-sic parameter perturbations and the filtering ability of the syntheticgene network can be estimated by Monte Carlo simulation as follows
ER 40
0�xT Q�xdt � EVð�x0ÞER 40
0�vT �vdt
� ð0:0188Þ2 < ð0:05Þ2:
Obviously, the effect of disturbances is attenuated more signifi-cantly than the prescribed attenuation level of q = 0.05(q 2 =0.0025). The conservative is mainly due to the inherent conserva-tiveness in solving LMIs in (21).
If we select a set of kinetic parameter ki from the infeasibleparameter sets (the pink region in Fig. 4) and degradation rate ri
out of the design-specified ranges in (28), for instance, thekinetic parameter k2 k3 k4½ � ¼ 2 159 498½ � and degradationrate r1 r2 r3 r4½ � ¼ 1:8843 1:4872 0:10967 0:52228½ �. Thesimulation results under intrinsic parameter fluctuations andrandom external disturbances with random initial conditions ofnormal distribution with unit standard deviation are shown inFig. 6. It can be seen that the model matching cannot be achieved
0 20 400
5
10
15
20
Tet
R c
once
ntra
tion
time (min)0 20 40
5
10
15
20
LacI
con
cent
ratio
n
time (min)
0 20 40
102
CI c
once
ntra
tion
time (min)0 20 40
0
10
20
30
40
EY
FP
con
cent
ratio
n
time (min)
x1−ref
x1
x2−ref
x2
x3−ref
x3 x
4−ref
x4
Fig. 6. Monte Carlo simulation results of the synthetic transcriptional cascade withinfeasible parameters for 30 rounds. We select a set of kinetic parameter ki from theinfeasible parameter sets and the degradation rate ri out of the design-specifiedranges, i.e., the pink region in Fig. 4. The expression of the synthetic transcriptionalcascade cannot match the desired reference model (red dashed curve) underexternal disturbances and parameter fluctuations with random initial conditions.The infeasible kinetic parameters are ki = [2 159 498] and the degradation rates areri = [1.8843 1.4872 0.10967 0.52228]. (For interpretation of the references to colourin this figure legend, the reader is referred to the web version of this article.)
Fig. 7. The synthetic coupled repressilator. The repressor protein TetR inhibits thetranscription of the repressor gene cI, whose protein product in turn inhibits theexpression of the repressor gene lacI. Finally, the repressor protein LacI inhibits tetRexpression.
0 20 400
5
10
15
20
Tet
R c
once
ntra
tion
time (min)
0 20 405
10
15
20
LacI
con
cent
ratio
n
time (min)
0 20 400
5
10
15
20
25
CI c
once
ntra
tion
time (min)
0 20 400
10
20
30
40E
YF
P c
once
ntra
tion
time (min)
x1−ref
x1
x2−ref
x2
x3−ref
x3
x4−ref
x4
Fig. 5. Monte Carlo simulation results of the synthetic transcriptional cascade withfeasible parameters for 30 rounds. The 4-genes synthetic transcriptional cascadematches the reference dynamic model (red dashed curve) under external distur-bances and parameter fluctuations with random initial conditions. The feasiblekinetic parameters are ki = [18 309 713] (which lie in the green cubic region inFig. 4) and the degradation rates are in the design-specified ranges with ri = [1.06912.0071 0.39826 0.18203].
30 B.-S. Chen et al. / Mathematical Biosciences 230 (2011) 23–36
because the robust matching conditions in (i)–(iv) are disobeyed,and the filtering ability to attenuate disturbances is estimated byMonte Carlo simulation as follows
ER 40
0�xT Q�xdt � EVð�x0ÞER 40
0�vT �vdt
� ð2:1811Þ2 > ð0:05Þ2:
Apparently, the design specification of filtering ability is violated.Thus, the proposed model matching design method for the robustsynthetic gene network can be validated by the simulation example.
4.2. Case 2: Synthetic coupled repressilator design
The second example is the synthetic coupled repressilator[40,41] shown in Fig. 7. The repressor protein TetR inhibits thetranscription of the repressor gene cI, whose protein product inturn inhibits the expression of the repressor gene lacI. Finally, therepressor protein LacI inhibits tetR expression. The negative feed-back loop leads to temporal oscillations in the concentration ofeach of its components which can be seen from a simple modelof transcriptional regulation. The dynamic equations for the syn-thetic coupled repressilator under stochastic external disturbancesare given as
dx1
dt¼ k1a
lþ xm3� r1x1 þ v 01;
dx2
dt¼ k2a
lþ xm1� r2x2 þ v 02;
dx3
dt¼ k3a
lþ xm2� r3x3 þ v 03;
ð29Þ
where x1 x2 x3½ �T ¼ xTetR xCI xLacI½ �T denote the protein con-centrations of TetR, CI and LacI, respectively. k1, k2, k3 are the kineticparameters which represent regulation abilities of the gene net-work. r1, r2, r3 are the degradation rates of the corresponding pro-teins in the host cell. The parameters are set as a = 1.8, l = 1.3,and the Hill coefficient m = 4. v 01; v 02; v 03 denote the correspondingstochastic external disturbances.
We want to design the three kinetic parameters and three deg-radation rates from the biologically allowable range to satisfy thefollowing four design specifications to achieve robust oscillationmatching design.
(i) The kinetic parameters and degradation rates are to bedesigned in the following allowable ranges.
k1 2 ½0:1 2 �; r1 2 ½0:01 1 �;k2 2 ½0:1 2 �; r2 2 ½0:01 1 �;k3 2 ½0:1 2 �; r3 2 ½0:01 1 �:
(ii) The standard deviations of parameter fluctuations to be tol-erated in the host cell are assumed as follows
Dk1 ¼ 0:1; Dr1 ¼ 0:01;Dk2 ¼ 0:1; Dr2 ¼ 0:01;Dk3 ¼ 0:1; Dr3 ¼ 0:01:
(iii) The non-linear reference model of the synthetic gene net-work is given as follows
_xr1 ¼1
1þ x7r3
� 0:5xr1 þ 0:2ur ;
_xr2 ¼1
1þ x7r1
� 0:5xr2 þ 0:2ur ;
_xr3 ¼1
1þ x7r2
� 0:5xr3 þ 0:2ur ;
ð30Þ
B.-S. Chen et al. / Mathematical Biosciences 230 (2011) 23–36 31
where ur is the reference input. The desired oscillations to bematched by the synthetic network are shown with blackdashed curves in Fig. 8.
(iv) The disturbance attenuation level is prescribed as q = 0.8(q2 = 0.64).
After applying the global linearization technique, we can obtainAj , Arj, and Cij with 5 vertices (M=5), which are shown in AppendixD. In order to deal with the oscillation tracking design problem, weconsider the eigenvalue-shift method with �xsðtÞ ¼ e�kt�xðtÞ and�v sðtÞ ¼ e�kt �vðtÞ as an expedient scheme for a positive valuek = 0.616. The next step is to solve LMIs in (25) by tuning the kineticparameters ki and degradation rates ri iteratively within the allow-able ranges to find a common positive definite matrix solution P.The weighting matrix is selected as Q ¼ diagð½0:01 0:01 0:010:01�Þ. Finally, we find that a P > 0 will be solved when the kineticparameters ki and degradation rates ri are selected from the follow-ing feasible parameter ranges for this robust oscillation trackingdesign problem.
k1 2 ½0:8;1�; k2 2 ½0:8;1�; k3 2 ½0:8;1�;r1 2 ½0:45;0:51�; r2 2 ½0:45;0:51�; r2 2 ½0:45;0:51�:
ð31Þ
In order to confirm the design result, we choose a set of thefeasible parameters for kinetic parameters ki and degradation ratesri from the feasible ranges in (31) for the synthetic coupled repress-ilator in (29) as follows
½ k1 k2 k3 � ¼ ½0:9133 0:91330:9133 �;½ r1 r2 r3 � ¼ ½0:46637 0:46637 0:46637 �:
ð32Þ
Thirty simulation results for the synthetic coupled repressilatorby (32) are shown in Fig. 8 with random initial condition of normaldistribution under intrinsic parameter fluctuations and externaldisturbances v0ðtÞ ¼ ½0:3n1ðtÞ 0:3n2ðtÞ 0:3n3ðtÞ �T where ni(t), i =1, . . . ,3 denote the standard white noises with zero mean and unitvariance. The desired oscillation can be properly matched by the
0 5 10 15
0.8
1
1.2
1.4
1.6
1.8
2
2.2
time (min)
Prot
ein
conc
entra
tion
TetRCILacI
Fig. 8. Monte Carlo simulation results of the synthetic coupled repressilator withfeasible parameters for 30 rounds. The 3-genes synthetic coupled repressilator canmatch the non-linear reference model (black dashed curves) under externaldisturbances and parameter fluctuations with random initial conditions. In theseMonte Carlo simulations, the feasible kinetic parameters ki = [0.9133 0.91330.9133] and the degradation rates ri = [0.46637 0.46637 0.46637] are selected forthe synthetic coupled repressilator from the design-specified ranges in (31). Thesesimulations confirm that the proposed design scheme can achieve robust oscillationmatching for the synthetic coupled repressilator. (For interpretation of thereferences to colour in this figure legend, the reader is referred to the web versionof this article.)
synthetic coupled repressilator in (32) under the intrinsic parame-ter perturbations and external disturbances. If we select a set ofkinetic parameter ki and degradation rates ri out of the design-spec-ified ranges in (31), for example, ½ k1 k2 k3 � ¼ ½1:5 1:5 1:5 �and ½ r1 r2 r3 � ¼ ½ 0:4 0:4 0:4 �. The simulation results of thesynthetic coupled repressilator under intrinsic parameter fluctua-tions and external disturbances with random initial conditions areshown in Fig. 9. It can be seen that the model matching cannot beachieved because the robust matching conditions in (i)–(iv) aredisobeyed.
5. Discussion
Due to intrinsic perturbations and extrinsic disturbances in thehost cell, the synthetic gene networks engineered so far in bacteriato behave in a particular way seem decay rapidly after a short per-iod of activity [37,42]. Therefore, the development of a robust de-sign scheme is an important topic for synthetic gene network towork properly and robustly in spite of intrinsic and external noises.In general, noise originates from many sources, including environ-mental disturbances, fluctuations in gene expression, cell cyclevariations, differences in the concentrations of metabolites andcontinuous mutational evolution. The synthetic gene networkcan reduce significant intrinsic genetic noise [34], even at the levelof a single gene [43,44], by the circuit topology. In previous studies[45–48], robust gene circuit designs have been proposed toattenuate the parameter variations or noises. In this study, thestochastic parameter fluctuations to be tolerated and the environ-mental disturbances to be attenuated below a prescribed level aresimultaneously considered. By doing this, we attempt to reduce theeffect of noise by constructing the synthetic gene network withdesired behaviors. In this situation, noise is assumed to have a neg-ative influence on cellular behaviors. The noise should be avoidedor attenuated by choosing adequate kinetic parameters and degra-dation rates based on the feasible ranges. Murphy et al. attemptedto tune and control gene expression noise in synthetic gene net-
0 5 10 150
0.5
1
1.5
2
2.5
3
3.5
time (min)
Prot
ein
conc
entra
tion
TetRCILacI
Fig. 9. Monte Carlo simulation results of the synthetic coupled repressilator withinfeasible parameters for 30 rounds. We select a set of kinetic parameters ki fromthe infeasible parameter sets and the degradation rates ri out of the design-specifiedranges. The gene expressions of the repressilator cannot match the desired non-linear reference model (black dashed curve) under external disturbances andparameter fluctuations with random initial conditions. The infeasible kineticparameters ki = [1.5 1.5 1.5] and degradation rates ri = [0.4 0.4 0.4 ] are selectedin this simulation. (For interpretation of the references to colour in this figurelegend, the reader is referred to the web version of this article.)
32 B.-S. Chen et al. / Mathematical Biosciences 230 (2011) 23–36
work so that it can be suppressed in some cases and harnessed inothers [49], but our purpose is to design the kinetic parameters anddegradation rates to satisfy four design specifications that guaran-tee the desired behaviors of the engineered synthetic gene networkin the host cell.
Based on the design specifications, a simple design procedureis developed in this study. From the in silico design examples,the four design specifications can be achieved for the robust syn-thetic gene network according to the proposed robust modelmatching design scheme. Specifically, we can specify robust ki-netic parameters ki and degradation rates ci within the feasibleparameter ranges to achieve the desired transient and steadystates of the synthetic gene network. The kinetic parametersand degradation rates are approximately dependent on the pro-moters and protein decay rates, respectively. In synthetic biology,the promoter activities of different promoter architectures areprovided as a combinatorial promoter libraries [50]. The promoteractivities are determined largely by the position of single opera-tor. In bacteria, operators are classified as being in the core, prox-imal or distal regions of the promoter, and the repressor canrepress expression from all three subregions [51,52]. In eukary-otes, the chromatin structure also strongly influences expressionlevel [53,54]. Although we still engineered the synthetic gene net-work in the bacteria, a more clearly understanding of the rela-tionship between the promoter activities and the promoterstructure will help us engineer a complex synthetic gene networkin eukaryotes. As for the biological implementation, we could re-fer to standard biological parts in biological device datasheets toconstruct the genetic circuits with the fine-tuned kinetic param-eters ki and degradation rates ci. In this way, synthetic biologistscan efficiently design the gene circuit through registries of biolog-ical parts and standard datasheets.
Quantitative descriptions of devices in the form of standardized,comprehensive datasheets are widely used in many engineeringdisciplines. A datasheet is intended to allow an engineer to quicklydetermine whether the behavior of a device will meet the require-ments of a system in which a device might be used [55]. Such adetermination is based on a set of standard characteristics of de-vice behavior, which are the product of engineering theory andexperience. In the datasheets of engineering, the characteristicstypically reported are common across a wide range of device types,such as sensors, logic elements and actuators. Recently, biologicaldatasheets have been set as standards for characterization, manu-facture and sharing of information about modular biological de-vices for a more efficient, predictable and design-driven geneticengineering science [55,56]. Because datasheets of biological partsor devices are an embodiment of engineering standard for syn-thetic biology [55], a good device standard should define sufficientinformation about biological parts or devices to allow the design ofsynthetic gene networks with the optimal parameters. Datasheetsusually contain a formal set of input–output transfer functions, dy-namic behaviors, compatibility, requirements and other detailsabout a particular part or device [55,56]. Since parameters ki arecombinations of transcription and translation, they could be mea-sured from the input–output transfer functions and dynamicbehaviors of biological parts or devices in biological device data-sheets. When the biological parts and devices in datasheets be-come more complete in future, we can rapidly select from a vastlist the parts that will meet our design parameterski. Thereforewe can ensure that devices selected from datasheets can fit the fea-sible parameters, and systems synthesized from them can satisfythe requirements of design specifications for robust synthetic genenetworks.
In the present integrated circuit (IC) industry, due to highcomplexity and difficulty, some system design companies, likeIntel, respond for system design of VLSI systems, and some
implementation companies, like TSMC (Taiwan SemiconductorManufacturing Company), are responsible for manufacturing VLSIsystems. In future, there may exist some system design companiesfor system design of synthetic gene networks, and other imple-mentation companies will respond for manufacturing complexsynthetic gene networks. If this is the case, the development ofsynthetic design tools will become an important work for systemdesign of robust synthetic gene networks.
Since the design examples in silico meets the four design spec-ifications, the synthetic gene network can overcome differentkinds of internal parameter fluctuations and external distur-bances. The design specification (ii) includes the prescribed stan-dard deviations of stochastic parameter variations to be toleratedby the synthetic gene network in the host cell, and in the designspecification (iv), we introduce a disturbance filtering method toguarantee that the effect of disturbances on the matching erroris attenuated by a prescribed level q. However, in reality, theamplitudes of external disturbances are uncertain, and so it isnot easy to estimate their statistics outside the host cell. But fromthe energy perspective, the effect of these disturbances on match-ing error can be prescribed below a desired level q by the pro-posed design method to ensure the matching performancewithout the knowledge of the disturbance statistics. In addition,specification (iii) provides a desired system behavior of syntheticgene networks. Unlike the synthetic gene network design meth-ods in [25,26], which can only asymptotically achieve some de-sired steady states, the proposed design method can makesynthetic gene network track the transient and steady behaviorsprescribed beforehand by a desired reference model. Based onthe H1 matching performance and global linearization technique,we can also validate the robustness of the proposed design meth-od under the internal parameter fluctuations and external distur-bances. Therefore, the proposed matching design method mightbe more useful than the conventional methods for practical syn-thetic biological design and will have a great impact on the syn-thetic biology in the near future.
6. Conclusions
In this study, we have shown a stochastic model for dealingwith the dynamic properties of synthetic gene networks underparameter uncertainties and external disturbances in the hostcell. In order for the synthetic gene networks to achieve a de-sired response matching for a reference model under parameterfluctuations and environmental disturbances, four design specifi-cations are proposed in the design procedure of robust syntheticgene network. Finally, a systematic method for robust modelmatching design of synthetic gene networks is introduced to sat-isfy these design specifications. To simplify the design procedure,a global linearization technique is employed to avoid solving thecomplex non-linear stochastic filtering and tracking problems di-rectly, and to represent the non-linear gene network by interpo-lating a set of simple linearized gene networks. Therefore thedesign procedures of a robust model matching design of syn-thetic gene network can be simplified by solving a set of LMIsvia the help of LMI toolbox in Matlab. Two design examples insilico can confirm the robust model matching performance of adesired reference response.
Acknowledgments
The study is supported by National Tsing Hua University undergrant 98N2565E1 and by the National Science Council of Taiwanunder grants NSC 98-2221-E-007-113-MY3 and NSC 99-2745-E-007-001-ASP.
B.-S. Chen et al. / Mathematical Biosciences 230 (2011) 23–36 33
Appendix A
Let us choose a Lyapunov function Vð�xÞ > 0 with V(0) = 0, thenwe get
EZ 1
0�xT Q�xdt ¼ E Vð�xð0ÞÞ � Vð�xð1ÞÞ þ
Z 1
0�xT Q�xþ dVð�xÞ
dt
� �dt
� �:
ðA:1Þ
By the Ito’s formula [57] and E ddt wi ¼ Eni ¼ 0, we get
Eddt
Vð�xÞ ¼ E@Vð�xÞ@�x
� �T
Fð�x; k; rÞ þ Hð�xÞ�vð Þ(
þ 12
Xm
i¼1
�gTð�x;Dki;DriÞ@2Vð�xÞ@�x2
�gð�x;Dki;DriÞ): ðA:2Þ
Substituting (A.2) into (A.1), we have
EZ 1
0�xT Q�xdt¼E Vð�xð0ÞÞ�Vð�xð1ÞÞþ
Z 1
0�xT Q�xþ @Vð�xÞ
@�x
� �T
Fð�x;k;rÞ"(
þ @Vð�xÞ@�x
� �T
Hð�xÞ�vþ12
Xm
i¼1
�gT ð�x;Dki;DriÞ@2Vð�xÞ@�x2
�gð�x;Dki;DriÞ#
dt
):
ðA:3Þ
By the inequality (16), we have
EZ 1
0
�xT Q�xdt6E Vð�xð0ÞÞ�Vð�xð1ÞÞþZ 1
0
@Vð�xÞ@�x
� �T
Hð�xÞ�v"(
� 14q2
@Vð�xÞ@�x
� �T
Hð�xÞHTð�xÞ@Vð�xÞ@�x�q2 �vT �vþq2 �vT �v
#dt
)
<E Vð�xð0ÞÞþZ 1
0� 1
2qHTð�xÞ@Vð�xÞ
@�x�q�v
� �T"(
� 12q
HTð�xÞ@Vð�xÞ@�x�q�v
� �þq2 �vT �v
�dt�
6E Vð�x0Þþq2Z 1
0�vT �vdt
� �: ðA:4Þ
This is the inequality of robust model matching in (13). Then theattenuation level q in (13) is achieved. If we specify k 2 [k‘,ku]and r 2 [r‘, ru] such that the inequality in (16) is satisfied, then thespecifications (i)–(iv) are satisfied for non-linear stochasticsynthetic network in (15).
Appendix B
Now we will derive the sufficient condition to ensure that thelinearized augmented systems in (19) can attenuate the externaldisturbance below a prescribed attenuation level q in (13). Bychoosing a Lyapunov function as Vð�xÞ ¼ �xT P�x for some P = PT > 0,we have
EZ 1
0
�xT Q�xdt¼ E �xTð0ÞP�xð0Þ��xTð1ÞP�xð1ÞþZ 1
0ð�xT Q�xþ d
dt�xT P�xÞdt
� �:
ðB:1Þ
By Ito formula [57] and E ddt wi ¼ 0, we have
EZ 1
0�xT Q�xdt¼ E �xTð0ÞP�xð0Þ��xT ð1ÞP�xð1Þ
þZ 1
0�xT Q�xþ
XM
j¼1
ajðxÞ Ajðk;rÞ�xþBj �vh iT
P�x�"
þ �xT P Ajðk;rÞ�xþBj �vh i
þXm
i¼1
�xT CTijPCij�x
� !#dt
)
< E �xð0ÞT P�xð0ÞþZ 1
0
XM
j¼1
ajðxÞ �xT PAjðk;rÞh� (
þATj ðk;rÞPþQ þ
Xm
i¼1
CTijPCij
� #�xþ �vT BT
j P�xþ �xT PBj �v!!
dt
):
ðB:2Þ
By the inequality in (20), we have
EZ 1
0
�xT Q�xdt< E �xð0ÞT P�xð0ÞþZ 1
0� 1
q2�xT PBjBT
j PT�xþ �vT BTj P�x
��þ �xT PBj �v �q2 �vT �v þq2 �vT �v
�dt�
6 E �xð0ÞT P�xð0ÞþZ 1
0� 1
qBT
j P�x�q�v� �T 1
qBT
j P�x�q�v� �"(
þq2 �vT �v�dt�6 E �xð0ÞT P�xð0Þþ
Z 1
0q2 �vT �vdt
� �:
ðB:3Þ
This is the robust disturbance attenuation with a prescribed level qin (13). When �xð0Þ ¼ 0, it is reduced to the robust disturbance atten-uation in (11). Therefore, if we select k 2 [k‘,ku] and r 2 [r‘,ru] suchthat the inequalities in (20) are satisfied, then the specifications(i)–(iv) are satisfied for non-linear stochastic synthetic genenetwork in (7).
Appendix C
The global linearization technique can be employed to trans-form the non-linear stochastic gene network of (2) into an equiva-lent interpolation of a set of globally linearized gene networks. Inthis design example, the global linearizations are bound by apolytope consisting of 6 vertices, shown as follows
dx ¼ ðAjðk; rÞxþ BjvÞdt þX4
i¼1
Cijxdwi; j ¼ 1; . . . ;6; ðC:1Þ
where
A1 ¼
�1:0691 0 0 0�0:030482 �2:0071 0 0
0 �0:47223 �0:39826 00 0 �0:18915 �0:18203
26664
37775;
A2 ¼
�1:0691 0 0 0�0:079495 �2:0071 0 0
0 �2:9315 �0:39826 00 0 �0:32622 �0:18203
26664
37775;
A3 ¼
�1:0691 0 0 0�0:20458 �2:0071 0 0
0 �16:931 �0:39826 00 0 �0:55293 �0:18203
26664
37775;
A4 ¼
�1:0691 0 0 0�3:4286 �2:0071 0 0
0 �160:14 �0:39826 00 0 �2:3385 �0:18203
26664
37775;
34 B.-S. Chen et al. / Mathematical Biosciences 230 (2011) 23–36
A5 ¼
�1:0691 0 0 0�7:7445 �2:0071 0 0
0 16:964 �0:39826 00 0 �50:28 �0:18203
26664
37775;
A6 ¼
�1:0691 0 0 0�0:000211 �2:0071 0 0
0 1:2959 �0:39826 00 0 �78:005 �0:18203
26664
37775:
Bj ¼
1 0 0 00 1 0 00 0 1 00 0 0 1
26664
37775; j ¼ 1; . . . ;6:
C11 ¼
�0:05 0 0 00 0 0 00 0 0 00 0 0 0
26664
37775; C12 ¼
�0:05 0 0 00 0 0 00 0 0 00 0 0 0
26664
37775;
C13 ¼
�0:05 0 0 00 0 0 00 0 0 00 0 0 0
26664
37775;
C14 ¼
�0:05 0 0 00 0 0 00 0 0 00 0 0 0
26664
37775; C15 ¼
�0:05 0 0 00 0 0 00 0 0 00 0 0 0
26664
37775;
C16 ¼
�0:05 0 0 00 0 0 00 0 0 00 0 0 0
26664
37775;
C21 ¼
0 0 0 0�0:0084672 �0:05 0 0
0 0 0 00 0 0 0
26664
37775;
C22 ¼
0 0 0 0�0:022082 �0:05 0 0
0 0 0 00 0 0 0
26664
37775;
C23 ¼
0 0 0 0�0:056827 �0:05 0 0
0 0 0 00 0 0 0
26664
37775;
C24 ¼
0 0 0 0�0:9524 �0:05 0 0
0 0 0 00 0 0 0
26664
37775;
C25 ¼
0 0 0 0�2:1513 �0:05 0 0
0 0 0 00 0 0 0
26664
37775;
C26 ¼
0 0 0 0�5:8681e� 5 �0:05 0 0
0 0 0 00 0 0 0
26664
37775;
C31 ¼
0 0 0 0
0 0 0 0
0 �0:0076413 �0:05 0
0 0 0 0
2666664
3777775;
C32 ¼
0 0 0 0
0 0 0 0
0 �0:0050056 �0:05 0
0 0 0 0
2666664
3777775;
C33 ¼
0 0 0 0
0 0 0 0
0 �0:0032505 �0:05 0
0 0 0 0
2666664
3777775;
C34 ¼
0 0 0 0
0 0 0 0
0 �0:00039857 �0:05 0
0 0 0 0
266664
377775;
C35 ¼
0 0 0 0
0 0 0 0
0 4:2854e� 5 �0:05 0
0 0 0 0
266664
377775;
C36 ¼
0 0 0 0
0 0 0 0
0 7:4721e� 6 �0:05 0
0 0 0 0
266664
377775;
C41 ¼
0 0 0 0
0 0 0 0
0 0 0 0
0 0 �0:0013264 �0:05
266664
377775;
C42 ¼
0 0 0 0
0 0 0 0
0 0 0 0
0 0 �0:0022876 �0:05
266664
377775;
C43 ¼
0 0 0 0
0 0 0 0
0 0 0 0
0 0 �0:0038775 �0:05
266664
377775;
C44 ¼
0 0 0 00 0 0 00 0 0 00 0 �0:016399 �0:05
26664
37775;
C45 ¼
0 0 0 00 0 0 00 0 0 00 0 �0:35259 �0:05
26664
37775;
C46 ¼
0 0 0 00 0 0 00 0 0 00 0 �0:54702 �0:05
26664
37775:
B.-S. Chen et al. / Mathematical Biosciences 230 (2011) 23–36 35
Appendix D
The global linearization technique can be employed to trans-form the non-linear synthetic coupled repressilator (29) into anequivalent interpolation of a set of globally linearized repressilatornetworks. Also the non-linear reference model in (30) can betransformed into an equivalent interpolation of a set of globallylinearized reference models. In this design example, the globallinearizations are bound by a polytope consisting of 5 vertices,shown as follows
d�x ¼ ðAjðk; rÞ�xþ Bj �vÞdt þX3
i¼1
Cij�xdwi; j ¼ 1; . . . ;5; ðD:1Þ
where Ajðk; rÞ ¼Arj 00 Ajðk; rÞ
� �, Bj ¼
Brj 00 Bj
� �, Cij ¼
0 00 Cij
� �,
A1 ¼�0:46637 0 �0:013703
0:41972 �0:46637 0
0 �0:08117 �0:46637
264
375;
A2 ¼�0:46637 0 �0:035303
0:052859 �0:46637 0
0 �1:0817 �0:46637
264
375;
A3 ¼�0:46637 0 0:29891
�1:8016e� 5 �0:46637 0
0 �0:20433 �0:46637
264
375;
A4 ¼�0:46637 0 0:41972
�0:08014 �0:46637 0
0 �0:013653 �0:46637
264
375;
A5 ¼�0:46637 0 0:099206
�0:49461 �0:46637 0
0 0:058579 �0:46637
264
375;
Ar1 ¼�0:5 0 �1:4452
�1:0311 �0:5 0
0 �0:11258 �0:5
264
375;
Ar2 ¼�0:5 0 �1:7195
�1:786 �0:5 0
0 �0:082919 �0:5
264
375;
Ar3 ¼�0:5 0 �1:2062
�0:97277 �0:5 0
0 �0:1657 �0:5
264
375;
Ar4 ¼�0:5 0 �0:84548
�0:31308 �0:5 0
0 �0:52263 �0:5
264
375;
Ar5 ¼�0:5 0 �1:6181
�0:098182 �0:5 0
0 �1:5431 �0:5
264
375:
Bj ¼1 0 00 1 00 0 1
264
375; Brj ¼
0:2 0 00 0:2 00 0 0:2
264
375; j ¼ 1; . . . ;5:
C11 ¼�0:01 0 �0:0015003
0 0 0
0 0 0
264
375;
C12 ¼�0:01 0 �0:0038654
0 0 0
0 0 0
264
375;
C13 ¼�0:01 0 0:032728
0 0 0
0 0 0
264
375;
C14 ¼
�0:01 0 0:045956
0 0 0
0 0 0
2664
3775;
C15 ¼
�0:01 0 0:010862
0 0 0
0 0 0
2664
3775;
C21 ¼0 0 0
0:045956 �0:01 0
0 0 0
2664
3775;
C22 ¼
0 0 0
0:0057876 �0:01 0
0 0 0
2664
3775;
C23 ¼0 0 0
�1:9726e� 6 �0:01 0
0 0 0
2664
3775;
C24 ¼
0 0 0
�0:0087747 �0:01 0
0 0 0
2664
3775;
C25 ¼0 0 0
�0:054156 �0:01 0
0 0 0
2664
3775;
C31 ¼
0 0 0
0 0 0
0 �0:0088876 �0:01
2664
3775;
C32 ¼
0 0 0
0 0 0
0 �0:11844 �0:01
2664
3775;
C33 ¼0 0 00 0 00 �0:022373 �0:01
264
375;
C34 ¼0 0 00 0 00 �0:0014949 �0:01
264
375;
C35 ¼0 0 00 0 00 0:006414 �0:01
264
375:
36 B.-S. Chen et al. / Mathematical Biosciences 230 (2011) 23–36
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