Modelling and Control of Gene Regulatory Networks for Perturbation Mitigation

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1 Tis article as been accepted for publication in a future issue of tis journal, but as not been fully edited. Content may cange prior to final publication. Citation information: DOI.9/TCBB.., IEEE/ACM IEEE/ACM TRANSACTIONS ON COMPUTATIONAL BIOLOGY AND BIOINFORMATICS Modelling and Control of Gene Regulatory Networks for Perturbation Mitigation Matias Foo, Jongrae Kim and Declan G. Bates Abstract Syntetic Biologists are increasingly interested in te idea of using syntetic feedback control circuits for te mitigation of perturbations to gene regulatory networks tat may arise due to disease and/or environmental disturbances. Models employing Micaelis-Menten kinetics wit Hill-type nonlinearities are typically used to represent te dynamics of gene regulatory networks. Here, we identify some fundamental problems wit suc models from te point of view of control system design, and argue tat an alternative formalism, based on so-called S-System models, is more suitable. Using tools from system identification, we sow ow to build S-System models tat capture te key dynamics of an example gene regulatory network, and design a genetic feedback controller wit te objective of rejecting an external perturbation. Using a sine sweeping metod, we sow ow te S- System model can be approximated by a linear transfer function and, based on tis transfer function, we design our controller. Simulation results using te full nonlinear S-System model of te network sow tat te syntetic control circuit is able to mitigate te effect of external perturbations. Our study is te first to igligt te usefulness of te S-System modelling formalism for te design of syntetic control circuits for gene regulatory networks. Index Terms System identification, gene regulatory networks, feedback control systems, S-System model I. INTRODUCTION In complex engineering networks suc as transportation systems, power grids, irrigation networks, etc, te presence of external perturbations can ave serious adverse effects on te functioning of te overall system. Tese undesirable effects include gridlock in te movement of veicles, major power outages in residential and industrial areas, and unreliable water supply to farming areas. In view of tis, te problem of developing a compreensive teory of network control, particularly in te presence of perturbations, as recently been te subject of intensive studies tat ave provided many useful tools for te control of complex networks (see e.g. [], [], [3], [], []). Due to advances in tis area, syntetic biologists ave recently began to investigate te application of te aforementioned tools to te control of biological networks and systems. Some notable examples can be found in [6], [], [8], [9], [], were strategies based on feedback control teory ave been used to analyse te controllability, observability and stability Tis work was supported by Engineering and Pysical Sciences Researc Council (EPSRC) and Biotecnology and Biological Sciences Researc Council (BBSRC) via researc grant BB/M98/. M. Foo and D.G. Bates are wit Warwick Integrative Syntetic Biology Centre, Scool of Engineering, University of Warwick, Coventry, CV AL, UK. J. Kim is wit Scool of Mecanical Engineering, University of Leeds, Leeds, LS 9JT, UK. M.Foo@warwick.ac.uk, menjkim@leeds.ac.uk, D.Bates@warwick.ac.uk of biological networks suc tat appropriate sets of control design rules can be developed. In tis paper, we focus our attention on te control of gene regulatory networks. Te ability to control te dynamics of gene regulatory networks using feedback, especially in te presence of perturbations, as many potential applications in te field of syntetic biology, were syntetic circuits can be developed to implement te proposed controllers and ence curb te effect of external perturbations due to disease or environmental canges. We investigate wat types of network models are most appropriate to describe gene regulatory networks for te purposes of feedback controller design, and sow ow system identification tecniques can be used to build suc models based on available gene expression data. Using te identified models, we design a feedback controller tat can be implemented genetically in order to mitigate te effect of perturbations tat enter te network. Te paper is organised as follows. In Section II, we present an example gene regulatory network for wic we need to build a model for te purposes of control system design. In Section III, we evaluate different types of possible models for gene regulatory networks from te perspective of controller design. Based on tis analysis, in Section IV we propose a system identification approac for building models of gene regulatory networks based on te so-called S-System modelling formalism. Te corresponding controller design procedure for perturbation mitigation is described and closedloop simulation results are provided in Section V. Conclusions are given in Section VI. An early version of tis work was presented in []. II. EXAMPLE GENE REGULATORY NETWORK Te DREAM in silico gene regulatory network callenge was establised to serve as a bencmark to assess different proposed approaces to infer gene regulatory networks from given experimental data [], [3], []. Typically, time-series data for eac gene (or node) in te network are provided and te aim is to infer te underlying network, i.e. identify interconnecting edges, te direction of information flow, etc. Te provided gene regulatory networks are typically subsets of actual transcriptional networks in model organisms suc as E. coli and S. cerevisiae, and ence are representative of real biological systems. In tis paper, we coose te DREAM3 Size data set (ereafter we use te term DREAM3 to denote tis network), wic consists of mrna temporal data on a network composed of interconnecting genes tat is a subset of a S. Tis work is licensed under a Creative Commons Attribution 3. License. For more information, see ttp://creativecommons.org/licenses/by/3./.

2 Tis article as been accepted for publication in a future issue of tis journal, but as not been fully edited. Content may cange prior to final publication. Citation information: DOI.9/TCBB.., IEEE/ACM IEEE/ACM TRANSACTIONS ON COMPUTATIONAL BIOLOGY AND BIOINFORMATICS (A) (B) (C) Disturbance Disturba 8 U System Identification U 9 8 U U3 8 U U 9 6 Control Design U 9 6 U3 6 U3 Controller K - Set-point ++ Fig. : (A) DREAM3 gene regulatory network. Purple circles represent genes and red rectangles represent external inputs. Te arrow denotes te direction of te regulation. (B) Using system identification, te types of regulation in te network are identified. Arrow ead indicates activation and Bar ead indicates inibition. (C) Proposed control design configuration for rejecting te effect of perturbation. Te patway igligted in yellow indicates te series of regulations involved from te control action, U3 to te output gene, N cerevisiae gene regulatory network. As te dataset does not include separate protein data, in te following, we make te following two assumptions: (i) te temporal evolution of te protein is similar to te mrna and (ii) te protein is linearly translated from mrna. Following tese two assumptions, we can lump te protein dynamics into te transcription rate of te mrna at steady state, and tis results in a complete network tat can be described solely using mrna levels. In tis DREAM3 data set, information regarding te interconnectivity between eac gene is provided, wile te regulation type (i.e. activatory or inibitory) is unknown. Te depiction of tese interactions is sown in Fig. (A). To facilitate te controller design procedure, a model describing te dynamics of te DREAM3 network is required, and in te following section, we discuss te selection of an appropriate modelling formalism for te DREAM3 gene regulatory network. or inibitory) between eac gene in te network is known. In te event tat te type of regulation is unknown, ten tis model structure is not suitable as te structure of an activation or an inibition type of regulation is different and arbitrarily assigning tem in te model building stage could tus lead to poor model accuracy. Perturbation D 3 III. M ODEL F ORMALISMS FOR C ONTROLLER D ESIGN A. Micaelis-Menten and Hill-type models Model structures employing Micaelis-Menten and Hilltype nonlinearities are commonly used to describe te dynamics of gene regulatory networks. If te regulation type and te cooperative binding are known, te modeller can eiter specify k NP Fa KM + NP () for an activation type of regulation or Fi k KM + NP () for an inibition type of regulation. In bot Eqns. () and (), NP is te transcription factor, k and KM are associated wit te Micaelis-Menten constants and is te Hill coefficient. In te context of network inference, tis type of model structure can be used only if te type of regulation (activatory 6 Controller U K - Set-point ++ Fig. : Model of a gene regulatory network taken from [] wose dynamics are represented using Micaelis-Menten kinetics and Hill-type nonlinearities. For tis illustration, te controller, K is a simple proportional-integral (PI) controller wit te controller gains, K p. and KI.. Te patway igligted in yellow indicates te series of regulations involved from te control action, U to te output gene, N3 A more fundamental problem in te context of syntetic biology is tat models of tis type are often not suitable for subsequent use in te design of syntetic controllers. For example, let us consider Eqn. () and assume tat our Tis work is licensed under a Creative Commons Attribution 3. License. For more information, see ttp://creativecommons.org/licenses/by/3./.

3 Tis article as been accepted for publication in a future issue of tis journal, but as not been fully edited. Content may cange prior to final publication. Citation information: DOI.9/TCBB.., IEEE/ACM IEEE/ACM TRANSACTIONS ON COMPUTATIONAL BIOLOGY AND BIOINFORMATICS 3 control action (i.e. output of te controller) is given by N P. If N P, ten F a k N P /N P k, wic renders te control action ineffective. It is tus imperative tat te value of sould be sufficiently large to ensure proper control, but as we will sow below, obtaining a reliable estimate of from time series data is often problematic. To illustrate te problem, we consider a model of a simple gene regulatory network taken from [], consisting of seven interconnecting genes, as sown in Fig., based on a subset of an E. coli gene regulatory network. Assume tat an external perturbation enters te network troug gene, its effect on gene 3 is measured, and fed back to a controller tat regulates gene 6 troug te input U. Using te standard modelling framework employing Micaelis-Menten kinetics and Hilltype nonlinearities, te associated Ordinary Differential Equations (ODEs) describing Fig. are given as follows: dn dn dn 3 k, (, + D ) + γ N k, (, + N ) + k,3n 3 (,3 + N 3 ) + k,n (, + N ) + γ N k,n (, + N ) + k,6,6 + N + k,n (, + N ) + k,8n (,8 + N ) + γ 3N 3 dn k,9 (,9 + N ) + γ N dn k, (, + N ) + γ N dn 6 dn k,u (, +U ) + γ 6N 6 k,n, + N + k,3n6,3 + N6 + γ N (3) were k, j,, j wit j,... and are te parameters associated wit te Micaelis-Menten coefficients and Hill-type nonlinearities, and γ is associated wit te degradation term. Witout loss of generality, for te purposes of illustration, we coose. Te rest of te parameters describing Eqn. (3) are sown in Table I. Tese parameters are estimated from available experimental data in [], were one data set is used for parameter estimation and an independent data set is used for model validation. Te parameters are estimated using te prediction error metod wit quadratic criterion, i.e., ˆΘ argmin Θ L T i L t [N i (t) ˆN i (t,θ)] () were T is te number of genes, L is te lengt of te data, Θ {k, j,, j γ j } wit j denotes te appropriate index describing te parameters in Eqn. (). N i and ˆN i represent te real experimental data and simulated data from Eqn. (3) respectively. Eqn. () is solved using MATLAB function fminsearc, wic uses te Nelder-Mead simplex algoritm. For te controller, we coose a standard proportional-integral (PI) controller wit te proportional gain, K P. and te integral gain K I., were tese parameters can be selected using standard rules, suc as te Ziegler-Nicols tuning rules (see e.g. [6]). TABLE I: Parameters for te network model sown in Fig. using Micaelis-Menten wit Hill-type nonlinearities model structure. Gene Parameter Values N k,.36,,.9, γ -.6, N k,.6,,.93, k,3.3,,3.69, k,.,,.88, γ -.36 N 3 k,.,,.88, k,6.8,,6.99, k,.8,,.666, k,8.68,,8., γ 3-3.8, N k,9.93,,9.699, γ -.6 N k,.6,,.96, γ -.66 N 6 k,.6,,.89, γ 6 -. N k,.9,,.9, k,3.36,3.986, γ In our simulation, sown by te solid blue line in Fig. 3, wen te perturbation enters te network at time s it causes te expression level of N 3 to drop from its intended reference value of.8 (Fig. 3(A)). Upon sensing tis drop in te expression level, te controller asserts appropriate control action, U (Fig. 3(C)) in its attempt to bring te expression level of N 3 back to.8. However, as sown in Fig. 3(A), a full recovery of te output to its intended reference value is not acievable. Tis is because in te controller s attempt to perform te needed recovery, te exerted control action U becomes larger tan,, tus te term k, U/(, +U) k,.6, wic is sown in Fig. 3(D). Tis implies no appropriate control action can be given to te network to counter te effect of te perturbation, resulting in a large error between te output and reference value (Fig. 3(B)). In reality, owever, tis may not necessarily be te case - te apparent limitation is due to te estimated value of, from te available experimental data. If te value of, is sufficiently larger tan U, te saturation issue is avoided. In addition, a closer look at te series of regulation along te patway igligted in yellow sown in Fig. indicates tat te values of,8 and,3 also need to be sufficiently large in order to acieve a proper control action and recover te levels of N 3. Te problems identified above are due to te values of,,,8 and,3 tat are estimated from te available experimental data. Tese estimated values are relatively small wen compared to te necessary control action, leading to saturated responses and large errors. Tus, a natural question arises as to weter or not tese values (sown in Table I) represent reliable estimates of tese parameters. For te network sown in Fig., te estimated values of,,,8 and,3 sown in Table I are te result of using as te initial values for te parameters in te optimisation problem defined in Eqn (). If a different set of initial values is used for te optimisation, do we obtain similar parameter values to tose sown in Table I particularly for,,,8 and,3? To investigate tis, we repeated te parameter estimation using.,., and as initial values for te optimisation, and te results are sown in Figs. (A), (C) and (E). Tis work is licensed under a Creative Commons Attribution 3. License. For more information, see ttp://creativecommons.org/licenses/by/3./.

4 Tis article as been accepted for publication in a future issue of tis journal, but as not been fully edited. Content may cange prior to final publication. Citation information: DOI.9/TCBB.., IEEE/ACM IEEE/ACM TRANSACTIONS ON COMPUTATIONAL BIOLOGY AND BIOINFORMATICS (A) N (C) 3 Reference, > U, < U Control action, U 6 8 (B).3.. (D).8 Error k, U/(, + U) 6 8 Fig. 3: Feedback control response wen perturbation enters te gene regulatory network sown in Fig.. (A) Comparison wit te output and reference values. (B) Error signal between te reference and output values. (C) Control action, U. (D) Te time series of k, U/(, +U). TABLE II: Estimated parameters given different initial values for optimisation as sown in Fig. (A), (C) and (E). Initial Gene Parameter Values Value. N 3 k,.66,,., k,6.6,,6.39, k,.9,,.3, k,8.83,,8.8, γ , N 6 k,.39,,.8, γ N k,.93,,.6, k,3.88,3., γ N 3 k,.836,,.83, k,6.3,,6., k,.,,., k,8.868,,8.9, γ 3-3.6, N 6 k,.,,.9, γ N k,.9,,.33, k,3.86,3.68, γ N 3 k,.,,.88, k,6.8,,6.99, k,.8,,.666, k,8.68,,8., γ 3-3.8, N 6 k,.6,,.89, γ 6 -. N k,.9,,.9, k,3.36,3.986, γ N 3 k,.8,,.9, k,6.396,,6.36, k,.93,,.36, k,8.98,,8.69, γ , N 6 k,.,, 9.99, γ 6 -. N k,.9,,.638, k,3.6,3.3, γ N 3 k,.8,,., k,6.6,,6., k,.9,,.99, k,8.83,,8., γ , N 6 k,.9,,.9, γ N k,.,,.669, k,3.96,3 99.8, γ Te plots sow tat te estimated parameter values are very different to te ones sown in Table I. Using terminology from te field of system identification, tere is no consistent estimate of te model parameters, as given different initial values for te optimisation, te optimiser can find different sets of parameters (see Table II) tat are equally well able to reproduce te experimental data, as sown in Figs. (A), (C) and (E). (A) (C) (E) N 3 K ~ M ~ N 6 N ~. ~. ~ Exp. Data (B) (D) (F) N 3 p: p p: p N N p ~. p:. p p ~ Exp. Data Fig. : Comparison of model and experimental data for different sets of estimated parameter given different initial values for optimisation. Te initial values used for optimisation are.,.,, and. Only genes in te igligted patway in Fig. are sown. Te experimental data sown ere is an independent data set tat is not used for parameter estimation. Left panel: Subfigures (A), (C) and (E) sow te plots using Micaelis-Menten wit Hill-type nonlinearities model structure for genes 3, 6 and respectively. Here, te estimated values of are close to te initial set of parameters used for optimisation. Rigt panel: Subfigures (B), (D) and (F) sow te plots using S-System model structure for genes 3, 6 and respectively. Te notation p denotes te parameter set obtained wen initial value of is used for te optimisation (sown in Table I). Te notation p :.,, p indicates te estimated parameters using initial values of., and are similar to p. From Table II, we note tat tere is one set of parameters tat includes large values of,,,8 and,3. Using tese larger values of,.9,,8. and,3 99.8, we repeat te simulation of te feedback controller sown in Fig.. As sown by te solid red line in Fig. 3(A), te same controller is now able to exert a proper control action to mitigate te effect of te perturbation, as te value of, is now larger tan te control action, U (Fig. 3(C)) and no issues wit saturation are observed (Fig. 3(D)). Te results sown ere suggest tat for tis typical experimental data set and network structure, te estimated values of te model parameters, in particular,,,8 and,3, are not consistent. Tis clearly poses a significant problem wen designing a controller to mitigate te effects of perturbations on tis network, since different estimated values of,,,8 and,3 lead to very different closed-loop beaviour of te control system. In ligt of tis, coupled wit te previously mentioned need for a priori knowledge of regulation type to use te Micaelis-Menten wit Hill-type nonlinearities model structure, an alternative modelling formalism is clearly Tis work is licensed under a Creative Commons Attribution 3. License. For more information, see ttp://creativecommons.org/licenses/by/3./.

5 Tis article as been accepted for publication in a future issue of tis journal, but as not been fully edited. Content may cange prior to final publication. Citation information: DOI.9/TCBB.., IEEE/ACM IEEE/ACM TRANSACTIONS ON COMPUTATIONAL BIOLOGY AND BIOINFORMATICS required in order to allow for te rational design of feedback controllers. Te alternate model formalism needs to ave a general structure tat can accommodate bot activatory and inibitory regulations, and more importantly, te estimated model parameters from experimental data sould be consistent, so tat it can be reliably used for controller design. B. S-System models Te so-called S-System modelling formalism as been proposed as an alternative approac to describe te dynamics of gene regulatory networks. Te S-System modelling framework was originally developed from te field of biocemical system teory (see e.g. [], [8]), and wen it as been used to describe te dynamics of gene regulation (see e.g. [9], []), it as been sown to be as accurate as Micaelis-Menten wit Hill-type nonlinearity models (see []). In particular, te autors in [] rigorously analysed te validity range of te concentrations produced by bot S-System and Micaelis- Menten models to determine wic model differs most from te true concentration obtained via experiment. It was found tat, not only were S-System models as accurate as Micaelis- Menten type models witin te same concentration range, but te S-System models were more accurate over a wider range of concentrations. Based on tis and oter analyses, te autors suggested tat te S-System model formalism better represents te actual biocemical system. Te S-System models we consider in tis work ave te following form: dn i a i M j N p i, j j + b i M N q M 3 i, j j + j j c i, j U j () were i denotes te number of biocemical component, a i >, b i < and c i, j (,+ ) are constants, N i represents te biocemical component, M and M are te total number of components involved in te interaction, U j is te external input and M 3 is te number of input. Te power exponent terms, p i, j and q i, j are associated wit te production and degradation terms respectively. For simplicity, we assume q i, j trougout tis paper, so tat a positive value for te parameter p i, j represents activation wile a negative value represents inibition. Tus, te S-System model as a general structure tat can accommodate eiter an activation or inibition type of regulation via te sign of p i, j, and no prior knowledge of te type of regulation at eac node in te network is required in te model building process. Te S-System model describing te gene regulatory network sown in Fig. is given as follows: dn b N + c D + d dn a N p, N p, 3 N p,3 + b N dn 3 a 3 N p 3, N p 3, N p 3,3 N p 3, + b 3 N 3 dn a N p, + b N dn a N p, + b N dn 6 b 6 N 6 + c 6 U dn a N p, N p, 6 + b N (6) Note tat for dn /, a constant value denoted by d is added to te model to ensure te overall mrna level stays positive since D is negatively correlated wit N and b is negative due to te degradation term. Like before, we use one set of experimental data for parameter estimation and an independent set of data for model validation. Te parameters are estimated using te prediction error metod wit quadratic criterion (Eqn. ()) wit Θ {a i,b i,c i,d, p i, j } were i and j denote te appropriate indices in Eqn. (6). Te estimated parameters, using as te initial value for all parameters in te optimisation, are given in Table III. TABLE III: Parameters for te network model sown in Fig. using S-System model structure. Gene Parameter Values N b -.389, c -.88, d., N a.9, p, -.9, p,., p,3.36, b -. N 3 a 3.688, p 3,.3, p 3, -.68, p 3,3., p 3,.396, b 3-6.3, N a.69, p, -.893, b N a., p, -.8, b -.8 N 6 b , c 6.33 N a.96, p,., p,.8 b We repeat te feedback control design using te same configuration sown in Fig.. Te feedback control response wen a perturbation enters te gene regulatory network is sown in Fig.. Wen te S-System model is used, te controller is able to produce an appropriate control action to attenuate te effect of te disturbance. Tere is no saturation issue observed, unlike in te scenario were te Micaelis- Menten wit Hill-type nonlinearities model structure is used. We proceed furter to ceck weter te estimated parameters for te S-System model are consistent or not. As before, we coose te initial parameter values for te optimisation to be.,., and. Te resulting estimated parameters are given in Table IV. Te results sown in Figs. (B), (D) and (F) indicate tat, using tis model structure, te estimated parameters are now consistent. Denoting p as te estimated parameter set obtained wen is used as te initial value for optimisation, we observe tat wen initial values of. and are used, te estimated parameters are close to p (see Table IV). Wen initial values of. and are used, te estimated parameters are not close to p, but in tis case te model responses do not reproduce te experimental data. Taken altogeter, tese results suggest tat we are able to obtain consistent estimates of te model parameters from experimental data wen using te S-System model structure, making tis modelling formalism muc more suitable for use in te design of feedback controllers for perturbation mitigation. Tis work is licensed under a Creative Commons Attribution 3. License. For more information, see ttp://creativecommons.org/licenses/by/3./.

6 Tis article as been accepted for publication in a future issue of tis journal, but as not been fully edited. Content may cange prior to final publication. Citation information: DOI.9/TCBB.., IEEE/ACM IEEE/ACM TRANSACTIONS ON COMPUTATIONAL BIOLOGY AND BIOINFORMATICS 6 TABLE IV: Estimated parameters given different initial values for te optimisation as sown in Figs. (B), (D) and (F). Initial Gene Parameter Values Value. N 3 a 3.8, p 3,.3, p 3, -.98, p 3,3., p 3,., b 3 -.9, N 6 b 6 -., c 6.99 N a.6, p,., p,. b -.. N 3 a 3., p 3,., p 3, -.9, p 3,3.8, p 3,.3888, b 3 -.3, N 6 b 6 -.8, c 6.3 N a.3, p,.3, p,.68 b -.36 N 3 a 3.688, p 3,.3, p 3, -.68, p 3,3., p 3,.396, b 3-6.3, N 6 b , c 6.33 N a.96, p,., p,.8 b N 3 a 3.6, p 3,.9, p 3, -., p 3,3.9, p 3,.36, b , N 6 b , c 6.33 N a.3, p,.3, p,.36 b N 3 a 3 8.3, p 3,., p 3, -.6, p 3,3.9, p 3,.93, b 3 -., N 6 b 6-9.8, c N a.89, p,., p, 6.6 b -.3 [a.u] (A) N (C) Control action, U 8 6 Reference Output N (B).3.. (D) Error 6 8 c 6 U 6 8 Fig. : Feedback control response wen a perturbation enters te gene regulatory network tat is modelled using te S- System formalism. (A) Output and reference values. (B) Error signal between te reference and output values. (C) Control action, U. (D) Te time series of c 6 U. IV. IDENTIFICATION OF AN DREAM3 NETWORK USING S-SYSTEM MODEL In te previous section, we ave illustrated wy te S- System model formalism is a more appropriate way to model gene regulatory networks for te purposes of control system design. We now proceed to use te S-System model structure to identify, model, and design a biologically implementable perturbation mitigation controller for te DREAM3 network. Fig. (A) sows te interconnection between te genes in te DREAM3 gene regulatory network. In contrast to te network sown in Fig., ere no information is provided regarding te type of regulation between te interconnecting genes, and terefore we use system identification tecniques (see e.g. []) to infer te type of regulation witin te network. Note tat, since no information regarding te type of regulation between te interconnecting genes is available, te Micaelis- Menten wit Hill-type nonlinearities model structure cannot be used in tis case. System identification tecniques ave been used to build models of gene regulatory networks in several previous studies, including [3], [], [], were linear black box network models were considered and te directions and te types of regulation were identified based on available data on gene expression profiles. In tis paper, we consider a nonlinear grey box S-System model, given tat we ave prior knowledge about te network interconnections, and focus our attention on te identification of te type of regulation between te interconnecting genes. We use one data set for parameter estimation and anoter independent data set for model validation. Note tat bot te estimation and validation data sets used are te provided temporal profiles from te DREAM3 gene regulatory network callenge. Te S-System model for te DREAM3 gene regulatory network following Fig. (A) is given by dn a N p, N p, N p,3 + b N dn b N + c U dn 3 a 3 N p 3, N p 3, + b 3 N 3 dn a N p, 9 + b N dn a N p, + b N dn 6 a 6 N p 6, + b 6 N 6 dn a N p, 8 + b N dn 8 b 8 N 8 + c 8 U dn 9 b 9 N 9 + c 9 U 3 + d 9 dn a N p, + b N () Again note tat for dn 9 /, a constant value denoted by d 9 is added to te model to ensure tat te overall mrna level stays positive since U 3 is negatively correlated wit N 9 and b 9 is negative due to te degradation term. Te parameters are estimated using Eqn. () wit Θ {a i,b i,c i,d 9, p i, j } and T. Using as te initial value for all parameter in te optimisation, te estimated parameters of Eqn. () are given in Table V. Fig. 6 sows te comparison between te S-System model and te real data on te validation data set. Te initial conditions for solving te ODEs are te first data points of eac gene taken from te experimental data set. From te estimated parameters sown in Table V, we are able to determine te type of regulation in te network, were a positive value of te power term denotes activation Tis work is licensed under a Creative Commons Attribution 3. License. For more information, see ttp://creativecommons.org/licenses/by/3./.

7 Tis article as been accepted for publication in a future issue of tis journal, but as not been fully edited. Content may cange prior to final publication. Citation information: DOI.9/TCBB.., IEEE/ACM IEEE/ACM TRANSACTIONS ON COMPUTATIONAL BIOLOGY AND BIOINFORMATICS TABLE V: Estimated parameters for te DREAM3 S-System model. Gene Parameter Values N a., p,.3, p,.9, p,3 -.89, b -.3 N b -.8, c.96 N 3 a 3.8, p 3, -., p 3,.393, b N a.3, p, -.6, b -.3 N a.99, p,.6, b -. N 6 a 6.6, p 6, -., b N a.6, p,., b -.3 N 8 b , c 8.8 N 9 b , c , d 9.33 N a.39, p, -.69, b -.8,,...,. Table VI sows te computed MSE for bot te estimation and validation data sets. TABLE VI: MSE for bot estimation and validation data sets. MSE MSE Gene (Estimation) (Validation) N.9. N.3. N 3..3 N.9. N..3 N N.9.6 N N N..8 MSE T... N N 3. N. N. N N. N 6. N 8 Exp. Data Model Te total MSE, MSE T, is obtained by summing all te individual MSE from eac genes. In general, te MSE values are small and similar between te two data sets. Wit te regulation types in te DREAM3 network as identified, te network interactions are as sown in Fig. (B). A. Modelling of DREAM3 wit Micaelis-Menten wit Hilltype nonlinearties Now tat te regulation types between eac node (activation or inibition) ave been identified, we can also use Micaelis- Menten wit Hill-type nonlinearities to model te DREAM3 network, as follows:.. N 9 N dn k,n, + N + k,n, + N + k,3,3 + N + γ N.. Fig. 6: Comparison between S-System model and DREAM3 data on te validation data set tat is not used for parameter estimation. wile a negative value of te power term denotes inibition. Reassuringly, all te a priori known degradation terms were identified to ave negative values, in accordance wit current biological data on te network. Te comparison between te S-System model and te real data on te validation data set sows good agreement, suggesting a good level of accuracy of te model. To quantify tis, we calculate te Mean Square Error (MSE) for eac gene between te S-System model and te real data. Te MSE is computed using, MSE L L t [N i (t) ˆN i (t,θ)] (8) were L is te lengt of te data, N i and ˆN i respectively represent te experimental and te simulated data and i dn k,u, +U + γ N k, dn 3, + N + k,6n,6 + N + γ 3 N 3 dn k,, + N9 + γ N dn dn 6 dn k,8n,8 + N + γ N k,9n,9 + N + γ 6 N 6 k,n8, + N8 + γ N dn 8 k,u, +U + γ 8 N 8 dn 9 k,, +U3 + γ 9 N 9 dn k,3,3 + N + γ N (9) We want to investigate weter te Micaelis-Menten wit Hill-type nonlinearities model would encounter te same problem of inconsistent parameter estimates as igligted in Section III-A. For te purposes of illustration, we focus Tis work is licensed under a Creative Commons Attribution 3. License. For more information, see ttp://creativecommons.org/licenses/by/3./.

8 Tis article as been accepted for publication in a future issue of tis journal, but as not been fully edited. Content may cange prior to final publication. Citation information: DOI.9/TCBB.., IEEE/ACM IEEE/ACM TRANSACTIONS ON COMPUTATIONAL BIOLOGY AND BIOINFORMATICS 8 only on te igligted patway tat involves te series of regulation from te control action to te output gene (see Fig. (C)) and as before set. We repeat te parameter estimation exercise (i.e., using Eqn. ()) were we coose.,., and as te initial values for te optimisation for bot Micaelis-Menten wit Hill-type nonlinearities and S-System model structures, focusing only on genes, and 9. Te results are sown in Fig. and te estimated model parameters are given in Tables VII and VIII TABLE VII: Estimated parameters given different initial values for te optimisation as sown in Figs. (A), (C) and (E). Initial Gene Parameter Values Value. N k,.68,,.69, k,.86,,., k,3.6,,3.96, γ , N k,.,,., γ -.6 N 9 k,.38,,., γ N k,.6,,.69, k,.,,.9, k,3.8,,3.9633, γ -.968, N k,.3,,., γ -.6 N 9 k,.3,,., γ 9 -. N k,.6868,,.9, k,.99,,.63, k,3.98,,3., γ -.99, N k,.,,.8, γ -.6 N 9 k,.33,, 8.3, γ N k,.39,,.93, k,.,, 9.33, k,3.6,,3.99, γ -.6, N k,.,, 9.6, γ -.63 N 9 k,.36,,.3, γ 9 -. N k,.99,,.88, k,.369,, 83.9, k,3.,,3.68, γ -.66, N k,.,, , γ -.63 N 9 k, 9.33,,.3, γ (A). (C).. N (E)..8.6 N N 9 ~. ~. ~ ~ ~ Exp. Data (B). (D).. N (F)..8.6 N N 9 p:. p p:. p p: p p ~ p ~ Exp. Data Fig. : Comparison of model and experimental data for different sets of estimated parameter given different initial values for optimisation. Te initial values used for optimisation are.,.,, and. Only genes in te igligted patway sown in Fig. (C) are sown. Te experimental data sown ere is an independent data set tat is not used for parameter estimation. Left panel: Subfigures (A), (C) and (E) sow te plots using Micaelis-Menten wit Hill-type nonlinearities model structure for genes, and 9 respectively. Here, te estimated values of are close to te initial values for optimisation. Rigt panel: Subfigures (B), (D) and (F) sow te plots using S-System model structure for genes, and 9 respectively. Te notations p and p :.,.,,, p follow te same interpretation given in previous section. TABLE VIII: Estimated parameters given different initial values for te optimisation as sown in Figs. (B), (D) and (F). Initial Gene Parameter Values Value. N a.8, p,.3, p,.39, p,3 -.9, b -.3, N a., p, -., b -.66 N 9 b 9 -.3, c 9 -.3, d 9.. N a.89, p,.6, p,.9, p,3 -.6, b -.33, N a., p, -.86, b -.6 N 9 b 9 -.3, c 9 -.3, d 9.6 N a., p,.3, p,.9, p,3 -.89, b -.3, N a.3, p, -.6, b -.3 N 9 b , c , d 9.33 N a.39, p,.9, p, 9.989, p, , b -.6, N a., p, -.639, b -. N 9 b , c , d 9.33 N a.38, p,.399, p,.689, p, , b -.6, N a.3, p, -.36, b -.6 N 9 b 9 -., c , d As sown in Fig., te estimated parameters using te Micaelis-Menten wit Hill-type nonlinearities model are not consistent, as different sets of parameter are able to reproduce te dynamics of te experimental data equally well. For te S- System model, owever, we obtain consistent estimates of te model parameters for genes and 9 wen te initial values used for optimisation are.,. and, wile for initial values of and, te resulting parameters cannot reproduce te experimental data. For gene, we obtain consistent estimates of te model parameters wen te initial values used for optimisation are, and, wile for initial values of. and. tere is again poor agreement between model responses and experimental data. B. Discussion on te parameter estimates of te model structures Troug our analysis of different modelling formalisms for te gene regulatory networks considered ere, we ave illustrated te inconsistent estimates of te model parameters obtained wen using Micaelis-Menten wit Hill-type nonlinearities model. Tis means tat tese model parameters are not identifiable from te available experimental data. One reason for tis could be tat tese experimental data do not excite te relevant dynamics (in particular te saturation region) tus making te data not informative enoug to obtain a Tis work is licensed under a Creative Commons Attribution 3. License. For more information, see ttp://creativecommons.org/licenses/by/3./.

9 Tis article as been accepted for publication in a future issue of tis journal, but as not been fully edited. Content may cange prior to final publication. Citation information: DOI.9/TCBB.., IEEE/ACM IEEE/ACM TRANSACTIONS ON COMPUTATIONAL BIOLOGY AND BIOINFORMATICS 9 consistent estimate. Tis inconsistent estimate is related to te notion of practical parameter identifiability (see e.g. [6], []) were te available experimental data is unable to excite te relevant dynamics to provide consistent estimate for a given model structure, as observed ere. Te problem of inconsistent parameter estimates is also observed in [8], were te autors attempt to build a compreensive network model for te plant circadian system, and te interactions between genes are modelled using te Micaelis-Menten wit Hill-type nonlinearities model structure. Te model parameters are estimated from experimental data, wic are te temporal profiles of te circadian genes and proteins and a total of eigt different parameter sets are found to be able to reproduce te experimental data. Te estimated values of te Micaelis- Menten coefficients ( ) from tese eigt sets of parameters cover a large range of possible values (from. to 9). Altoug its relevance from te point of view of control system design as not to-date been considered, te problem of obtaining consistent estimates of parameters in te Micaelis- Menten model structure as been previously investigated (see te review paper [9] and references terein). In [3] and [3], different metods for fitting te Micaelis-Menten equation were analysed, and bot studies concluded tat different fitting metods will give different estimates of te parameters unless te experimental data is free from error (wic in biological reality it never is). Different approaces to estimate te Micaelis-Menten coefficients ave also been studied in [3], [33] and [3], and tose studies concluded tat it is difficult to obtain a consistent estimate of te Micaelis- Menten coefficients unless particular design considerations are taken into account. On te oter and, for te parameters of te S-System model, our two illustrative examples indicate tat tese parameters are locally identifiable [3], as we are able to obtain consistent parameter estimate wen different initial values are used for te optimisation. Te identifiability of model parameters using a power law type of model structure (tat includes te S-System model) as been investigated in [36]. Teir analyses sow tat wile in general it is practically callenging to obtain consistent estimate for all te parameters in te model, one can obtain consistent estimates of te model parameters under certain conditions. Recent work by [3] also sows tat wit an appropriate coice of optimiser, one can obtain consistent parameter estimates using te S-System model structure. V. DESIGN OF A FEEDBACK CONTROLLER FOR PERTURBATION MITIGATION Here, we sow ow te S-System model of te considered gene regulatory network can be used to design a controller for perturbation mitigation. To acieve an implementable design, a genetic-based controller is required, and tere are frameworks available for suc designs (see e.g. [38], [39]). In tis paper, we employ a frequency domain control design metodology, motivated by te design framework proposed in [39]. In order to design controllers in te frequency domain, a linear model is required. As te S-System is a nonlinear model, we linearise it to obtain a transfer function model using te sine sweeping metod (see e.g. [], []). A. Sine sweeping metod In te sine sweeping metod, sinusoidal input signals over te frequency range of interest are given as te inputs to te system. Te output responses witin te frequency range are ten analysed in terms of teir magnitude and pase relative to te input signal. By collecting tese magnitude and pase values, te frequency response and transfer function model of te system can be easily obtained. Here, we summarise te procedure for obtaining a transfer function model using te sine sweeping metod metod and refer readers to [], [] for complete details. Consider a sinusoidal input u(t) Asin(ω t), were A is te amplitude and ω is te frequency. For any linear time invariant system, te output would be also sinusoidal wit te same frequency but wit scaled amplitude and a pase sift. In practice, te output response is subject to transient effects, as well as te effects of nonlinearities and disturbances d(t), yielding, y(t) Bsin(ω t + φ) + d(t) + transient + nonlinearities () were B A G( jω ), φ G( jω ) tan Im G( jω ) and Re G( jω ) G( jω ) is te transfer function relating te input and output wit j denotes te imaginary number. Te effects of transients and nonlinearities can be removed by neglecting te initial part of te data and assuming tat te linear dynamics make te dominant contribution to te overall response. To reduce te effect of d(t) on y(t), one can use a correlation metod [], were te idea is to correlate y wit a sine and cosine of te same frequency and average it over te lengt of te data (see Fig. 8). y(t) sin ω t cos ω t From Fig. 8, we obtain, Fig. 8: Correlation metod. I S ( ) y(t)sin(ω t) t I S ( ) I C ( ) I C ( ) y(t)cos(ω t) () t Substituting Eqn. () into (), and after some algebraic manipulation, we arrive at Tis work is licensed under a Creative Commons Attribution 3. License. For more information, see ttp://creativecommons.org/licenses/by/3./.

10 Tis article as been accepted for publication in a future issue of tis journal, but as not been fully edited. Content may cange prior to final publication. Citation information: DOI.9/TCBB.., IEEE/ACM IEEE/ACM TRANSACTIONS ON COMPUTATIONAL BIOLOGY AND BIOINFORMATICS I S ( ) A G( jω ) cosφ A G( jω ) cos(ω t + φ) + d(t)sin(ω t) t t I C ( ) A G( jω ) sinφ A G( jω ) sin(ω t t + φ) + d(t)cos(ω t) () t From Eqn. (), te second term for bot I S ( ) and I C ( ) will go to zero as. Assuming d(t) is a stationary stocastic process wit zero mean value and covariance function R d (l) suc tat l l R d(l) <, te tird term for bot I S ( ) and I C ( ) will be zero as, since te variance of te tird term decays at a rate of / (see [] for details). From te remaining terms of Eqn. (), te magnitude, G( jω ) and te pase, G( jω ) can be estimated using te following equations, i.e. G( jω ) IS () + IC () A G( jω ) tan I C( ) I S ( ) (3) For te DREAM3 network, we assume tat te input to te network is troug U 3 and te output of interest is te expression of gene N. We apply sinusoidal signals of te form 3sin(ωt) + 3 wit te frequency ω ranging from. rad/s to. rad/s. Despite using a nonlinear model, we note tat te output sinusoidal responses ave te same frequency as te input and no subarmonics are apparent, indicating a dominant linearity of te model. By computing te magnitude and pase values using Eqn. (3), te Bode plot of te DREAM3 network from input U 3 to output N is obtained and sown in Fig. 9. Magnitude [db] Pase [ ] ω [rad/s] Fig. 9: Bode plot of DREAM3 network from input U 3 to output N. From te Bode plot, we note te following: (i) At low frequency, te magnitude of te system is about -.db. (ii) Te corner frequency is. rad/s. (iii) At te corner frequency, te slope is close to -db/dec and te pase is approximately -9, suggesting a second order system wit repeating poles. Tus, te transfer function relating input U 3 to output N can be approximated by N (s) U 3 (s). ( +. s.9 ) s +.s +. () From te sine sweeping metod, te linear transfer function of te gene regulatory from U 3 to N is given by Eqn. (). We compare te accuracy of te linear model wit te nonlinear S- System model troug a step response comparison, as sown in Fig.. Since te base signal level used in te sine sweeping metod is 3, te input is stepped from 3 to. N [a.u] Step response S System Model Linearised Model Fig. : Step response comparison between te linear model obtained troug sine sweeping metod and te full nonlinear S-System model. From Fig., we observe similar performance between te two models in terms of teir transient responses, i.e. similar rise time and settling time. On te oter and, te steady state levels between te two models are different wit te linear model aving a iger steady state level compared to te nonlinear model. Neverteless, te difference between tese two steady state level is relatively small, indicating acceptable accuracy of te linear model in approximating te nonlinear S-System model relating input U 3 to output N. Wit tis transfer function identified, we can proceed wit te design of te controller using a frequency domain approac. B. Design of a genetic pase lag controller Here, we illustrate te design of te genetic pase lag controller. A pase lag controller is cosen, as tis type of controller is typically used to improve disturbance rejection and reduce steady state errors. Te pase lag controller as te following form: Tis work is licensed under a Creative Commons Attribution 3. License. For more information, see ttp://creativecommons.org/licenses/by/3./.

11 Tis article as been accepted for publication in a future issue of tis journal, but as not been fully edited. Content may cange prior to final publication. Citation information: DOI.9/TCBB.., IEEE/ACM IEEE/ACM TRANSACTIONS ON COMPUTATIONAL BIOLOGY AND BIOINFORMATICS K(s) K s + a P + K K (s + a P + K K ) s + a P () were te zero of te controller z (a P + (K /K )) and te pole of te controller p a P, wit te gain of te controller being K. As bot te gain and pase margins of te system obtained from te Bode plot are infinite, our primary focus is on improving te transient dynamics of te disturbance rejection and reducing te steady state error. Te transfer function given in Eqn. () is a type system, and wit te use of a pase lag controller, tere is no integrator in te open loop gain to eliminate te steady state error. As suc, wen coosing te pole of te pase lag controller, we try to place te pole, a P as close as possible to te origin. Likewise, te static error constant, K p.k sould be cosen as large as possible to reduce te steady state error. Te coice of te design parameters are constrained by te acievable biological values and following te range of allowable values given in [39]; te following allowable parameter ranges are adered to:. a P., K <.3 and K <.8. C. Simulation Results Wile te design of te controller is carried out using te linear model, for implementation, we carried out our simulation using te nonlinear S-System network model. In most gene regulatory network perturbation mitigation problems, we are interested in maintaining te steady state level of a particular gene of interest in te presence of a perturbation. Biologically, tis can be interpreted as maintaining te level of expression of a gene of interest to ensure optimal biological function. Tus, in tis simulation example, we are interested in maintaining te steady state level of N at its desired reference value in te presence of a perturbation. Here, we assume tat te perturbation enters te network troug U and our control action is provided by U 3 as depicted in Fig. (C). In te absence of a perturbation, te steady state level of N is.86, tus, our control objective is to maintain te steady state level of N close to.86 in te presence of a perturbation. In our simulation, a perturbation in te form of a step response wit amplitude of enters te network at time s. As can be seen in Fig. (A), witout control, te steady state level of N increase to.63 and is unable to return to its desired value. In te design of te pase lag controller, te following values are cosen. To ave te pole close to te origin, we coose a P.. To ave te static error constant as large as possible, we coose K.. For K, we initially consider two cases, i.e. K. (controller s zero close to origin) and K (controller s zero far from te origin). Te simulation results are sown in Fig. (B). For a small value of K, we see tat te performance of te system is slow and at time 6s, tere is still a noticeable steady state error, i.e... On te oter and, for a large value of K, we see (A) [a.u] (B) [a.u].6.. N (witout control) 3 6 Reference N (wit control) Large K Small K.6 Optimised K Fig. : (A) N set-point regulation (witout control). (B) N set-point regulation (wit control). Black solid line: Set-point. Red dotted line: N response to small K. Blue dased line: N response to large K. Green das-dotted line: N response to optimised K. a significant improvement in te performance, were we get a faster response and an almost zero steady state error, i.e..8. Magnitude [db] Pase [ ] (A) (B) Wit control ω [rad/s] (C) (D) Witout control ω [rad/s] Fig. : (A) & (B) Gain and pase plots of system wit control. Red dotted line: Small K, Blue dased line: Large K. Green das-dotted line: Optimised K. (C) & (D) Gain and pase plots of system witout control. Te Bode plots of te system wit and witout control are sown in Fig.. For a small value of K, we note tat te pase margin of te system is 9. On te oter and, for a large value of K, despite te good performance, we note tat te pase margin of te system reduces from 9 to, wic is less tan typically specified values. Tus, a compromise between te transient performance and overall stability robustness needs to be performed wen designing te controller, and tis trade-off can be effectively managed troug te coice of te controller parameter K. According to standard specifications, te pase margin is typically required to be between to 6 (see e.g. [6]) to acieve Tis work is licensed under a Creative Commons Attribution 3. License. For more information, see ttp://creativecommons.org/licenses/by/3./.

12 Tis article as been accepted for publication in a future issue of tis journal, but as not been fully edited. Content may cange prior to final publication. Citation information: DOI.9/TCBB.., IEEE/ACM IEEE/ACM TRANSACTIONS ON COMPUTATIONAL BIOLOGY AND BIOINFORMATICS satisfactory performance. To find te optimal value of K tat can acieve fast response, small steady state error and acieve a pase margin in te aforementioned range, we proceed as follows. Te transfer functions of te process and te lag compensator are given by Eqns. () and () respectively. Rewriting tem ere togeter wit te substitution of a P. and K., as well as defining G OL (s) as te open loop gain transfer function, we ave te following expression. [ ][(..(s +. + K G OL (s) (. s + ) s +.. ) )] (6) Replacing s jω, and after some algebraic manipulation we ave G OL ( jω) Q(T jω + ) (T jω + ) (T 3 jω + ) () were Q (. + 3K ), T /(. + K. ), T /. and T 3 /.. Te magnitude and pase of G OL ( jω) can be computed as follows, G OL ( jω) log Q + log T jω + log T jω + + log T 3 jω + G OL ( jω) tan (T ω) tan (T ω) + tan (T 3 ω) (8) and we are now left wit te task to find K and ω to acieve our desired pase margin. From te Bode plot in Fig. (A), we observe tat to acieve te desired pase margin would require te gain cross over frequency of G OL ( jω) to be around te frequency. rad/s. Wit ω., solving K suc tat G OL ( jω) and G OL ( jω)+8 6 are satisfied, we obtain te optimal K.8. As sown by te green das-dotted line in Fig. (B), wit K.8, te magnitude plot as sifted to te left. Tis left sift in magnitude canges te gain cross over frequency from. rad/s to te one we specified, i.e.. rad/s. On te oter and, te pase plot is similar to te case wen using large K. Neverteless, more importantly, te Bode plot sown in Fig. (A) and (B) sows tat te new pase margin is. wen using K.8, wic is witin te preferred range and a significant improvement compared to using large K. VI. CONCLUSIONS Altoug several modelling formalisms are now available for te representation of gene regulatory networks, te question of teir suitability for te design of syntetic feedback control systems as so far received little attention in te literature. In tis paper, we sow tat standard modelling approaces employing Micaelis-Menten models wit Hilltype nonlinearities are not appropriate for use in te design of syntetic controllers, for two reasons. Firstly, suc models require te type of regulation between interacting genes in te network to be known a priori, wic is igly unlikely to be te case in general. Even more problematically, te values of te particular parameters in suc models on wic te controller design depends cannot in general be reliably identified from standard time-series data. As an alternative approac, we propose te use of te S- System modelling formalism. Wile te use of te S-System modelling formalism for describing te dynamics of gene regulatory networks is well establised, its usefulness for te purposes of control design as not so far been investigated. Here, we sowed tat using tis modelling formalism combined wit standard system identification procedures allows us to establis te type of regulation between eac gene, obtain consistent estimates of model parameters, and ence derive a model tat is suitable for te design of a syntetic genetic feedback controller. Given tat te design of te considered genetic feedback controller is carried out in frequency domain, we sowed tat te nonlinear S-System model can be approximated by a second order linear transfer function using te sine sweeping metod. Based on tis transfer function model, we designed a genetic pase lag feedback controller, wose structure and parameter values can be readily implemented biologically. Simulation results sow satisfactory performance of te controller in mitigating external network perturbations. Te proposed modelling and control system design approac considered ere as been tailored to te problem of mitigating external perturbations in gene regulatory network. However, te proposed approac can be readily extended to address oter control problems (e.g. reference tracking) and sould ave wide potential application to network control problems trougout te field of syntetic biology. VII. ACKNOWLEDGEMENTS We gratefully acknowledge te financial support EPSRC and BBSRC via researc grants BB/M98/ and from te Scool of Engineering of te University of Warwick. Te autors would also like to tank Prof. Micael Cappell from Scool of Engineering, University of Warwick for useful discussions on parameter identifiability. REFERENCES [] M. Cantoni, E. Weyer, Y. Li, S.K. Ooi, I. Mareels, and M. Ryan, Control of large-scale irrigation networks, Proceedings of te IEEE, vol. 9, no., pp. 9,. [] R.F. Arritt and R.C. Dugan, Distribution system analysis and te future smart grid, IEEE Transactions on Industry Applications, vol., no. 6, pp. 33 3,. [3] S.P. Cornelius, W.L. Kat, and A.E. Motter, Realistic control of network dynamics, Nature Communications, vol., no. 9,. [] D. Scwanenberg, B.P.J. Becker, and M. Xu, Journal of ydroinformatics, Te open real-time control (RTC)-Tools software framework for modeling RTC in water resources sytems, vol., no., pp. 3 8,. [] M. Hajiamadi, B. 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(Hons.) and M.Eng.Sc. degrees in electronic engineering from Multimedia University, Malaysia, in and, respectively, and te P.D. degree from Te University of Melbourne, Melbourne, Australia, in. He was a Post-Doctoral Researc Fellow wit te Asia Pacific Center for Teoretical Pysics (APCTP), Poang, Sout Korea from -. He is currently a Researc Fellow at Warwick Integrative Syntetic Biology Centre, Scool of Engineering, University of Warwick, UK. His researc interests include dynamical system modelling, application of control system and control teory for syntetic biology. Jongrae Kim received te P.D. degree in aerospace engineering from Texas A&M University, College Station, TX, USA, in. Currently, e is an Associate Professor wit te Institute of Design, Robotics & Optimisation (idro) and Aerospace Systems Engineering in te Scool of Mecanical Engineering at te University of Leeds, Leeds, UK. He was a Postdoctoral Researcer wit te University of California, Santa Barbara, CA, USA, in and 3 and a Researc Associate wit te University of Leicester, Leicester, UK. from -. He was a Lecturer in Biomedical Engineering/Aerospace Sciences, University of Glasgow, Glasgow, UK. from -. His main researc interests are in te area of robustness analysis, optimal control and estimation, large-scale network analysis, system identification, dynamics, robotics, systems biology, syntetic biology, and neuroscience. Tis work is licensed under a Creative Commons Attribution 3. License. For more information, see ttp://creativecommons.org/licenses/by/3./.

14 Tis article as been accepted for publication in a future issue of tis journal, but as not been fully edited. Content may cange prior to final publication. Citation information: DOI.9/TCBB.., IEEE/ACM IEEE/ACM TRANSACTIONS ON COMPUTATIONAL BIOLOGY AND BIOINFORMATICS Declan G. Bates received te B.Eng degree in Electronic Engineering and a P.D. degree in Robust Control Teory from te Scool of Electronic Engineering, Dublin City University, Ireland, in 99 and 996 respectively. On completing is PD e joined te Control and Instrumentation Researc Group led by Prof. Ian Postletwaite in te Department of Engineering at Leicester University, were e worked as a post-doctoral researc associate, lecturer, and senior lecturer, before being appointed to a Personal Cair in Control Engineering. In e was appointed Professor of Biological Systems Engineering in te College of Engineering, Matematics and Pysical Sciences of te University of Exeter and in 3 e moved to te University of Warwick as Professor of Bioengineering. His researc is focussed on te modelling, analysis, design and control of complex biological and medical systems. He is Co-Director of te ESPRC/BBSRC Warwick Integrative Syntetic Biology Centre (WISB), and Co-Director at Warwick of te EPSRC/BBSRC Centre for Doctoral Training in Syntetic Biology. Tis work is licensed under a Creative Commons Attribution 3. License. For more information, see ttp://creativecommons.org/licenses/by/3./.