This paper is concerned with validating a mathematical model of regulation in the tryptophan operon using global optimization. Although a number of models for this biochemical network are proposed, in many cases only qualitative agreement between the model output and experimental data was demonstrated, since very little information is currently available to guide the selection of parameter values for the models. This paper presents a model validating method using both multiple experimental data and global optimization

Bates, D.G.

Heslop-Harrison, P.

Kim, J.-R.

Kim, J.-S.

Postlethwaite, I.

English

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       Kim, J.-R. and Kim, J.-S. and Postlethwaite, I. and Heslop-Harrison, P. and Bates, D.G. (2009) Validation of a model of regulation in the tryptophan operon against multiple experiment data using global optimisation. In: ICCAS-SICE International Joint Conference 2009: Fukuoka International Congress Center, Fukuoka, Japan, August 18-21, 2009. Society of Instrument and Control Engineers / IEEE Computer Society, Fukuoka, Japan, pp. 209-211. ISBN 9784907764340  http://eprints.gla.ac.uk/24594/  Deposited on: 26 January 2010   Enlighten – Research publications by members of the University of Glasgow http://eprints.gla.ac.uk - 209 -Validation of a model of regulation in the tryptophan operon against multipleexperiment data using global optimisationJongrae Kim1, Jung-Su Kim2, Ian Postlethwaite3, Pat Heslop-Harrison3, and Declan G. Bates31 Dept. of Aerospace Engineering, University of Glasgow, Glasgow, UK, jkim@aero.gla.ac.uk2 Dept. of Control & Instrumentation Eng., Seoul National University of Technology, Seoul, Korea, jungsu@snut.ac.kr3 Systems Biology Lab, Depts. of Engineering and Biology, University of Leicester, LE1 7RH, UK,{ixp,phh4,dgb3}@le.ac.ukAbstract: This paper is concerned with validating a mathematical model of regulation in the tryptophan operon usingglobal optimization. Although a number of models for this biochemical network are proposed, in many cases onlyqualitative agreement between the model output and experimental data was demonstrated, since very little informationis currently available to guide the selection of parameter values for the models. This paper presents a model validatingmethod using both multiple experimental data and global optimization.Keywords: Model validation, global optimization, tryptophan regulation, parameter identiﬁcation1. INTRODUCTIONMany cellular regulation systems employ multiplefeedback loops to allow fast and efﬁcient adaptation touncertain environments. The various feedback mecha-nisms used by prokaryotes such as E. coli to regulatethe expression of proteins involved in the production ofthe amino acid tryptophan combine to form an extremelycomplex, but highly effective, feedback control system.This system has been the subject of numerous modellingstudies in recent Systems Biology research, with the re-sult that a plethora of different mathematical models oftryptophan regulation may now be found in the literature,see for example [1, 3-5] and references therein. In eachof these modelling studies, the dynamics of the proposedmodel were compared with an extremely limited set ofexperimental data. In addition, in many cases only qual-itative agreement between the model output and experi-mental data was demonstrated, [3], since very little infor-mation is currently available to guide the selection of pa-rameter values for the models. Since many of these mod-els have been derived using diverse assumptions about theexact workings of the underlying feedback mechanismsinvolved, the lack of any strong validation (or invalida-tion) of a particular model has severely limited the degreeof insight achieved into the underlying design principlesof this system.In this paper, it is claimed that if there is a properlystructured model, there must be a parameter set whichmake the model produce responses close to the experi-ment data. We address this issue by showing how globaloptimisation methods may be used to validate SystemsBiology models against multiple experiment data. Wefocus on one particular model of the tryptophan controlsystem, [3], which includes regulation of the trp operonby feedback loops representing repression, feedback in-hibition, and transcriptional attenuation [3]. The modelalso incorporates the effect of tryptophan transport fromthe growth medium as well as the various time delays in-volved in the transcription and translation processes. Weuse global optimisation methods to investigate whether,for the proposed model structure, realistic (i.e. biologi-cally plausible) parameter values can be found so that themodel reproduces the dynamic response of the in vitrosystem to a number of different sets of experimental data.The experimental data is extracted from [6].2. MATHEMATICAL MODEL FORREGULATION OF TRYPTOPHANOPERON2.1 Mathematical modelThemathematicalmodel of the tryptophan control sys-tem considered in this study is taken from [3], and con-sists of the set of nonlinear differential equations (6). InEquation 6, R is total repressor concentration, O is to-tal operon concentration, P is mRNA polymerase (mR-NAP) concentration,OF (t) is free operon concentration,MF (t) is free mRNA concentration,E(t) is total enzymeconcentration, T (t) is tryptophan concentration, Kr isthe repression equilibrium constant, Kt is the the rateequilibrium constant between the total repressor and theactive repressor, μ is the growth rate, kp is the DNA-mRNAP isomerisation rate, b and c are the constants forthe transcriptional attenuation, kρ is the mRNA-ribosomeisomerisation rate, ρ is the ribosomal concentration, kd isthe mRNA destroying rate, D is the mRNA destroyingenzyme, γ is the enzymatic degradation rate constant,Kis the tryptophan production rate, which is proportionalto the active enzyme concentration,Ki is the equilibriumconstant for the Trp feedback inhibition of anthranilatesynthase reaction, which is modelled by a Hill equationwith the coefﬁcient, nH , g is the maximum tryptophanconsumption rate, the internal tryptophan consumption ismodelled by a Michaelis-Menten type with the constantKg , Text is the external tryptophan uptake, d, e, and fare the parameters for modelling the external tryptophanuptake rate, τp is the time taking that mRNAP bindingthe DNA moves away and frees the operon, τm is thetime taking that actually mRNA produces after mRNAPICROS-SICE International Joint Conference 2009August 18-21, 2009, Fukuoka International Congress Center, JapanPR0002/09/0000-0209 ¥400 © 2009 SICEAuthorized licensed use limited to: UNIVERSITY OF GLASGOW. Downloaded on January 26, 2010 at 06:07 from IEEE Xplore.  Restrictions apply. - 210 -dOF (t)dt= KrKr + T (t)T (t) +KtR�μO − kpP�OF (t)−OF (t− τp)e−μτp��− μOF (t) (6a)dMF (t)dt= kpPOF (t− τm)e−μτm�1− b�1− eT (t)/c��− kρρ�MF (t)−MF (t− τρ)e−μτρ�− (kdD + μ)MF (t)(6b)dE(t)dt= 12kρρMF (t− τe)e−μτe − (γ + μ)E(t) (6c)dT (t)dt= K KnHiKnHi + T nH (t)E(t)− g T (t)T (t) +Kg + dTexte+ Text [1 + T (t)/f ] − μT (t). (6d)bound to the DNA, τρ is the time taking that ribosomebinds to mRNA and initiates translation, and τe is anotherribosome binding rate delay for the enzyme. More detailabout the model can be found in [3].All 25 independent parameters are given in Table 1and the dependent parameters are calculated as follows:T¯ = Ki, kρ = 1/(ρτρ), kdD = ρkρ/30, Kg = T¯ /20,g = Tcr(T¯ + Kg)/T¯ , E¯A = E¯KnHi /(KnHi + T¯ nH ),G¯ = gT¯ /(T¯ + Kg), and K = (G¯ + μT¯ )/E¯A, whereT¯ and E¯ are the steady state of tryptophan and enzyme,respectively, and Tcr is the tryptophan consumption rate.2.2 Model structure: feedback loopsThis subsection describes the important regulatorymechanism in regulatory network in the tryptophanoperon and shows the mathematical model under con-sideration includes these mechanism properly. As pre-viously mentioned, there are three feedback mechanismsin the regulatory network: repression, feedback inhi-bition, and transcriptional attenuation [3]. Repressionmeans that when the produced tryptophan bounds tothe repressor, the repressor becomes active which pre-vents the mRNA polymerase from bounding to Operon.The term (active repressor:=)T (t)T (t) +KtR belonging toEquation 6a indicates the repression. Feedback inhibi-tion implies that the anthranilate synthase is feedbackinhibited by the produced tryptophan. The term (ac-tive anthranilate:=)KnHiKnHi + T¯ nHE¯ in Equation 6d de-notes the feedback inhibition. Transcriptional attenua-tion means that the produced tryptophanmakes transcrip-tion abort. Equation 6b represents the transcriptional at-tenuation. We can see that the mathematical model inEquation 6 has a proper structure in the sense that themodel includes most important regulatory network. Inorder to build a useful model for the network, the nextstep is to determine biochemically plausible parameterswith which the model response is close to the experi-ment data. In [3], they identiﬁed the nominal parametersbased on various biological reasonings and approxima-tions. In Figure 1, the dash-dot lines denotes the modelresponses according to three different experimental se-tups. As shown in the ﬁgure, the steady state values areclose to the experimental data but there are large discrep-ancies in the transient period, which means that the modelwith the nominal parameters taken from [3] is not goodenough. In the next section, we present a parameter iden-tiﬁcation method using global optimization and multipleexperiment data.3. MODEL VALIDATION USINGMULTIPLE EXPERIMENT DATA SET3.1 Multiple experiment dataThe experimental data is extracted from [6], which re-ports the results of a number of experiments with wildand mutant strains of the E. coli CY15000 strain. Theseexperiments consisted of growing bacteria in various me-dia which included tryptophan until the culture reacheda steady state. Then the bacteria were washed and putinto the same media without tryptophan. The response ofthe activity of the enzyme anthranilate synthase (the keyenzyme in tryptophan biosynthesis) to these nutritionalshifts was then measured as a function of time. The ac-tivity is proportional to the production rate of tryptophan.3.2 Parameter identiﬁcationThree sets of experimental data are used in this paperand the optimal parameter set is found by minimising thesquare sum of errors between the dynamics of the activeenzyme concentration produced by the model and the ex-perimental data as followsminpJ =N�j=1|x¯(tj)− ˜¯xi(tj)|2 (1)where p denotes the parameters in the model, the sub-script i the index of the experiment, ˜¯xi(tj) measurementat time tj , and x¯i(tj) model response at time tj . Thisnonlinear and nonconvex optimisation problem is solvedusing a hybrid Genetic Algorithm based on the one de-veloped by the authors in [2] for each of the three dif-ferent sets of experimental data. Results of our modelvalidation are shown in Figure 1. As can be seen fromthe ﬁgure, while the original model shows quite a pooragreement with the data, the optimised model is able toalmost exactly reproduce the responses of the in vitro sys-tem for each different experiment. Importantly, the opti-mal model parameters are all within biologically plausi-Authorized licensed use limited to: UNIVERSITY OF GLASGOW. Downloaded on January 26, 2010 at 06:07 from IEEE Xplore.  Restrictions apply. - 211 -0 20 40 60 80 100 120 140 160 18000.51Experiment Atime [min]EA  0 20 40 60 80 100 120 140 160 18000.511.5time [min]EAExperiment B  0 20 40 60 80 100 120 140 160 18000.511.5time [min]EAExperiment C  Fig. 1 Optimised (dashed line) versus original (dash-dotline) model responses for data from experiment A (x),experiment B (o) and experiment C (+), experimentaldata taken from [6].ble ranges. See the rightmost column in Table 1 (we pre-sented the resulting optimal parameter only for the casewhere the experiment data A was used). Thus the pro-posed method represents a strong validation of the pro-posed model of the tryptophan control system. Conse-quently, this supports the claim that the global optimiza-tion based method can search at least one parameter setwhich makes the model produce responses very close tothe experiment data as long as the assumed model struc-ture is proper.4. SUMMARY AND OUTLOOKThis paper presents a global optimization based pa-rameter identiﬁcation method using multiple experimentdata. Thanks to the global optimization, it was shown thatat least one plausible parameter set can be found with aproperly structured model.As mentioned, the structure of the model is assumed(i.e. ﬁxed) but the model shows robust performanceagainst different experimental setups, i.e. environmentalchanges. This observation implies that some parametersneed to change for the purpose of providing the modelwith robust functioning. This issue is one of our futureresearch topic along this line.REFERENCES[1] Sharad Bhartiya, Subodh Rawool, and K. V.Venkatesh. Dynamic model of escherichia coli tryp-tophan operon shows an optimal structural design.Journal of European Biochemistry, 270(12):2644–2651, June 2003.[2] Prathyush P. Menon, Jongrae Kim, Declan G. Bates,and Ian Postlethwaite. Clearance of nonlinear ﬂightcontrol laws using hybrid evolutionary optimisation.IEEE Transactions on Evolutionary Computation,10(6):689–699, December 2006.[3] Moise´s Santilla´n and Michael C. Mackey. Dy-namic reguiation of the tryptophan operon: A mod-eling study and comparison with experimental data.Proceedings of the National Academy of Sciences,98(4):1364–1369, February 2001.[4] Moise´s Santilla´n and Eduardo S. Zeron. Analyticalstudy of the multiplicity of regulatory mechanismsin the tryptophan operon. Bulletin of MathematicalBiology, 68(2):343–359, February 2006.[5] Zhi-Long Xiu, Zeng-Yi Chang, and An-Ping Zeng.Nonlinear dynamics of regulation of bacterial trpoperon: Model analysis of integrated effects ofrepression, feedback inhibition, and attenuation.Biotechnology Progress, 18(4):686–693, August2002.[6] Charles Yanofsky and Virginia Horn. Role of regula-tory features of the trp operon of Escherichia coli inmediating a response to a nutritional shift. Journal ofBacteriology, 176(20):6245–6254, October 1994.5. APPENDIXTable 1 Nominal parameter valuesUnit Original in [3] Optimal parametersd [·] 23.5 23.5e [·] 0.9 0.9f [·] 380 380R [μM] 0.8 1.357O [μM] 0.0033 0.0059P [μM] 2.6 3.22E¯ [μM] 0.378 0.338Tcr [μM/min] 22.7 14.07Kr [μM] 0.0026 0.0015Kt [μM] 60.34 64.55Ki [μM] 4.09 6.93nH [·] 1.2 1.00b [·] 0.85 0.53c [·] 0.04 0.0083ρ [μM] 2.9 3.33γ [1/min] 0.0 0.0113μ [1/min] 0.01 0.0264τp [min] 0.1 0.0267τm [min] 0.1 0.0277τρ [min] 0.05 0.0874τe [min] 0.66 1.1284OF (0) [μM] 4.8765×10−5 7.6444×10−5MF (0) [μM] 1.2037×10−4 0.4304×10−4E(0) [μM] 0.0119 0.0238T (0) [μM] 16.571 13.962Authorized licensed use limited to: UNIVERSITY OF GLASGOW. Downloaded on January 26, 2010 at 06:07 from IEEE Xplore.  Restrictions apply. 

Analytical study of the multiplicity of regulatory mechanisms in the tryptophan operon.

Clearance of nonlinear ﬂight control laws using hybrid evolutionary optimisation.

Dynamic model of escherichia coli tryptophan operon shows an optimal structural design.

Dynamic reguiation of the tryptophan operon: A modeling study and comparison with experimental data.

Nonlinear dynamics of regulation of bacterial trp operon: Model analysis of integrated effects of repression, feedback inhibition, and attenuation.

Role of regulatory features of the trp operon of Escherichia coli in mediatinga response to a nutritionalshift. Journalof Bacteriology,

Validation of a model of regulation in the tryptophan operon against multiple experiment data using global optimisation

Validation of a model of regulation in the tryptophan operon against multiple experiment data using global optimisation

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