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Modeling and Simulation of Genetic Regulatory Networks using Ordinary Differential Equations Hidde de Jong Projet HELIX Institut National de Recherche en Informatique et en Automatique Unité de Recherche Rhône-Alpes 655, avenue de lEurope Montbonnot, 38334 Saint Ismier CEDEX Email: Hidde.de-Jong@inrialpes.fr

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2 Overview 1. Analysis of genetic regulatory networks 2. Approaches towards modeling and simulation of genetic regulatory networks l overview l nonlinear differential equations l linear differential equations l piecewise-linear differential equations 4. Discussion: towards virtual cells

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3 Genome vGenome is genetic material in chromosomes of organism DNA in most organisms, RNA in some viruses vMany prokaryotic and eukaryotic genomes have been sequenced in recent years E. coli genome: 4300 genes

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4 Genes and proteins vGenes code for proteins that are essential for development and functioning of organism: gene expression DNA RNA protein protein and modifier molecule transcription translation post-translational modification

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5 vCellular processes involve interactions between proteins, genes, metabolites, and other molecules: l cell structure l metabolism l gene regulation l signal transduction Molecular interactions membrane metabolite enzyme gene transcription factor phosphorylated regulatory protein kinase

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6 Organism as biochemical system vOrganism can be viewed as biochemical system, structured by network of interactions between its molecular components

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7 Systems biology vChallenge of systems biology: understand how global behavior of organism emerges from local interactions between its molecular components vElements of systems biology: l High-throughput experimental techniques l Advanced computational techniques and powerful computers l Integrated application of experimental and computational tools "A transition is occurring in biology from the molecular level to the system level that promises to revolutionize our understanding of complex biological regulatory systems... " Kitano (2002), Science, 295(5560):564

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8 Model-driven analysis of biological systems vModel-driven analysis: integrated application of experimental and computational tools vModel composition versus model induction (reverse engineering) choose experiments simulate compare perform experiments construct and revise models predictions observations experimental conditions observations fit of models models experimental conditions biological system biological knowledge experimental data

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9 Genetic regulatory networks vGenetic regulatory network is part of biochemical network consisting (mainly) of genes and their regulatory interactions

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10 Experimental tools vStudy of large and complex genetic regulatory networks requires powerful experimental tools High-throughput, low-cost, reliable, precise vInformation obtained from experimental tools in genomics: l DNA sequence (genes) of organism l interactions between proteins and DNA (microarrays) l temporal variation of gene products (microarrays, mass spectometry)

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11 Computational tools vComputer support indispensable for dynamical analysis of genetic regulatory networks: modeling and simulation l precise and unambiguous description of network l systematic derivation of behavior predictions vFirst models of genetic regulatory networks date back to early days of molecular biology Regulation of lac operon (Jacob and Monod) vVariety of modeling formalisms exist… de Jong (2002), J. Comput. Biol., 9(1): 69-105 Hasty et al. (2001), Nat. Rev. Genet., 2(4):268-279 Smolen et al. (2000), Bull. Math. Biol., 62(2):247-292 Goodwin (1963), Temporal Organization in Cells

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12 Hierarchy of modeling formalisms vDifferential equations are major formalism for modeling genetic regulatory networks : nonlinear, linear, piecewise-linear differential equations Graphs Boolean equations Ordinary differential equations Stochastic master equations precision abstraction feasibility

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13 Nonlinear differential equation models Cellular concentration of proteins, mRNAs, and other molecules at time-point t represented by continuous variable x i (t) R 0 vRegulatory interactions modeled by differential equations where x [x 1,…, x n ]´ and f (x) is nonlinear rate law vNo analytical solution for most nonlinear differential equations Approximation of solution obtained by numerical simulation, given parameter values and initial conditions x(0) x 0 x f (x),. dxdx dtdt

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14 Model of cross-inhibition network x 1 = concentration protein 1 x 2 = concentration protein 2 x 1 = 1 f (x 2 ) 1 x 1 x 2 = 2 f (x 1 ) 2 x 2 1, 2 > 0, production rate constants 1, 2 > 0, degradation rate constants.. f (x) =, > 0 threshold n n + x n x f (x ) 0 gene 1 gene 2 1

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15 Phase-plane analysis vAnalysis of steady states in phase plane vTwo stable and one unstable steady state. System will converge to one of two stable steady states (differentiation) vSystem displays hysteresis effect: perturbation may cause irreversible switch to another steady state x2x2 x1x1 0 x 2 = 0. x 1 = 0. x 1 = 0 : x 1 = f (x 2 ) 1 1 x 2 = 0 : x 2 = f (x 1 ) 2 2..

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16 Construction of cross inhibition network vConstruction of cross inhibition network in vivo vDifferential equation model of network u = – u 1 + v β α1α1 v = – v 1 + u α2α2.. Gardner et al. (2000), Nature, 403(6786): 339-342

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17 Experimental test of model vExperimental test of mathematical model (bistability and hysteresis) Gardner et al. (2000), Nature, 403(6786): 339-342

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18 Bacteriophage infection of E. coli vResponse of E. coli to phage infection involves decision between alternative developmental pathways: lytic cycle and lysogeny Ptashne, A Genetic Switch, Cell Press,1992

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19 Simulation of phage infection vDifferential equation model of the regulatory network underlying decision between lytic cycle and lysogeny McAdams, Shapiro (1995), Science, 269(5524): 650-656

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20 Simulation of phage infection vNumerical simulation of promoter activity and protein concentrations in (a) lysogenic and (b) lytic pathways vCell follows one of two pathways for different initial conditions

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21 Evaluation nonlinear differential equations vPro: reasonably accurate description of underlying molecular interactions vContra: for more complex networks, difficult to analyze mathematically, due to nonlinearities vPro: approximate solution can be obtained through numerical simulation vContra: simulation techniques difficult to apply in practice, due to lack of numerical values for parameters and initial conditions

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22 Linear differential equation models Cellular concentration of proteins, mRNAs, and other molecules at time-point t represented by continuous variable x i (t) R 0 vRegulatory interactions modeled by differential equations where x [x 1,…, x n ]´ and f (x) is linear rate law vAnalytical solution exists for linear differential equations: x f (x) Ax b,. dxdx dtdt x(t) e At x 0 e A(t-τ) dτ b 0 t

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23 Model of cross-inhibition network x 1 = concentration protein 1 x 2 = concentration protein 2 1, 2 > 0, production rate constants 1, 2 > 0, degradation rate constants gene 1 gene 2 x 1 = 1 f (x 2 ) 1 x 1 x 2 = 2 f (x 1 ) 2 x 2.. f (x) = 1 x / (2 ), > 0, x 2 x f (x ) 0 2 1 x 1 = 1 x 1 1 1 x 2 1. x 2 = 2 2 x 1 2 x 2 2.

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24 Phase-plane analysis vAnalysis of steady states in phase plane vSingle unstable steady state. vLinear differential equations too simple to capture dynamic phenomena of interest: no bistability and no hysteresis x2x2 x1x1 0 x 1 = 0. x 2 = 0. x 1 = 0 : x 1 = f (x 2 ) 1 1 x 2 = 0 : x 2 = f (x 1 ) 2 2..

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25 Model induction vLinear differential equation models much used for induction of model of regulatory network from gene expression data network reconstruction, reverse engineering Given time-series of gene expression data, find A and b, such that solution of with noise term ξ, fits expression data vPowerful techniques for induction of linear model from experimental data x Ax b ξ,. Ljung (1995), System Identification, Prentice Hall, 1999

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26 SOS response in E. coli vSOS response of E. coli regulates cell survival and repair after DNA damage Gardner et al. (2003), Science, 301(5629): 102-105

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27 Induction of model of SOS network vReconstruction of subnetwork by inducing linear differential equation model from gene expression data Steady-state response of bacterium measured under genetic and physiological perturbations vMethod robust to measurement noise and upscalable Gardner et al. (2003), Science, 301(5629): 102-105

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28 Evaluation of linear differential equations vPro: analytical solution exists, thus facilitating qualitative analysis of complex systems vContra: too simple to capture important dynamical phenomena of regulatory network, due to neglect of nonlinear character of interactions vPro: powerful techniques for induction of model of network from gene expression data

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29 Piecewise-linear differential equation models Cellular concentration of proteins, mRNAs, and other molecules at time-point t represented by continuous variable x i (t) R 0 vRegulatory interactions modeled by differential equations where x [x 1,…, x n ]´ and f (x) is piecewise-linear (PL) Global solution obtained by piecing together local solutions of linear differential equations in regions D j. dxdx dtdt x f (x) A D m x b D m, D m R 0 A D 1 x b D 1, D 1 R 0 n n

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30 Model of cross-inhibition network x 1 = concentration protein 1 x 2 = concentration protein 2 1, 2 > 0, production rate constants 1, 2 > 0, degradation rate constants x 1 = 1 f (x 2 ) 1 x 1 x 2 = 2 f (x 1 ) 2 x 2.. f (x) = s ( x, ) = gene 1 gene 2 1, x < 0, x > x f (x ) 0 1

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31 Phase-plane analysis vAnalysis of dynamics in phase plane In every region D j, model simplifies to system of piecewise- affine differential equations All solutions, while being in D j, converge towards target steady state vDifferent regions have different target steady states x2x2 x1x1 0 2 1 x 1 = 1 s (x 2, 2 ) 1 x 1 x 2 = 2 s (x 1, 1 ) 2 x 2.. x 1 = 1 1 x 1 x 2 = 2 2 x 2.. in D 1 :, x 1 = 0 : x 1 = 1 1., x 2 = 0 : x 2 = 2 2. x 2 = 0. x 1 = 0. D1D1 x 1 = 1 x 1 x 2 = 2 2 x 2.. in D 2 :, x 1 = 0 : x 1 =c 0., x 2 = 0 : x 2 = 2 2. x 1 = 0. x 2 = 0. D2D2

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32 Phase-plane analysis vGlobal phase-plane analysis by combining analyses in local regions of phase plane Techniques for dealing with discontinuities due to step functions vPiecewise-linear model good approximation of nonlinear model, retaining properties of bistability and hysteresis x2x2 x1x1 0 x 2 = 0. x 1 = 0. 2 1 Gouzé, Sari (2003), Dyn. Syst., 17(4):299-316

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33 Initiation of sporulation in B. subtilis vB. subtilis can sporulate when environmental conditions become unfavorable de Jong et al. (2004), Bull. Math. Biol., 66(2):261-300

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34 Network underlying initiation of sporulation vInitiation of sporulation controlled by complex genetic regulatory network integrating environmental, cell-cyle and metabolic signals de Jong et al. (2004), Bull. Math. Biol., 66(2):261-300

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35 Genetic Network Analyzer (GNA) vQualitative simulation of initiation of sporulation using tool based on piecewise-linear differential equation models (GNA) de Jong et al. (2003), Bioinformatics, 19(3):336-344

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36 Qualitative simulation of sporulation vPredictions obtained through qualitative simulation consistent with observed behavior of B. subtilis cells under starvation vDecision between sporulation and vegetative growth outcome of competition between positive and negative feedback loops de Jong et al. (2004), Bull. Math. Biol., 66(2):261-300

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37 Evaluation of PL differential equations vPro: captures important dynamical phenomena of network, by suitable approximation of nonlinearities vPro: qualitative analysis of dynamics of complex systems possible, due to favorable mathematical properties vPro: powerful techniques for induction of model of network from gene expression data

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38 Conclusions vSeveral kinds of mathematical model of genetic regulatory networks vNonlinear models give reasonably accurate description of regulatory interactions, but difficult to apply in practice vLinear models have favorable mathematical and computational properties, but can only give rough picture of regulatory structure vPiecewise-linear models are compromise between nonlinear and linear models, satisfying biological applicability and computational feasibility

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39 Beyond modeling and simulation vIntegration of modeling and simulation with other computational and experimental tools: l Biological knowledge and databases l Selection of discriminatory experiments l Validation of model predictions with experimental data choose experiments simulate compare perform experiments construct and revise models predictions observations experimental conditions observations fit of models models experimental conditions biological system biological knowledge

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40 Beyond genetic regulatory networks vIntegration of genetic networks with metabolic and signal transduction networks Virtual cell or whole-cell simulation Tomita et al. (1999), Bioinformatics, 15(1):72-84

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