Mathematical Modeling of Genetic Regulatory Networks Hidde de Jong Projet HELIX INRIA Rhône-Alpes 655, avenue de l’Europe Montbonnot, 38334 Saint Ismier CEDEX Email: Hidde.de-Jong@inrialpes.fr
Overview 1. Genetic regulatory networks 2. Modeling and simulation of genetic regulatory networks 3. Modeling and simulation approaches: G differential equations G stochastic equations 4. Conclusions 2
Genes and proteins O Genes code for proteins that are essential for development and functioning of organism: gene expression DNA transcription RNA translation protein post-translational modification protein and modifier molecule 3
Regulation of gene expression O Regulation of gene expression on several levels transcriptional regulation translational regulation regulation of post-translational modification O Gene expression controlled by proteins produced by other genes: regulatory interactions 4
Genetic regulatory network O Genetic regulatory network consists of set of genes, proteins, small molecules, and their mutual regulatory interactions gene 1 repressors activator repressor complex gene 3 gene 2 O Development and functioning of organisms cell emerges from interactions in genetic regulatory networks 5
Bacteriophage λ infection of E. coli O Response of E. coli to phage λ infection involves decision between alternative developmental pathways: lytic cycle and lysogeny Ptashne, 1992 6
Genetic regulatory network phage λ O Choice between alternative developmental pathways controlled by network of genes, proteins, and mutual regulatory interactions McAdams & Shapiro, 1995 7
Computational approaches O Most genetic regulatory networks are large and complex Cells have many components that can interact in complex ways O Dynamics of large and complex genetic regulatory processes hard to understand by intuitive approaches alone O Mathematical methods for modeling and simulation are required: G precise and unambiguous description of network of interactions G systematical derivation of behavioral predictions O Practical application of mathematical methods requires user- friendly computer tools 8
Mathematical modeling approaches O Mathematical modeling has developed since the 1960s and is currently attracting much attention Bower and Bolouri, 2001; Hasty et al. , 2001; McAdams and Arkin, 1998; Smolen et al. , 2000; de Jong, 2002 O Two approaches to computer modeling and simulation discussed in this session: G differential equations G stochastic equations O Jean-Luc Gouzé will discuss class of piecewise-linear differential equations central to this project in more detail 9
Differential equation models O Cellular concentration of proteins, mRNAs, and other molecules at time-point t represented by continuous variable x i ( t ) ∈ R ≥ 0 O Regulatory interactions modeled by kinetic equations . x i = f i ( x ), 1 ≤ i ≤ n , where f i ( x ) is rate law O Rate of change of variable x i is function of other concentration variables x = [ x 1 ,…, x n ]´ O Differential equations are major modeling formalism in mathematical biology Segel, 1984; Kaplan and Glass, 1995; Murray, 2002 10
Negative feedback system O Gene encodes a protein inhibiting its own expression: negative feedback gene - mRNA protein O Negative feedback important for homeostasis , maintenance of system near a desired state Thomas and d’Ari, 1990 11
Model of negative feedback system gene x 1 = mRNA concentration x 2 = protein concentration – x 1 = κ 1 f ( x 2 ) - γ 1 x 1 . mRNA x 2 = κ 2 x 1 - γ 2 x 2 . protein κ 1 , κ 2 > 0 , production rate constants γ 1 , γ 2 > 0 , degradation rate constants f ( x 2 ) n θ f ( x 2 ) = , θ > 0 threshold θ n + x 2 n 0 θ x 2 12
Steady state analysis O No analytical solution of nonlinear differential equations describing feedback system . O System has single steady state at x = 0 κ 1 . x 1 . x 2 = 0 x 1 = 0 : x 1 = f ( x 2 ) γ 1 γ 2 . x 2 = 0 : x 1 = x 2 κ 2 . x 1 = 0 0 x 2 O Steady state is stable , that is, after perturbation system will return to steady state (homeostasis) 13
Transient behavior after pertubation O Numerical simulation of differential equations shows transient behavior towards steady state after perturbation Initial values x 1 (0), x 2 (0) correspond to perturbation x 1 . x 1 x 2 = 0 . x 2 x 1 = 0 0 x 2 0 t 14
Positive feedback system O Gene encodes a protein activating its own expression: positive feedback gene + mRNA protein O Positive feedback important for differentiation , evolution towards one of two alternative states of system 15
Model of positive feedback system gene x 1 = mRNA concentration x 2 = protein concentration + x 1 = κ 1 f ( x 2 ) - γ 1 x 1 . mRNA x 2 = κ 2 x 1 - γ 2 x 2 . protein κ 1 , κ 2 > 0 , production rate constants γ 1 , γ 2 > 0 , degradation rate constants f ( x 2 ) n x 2 f ( x 2 ) = 0 θ n + x 2 n θ x 2 16
Steady state analysis O No analytical solution of nonlinear differential equations describing feedback system O System has three steady states . x 2 = 0 κ 1 x 1 . x 1 = 0 : x 1 = f ( x 2 ) γ 1 γ 2 . x 2 = 0 : x 1 = x 2 κ 2 . x 1 = 0 0 x 2 O Two stable and one unstable steady state. System will tend to one of two stable steady states (differentiation) 17
Transient behavior after pertubation O Depending on strength of perturbation, transient behavior towards different steady states x 1 . x 2 = 0 x 1 x 1 x 2 . x 2 x 1 = 0 0 x 2 0 0 t t 18
Model of time-delay feedback system O Time to complete transcription and translation introduces time- delay in differential equations x 1 = mRNA concentration gene x 2 = protein concentration - τ x 1 = κ 1 f ( x 2 ) - γ 1 x 1 . mRNA τ x 2 = κ 2 x 1 - γ 2 x 2 . τ x 1 ( t ) = x 1 ( t - τ 1 ) , τ 1 > 0 time-delay protein τ x 2 ( t ) = x 2 ( t - τ 2 ) , τ 2 > 0 time-delay O Time-delay feedback systems may exhibit oscillatory behavior 19
More complex feedback systems O Gene encodes a protein activating synthesis of another protein inhibiting expression of gene: positive and negative feedback gene a gene b + - mRNA a mRNA b protein A protein B O Interlocking feedback loops give rise to models with complex dynamics: numerical simulation techniques necessary 20
Application of differential equations O Differential equations have been used to model a variety of genetic regulatory networks: G circadian rhythms in Drosophila (Leloup and Goldbeter, 1998) G λ phage infection of E. coli (McAdams and Shapiro, 1998) G segmentation of early embryo of Drosophila (Reinitz and Sharp, 1996) G cell division in Xenopus (Novak and Tyson, 1993) G Trp synthesis in E. coli (Santillán and Mackey, 2001) G induction of lac operon in E. coli (Carrier and Keasling, 1999) G developmental cycle of bacteriophage T7 (Endy et al. , 2000) G ... 21
Simulaton of phage ? infection O Kinetic model of the phage ? network underlying decision between lytic cycle and lysogeny McAdams & Shapiro, 1995 22
Simulaton of phage ? infection O Time evolution of promoter activity and protein concentrations in (a) lysogenic and (b) lytic pathways McAdams & Shapiro, 1995 23
Evaluation of differential equations O Pro : general formalism for which powerful analysis and simulation techniques exist O Contra : numerical techniques are often not appropriate due to lack of quantitative knowledge value of parameters and evolution of concentrations are not known O Contra : implicit assumptions of continuous and deterministic change of concentrations may not be valid on molecular level 24
Gene expression is discrete process O Gene expression is result of large number of discrete events: chemical reactions RNA polymerase DNA n-1 n n-1 n 0 1 2 3 4 0 1 2 3 4 DNA + RNAP → DNA 0 • RNAP DNA n-1 n n-1 n 0 1 2 3 4 0 1 2 3 4 DNA i • RNAP → DNA i+1 • RNAP 25
Gene expression is stochastic process O Gene expression is stochastic process: random time intervals τ between occurrence of reactions RNA polymerase DNA n-1 n n-1 n 0 1 2 3 4 0 1 2 3 4 O Time interval τ has probability distribution P( τ ) τ 26
Differential equations are abstractions O Differential equation models make continuous and deterministic abstraction of discrete and stochastic process G x i ( t ) ∈ R ≥ 0 is continuous variable G x i = f i ( x ) determines change in x i at t . . O Abstraction may not be warranted when modeling gene regulation on molecular level: low number of molecules O Therefore, more realistic stochastic models of gene regulation 27
Stochastic variables O Stochastic variables X i describe number of molecules of proteins, mRNAs, etc. G X i ( t ) ∈ N ≥ 0 is discrete variable . G P(X i ( t )) is probability distribution describing probability that at time- point t cell contains X i molecules of i P(X i ( t )) X i ( t ) 28
Stochastic master equations O Stochastic master equations describe evolution of state X = [ X 1 ,…, X n ]´ of regulatory system m m P ( X ( t + ∆ t )) = P ( X ( t )) (1 - ∑ α j ∆ t ) + ∑ β j ∆ t j = 1 j = 1 G m is the number of reactions that can occur in the system G α j ∆ t is the probability that reaction j will occur in [ t, t + ∆ t ] given that the system is in state X at t G β j ∆ t is the probability that reaction j will bring the system in state X from another state in [ t, t + ∆ t ] van Kampen, 1997 29
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