modeling and simulation of genetic regulatory networks using ordinary differential equations hidde...
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Modeling and Simulation of Genetic Regulatory Networks
using Ordinary Differential Equations
Hidde de Jong
Projet HELIXInstitut National de Recherche en Informatique et en Automatique
Unité de Recherche Rhône-Alpes655, avenue de l’Europe
Montbonnot, 38334 Saint Ismier CEDEX
Email: [email protected]
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Overview
1. Analysis of genetic regulatory networks
2. Approaches towards modeling and simulation of genetic
regulatory networks
overview
nonlinear differential equations
linear differential equations
piecewise-linear differential equations
4. Discussion: towards virtual cells
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Genome
Genome is genetic material in chromosomes of organism
DNA in most organisms, RNA in some viruses
Many prokaryotic and eukaryotic genomes have been sequenced in recent years
E. coli genome: 4300 genes
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Genes and proteins Genes 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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Cellular processes involve interactions between proteins, genes, metabolites, and other molecules:
cell structure
metabolism
gene regulation
signal transduction
Molecular interactions
membrane
metabolite
enzyme
genetranscription factor
phosphorylated regulatory protein
kinase
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Organism as biochemical system
Organism can be viewed as biochemical system, structured by network of interactions between its molecular components
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Systems biology
Challenge of systems biology: understand how global behavior of organism emerges from local interactions between its molecular components
Elements of systems biology:
High-throughput experimental techniques
Advanced computational techniques and powerful computers
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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Model-driven analysis of biological systems
Model-driven analysis: integrated application of experimental and computational tools
Model composition versus model induction (reverse engineering)
chooseexperiments
simulate
compare
perform
experiments
constructand revise
models
predictions observationsexperimental
conditions
observations
fit of models
models
experimental
conditions
biological
system
biological
knowledgeexperimental
data
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Genetic regulatory networks
Genetic regulatory network is part of biochemical network consisting (mainly) of genes and their regulatory interactions
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Experimental tools Study of large and complex genetic regulatory networks
requires powerful experimental toolsHigh-throughput, low-cost, reliable, precise
Information obtained from experimental tools in genomics: DNA sequence (genes) of organism interactions between proteins and DNA (microarrays) temporal variation of gene products (microarrays, mass spectometry)
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Computational tools
Computer support indispensable for dynamical analysis of genetic regulatory networks: modeling and simulation
precise and unambiguous description of network
systematic derivation of behavior predictions
First models of genetic regulatory networks date back to early days of molecular biologyRegulation of lac operon (Jacob and Monod)
Variety 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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Hierarchy of modeling formalisms
Differential equations are major formalism for modeling genetic regulatory networks :nonlinear, linear, piecewise-linear differential equations
Graphs
Boolean equations
Ordinary differential equations
Stochastic master equations
precisionabstraction
feasibility
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Nonlinear differential equation models
Cellular concentration of proteins, mRNAs, and other molecules at time-point t represented by continuous variable xi(t) R0
Regulatory interactions modeled by differential equations
where x [x1,…, xn]´and f (x) is nonlinear rate law
No analytical solution for most nonlinear differential equations
Approximation of solution obtained by numerical simulation, given parameter values and initial conditions x(0) x0
x f (x), .dxdt
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Model of cross-inhibition network
x1 = concentration protein 1
x2 = concentration protein 2
x1 = 1 f (x2) 1 x1
x2 = 2 f (x1) 2 x2
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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Phase-plane analysis
Analysis of steady states in phase plane
Two stable and one unstable steady state. System will converge to one of two stable steady states (differentiation)
System displays hysteresis effect: perturbation may cause irreversible switch to another steady state
x2
x1
0
x2 = 0 .
x1 = 0 .
x1 = 0 : x1 = f (x2)1
1
x2 = 0 : x2 = f (x1)2
2
.
.
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Construction of cross inhibition network
Construction of cross inhibition network in vivo
Differential equation model of network
u = – u1 + v β
α1v = – v
1 + u α2..
Gardner et al. (2000), Nature, 403(6786): 339-342
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Experimental test of model
Experimental test of mathematical model (bistability and hysteresis)
Gardner et al. (2000), Nature, 403(6786): 339-342
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Bacteriophage infection of E. coli
Response 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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Simulation of phage infection
Differential 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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Simulation of phage infection
Numerical simulation of promoter activity and protein concentrations in (a) lysogenic and (b) lytic pathways
Cell follows one of two pathways for different initial conditions
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Evaluation nonlinear differential equations
Pro: reasonably accurate description of underlying molecular interactions
Contra: for more complex networks, difficult to analyze mathematically, due to nonlinearities
Pro: approximate solution can be obtained through numerical simulation
Contra: simulation techniques difficult to apply in practice, due to lack of numerical values for parameters and initial conditions
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Linear differential equation models
Cellular concentration of proteins, mRNAs, and other molecules at time-point t represented by continuous variable xi(t) R0
Regulatory interactions modeled by differential equations
where x [x1,…, xn]´ and f (x) is linear rate law
Analytical solution exists for linear differential equations:
x f (x) Ax b, .dx
dt
x(t) eAt x0 eA(t-τ) dτ b 0
t
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Model of cross-inhibition network
x1 = concentration protein 1
x2 = concentration protein 2
1, 2 > 0, production rate constants 1, 2 > 0, degradation rate
constants
gene 1 gene 2
x1 = 1 f (x2) 1 x1
x2 = 2 f (x1) 2 x2
.
.
f (x) = 1 x / (2 ) , > 0, x 2
x
f (x )
0 2
1
x1 = 1 x1 11 x2 1.
x2 = 22 x1 2 x2 2.
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Phase-plane analysis
Analysis of steady states in phase plane
Single unstable steady state. Linear differential equations too simple to capture dynamic
phenomena of interest: no bistability and no hysteresis
x2
x1
0
x1 = 0 .
x2 = 0 .
x1 = 0 : x1 = f (x2)1
1
x2 = 0 : x2 = f (x1)2
2
.
.
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Model induction
Linear 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
Powerful techniques for induction of linear model from experimental data
x Ax b ξ, .
Ljung (1995), System Identification, Prentice Hall, 1999
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SOS response in E. coli
SOS 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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Induction of model of SOS network
Reconstruction of subnetwork by inducing linear differential equation model from gene expression data
Steady-state response of bacterium measured under genetic and physiological perturbations
Method robust to measurement noise and upscalable
Gardner et al. (2003), Science, 301(5629): 102-105
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Evaluation of linear differential equations
Pro: analytical solution exists, thus facilitating qualitative analysis of complex systems
Contra: too simple to capture important dynamical phenomena of regulatory network, due to neglect of nonlinear character of interactions
Pro: powerful techniques for induction of model of network from gene expression data
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Piecewise-linear differential equation models
Cellular concentration of proteins, mRNAs, and other molecules at time-point t represented by continuous variable xi(t) R0
Regulatory interactions modeled by differential equations
where x [x1,…, xn]´and f (x) is piecewise-linear (PL)
Global solution obtained by piecing together local solutions of linear differential equations in regions Dj
.dxdt
x f (x) ADm x bDm, Dm R0
AD1 x bD1, D1 R0
n
n
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Model of cross-inhibition network
x1 = concentration protein 1
x2 = concentration protein 2
1, 2 > 0, production rate constants 1, 2 > 0, degradation rate
constants
x1 = 1 f (x2) 1 x1
x2 = 2 f (x1) 2 x2
.
.
f (x) = s( x, ) =
gene 1 gene 2
1, x <
0, x >
x
f (x )
0
1
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Phase-plane analysis
Analysis of dynamics in phase plane
In every region Dj , model simplifies to system of piecewise-
affine differential equationsAll solutions, while being in Dj , converge towards target steady state
Different regions have different target steady states
x2
x1
0 2
1
x1 = 1 s (x2, 2) 1 x1
x2 = 2 s (x1, 1) 2 x2
.
.
x1 = 1 1 x1
x2 = 2 2 x2
.
.
in D1 :, x1 = 0 : x1 = 1 1
.
, x2 = 0 : x2 = 2 2.
x2 = 0 .
x1 = 0 .
D1
x1 = 1 x1
x2 = 2 2 x2
.
.
in D2 :, x1 = 0 : x1 =c 0.
, x2 = 0 : x2 = 2 2.x1 = 0 .
x2 = 0 .
D2
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Phase-plane analysis
Global phase-plane analysis by combining analyses in local regions of phase plane
Techniques for dealing with discontinuities due to step functions
Piecewise-linear model good approximation of nonlinear model, retaining properties of bistability and hysteresis
x2
x1
0
x2 = 0 .
x1 = 0 .
2
1
Gouzé, Sari (2003), Dyn. Syst., 17(4):299-316
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Initiation of sporulation in B. subtilis
B. subtilis can sporulate when environmental conditions become unfavorable
de Jong et al. (2004), Bull. Math. Biol., 66(2):261-300
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Network underlying initiation of sporulation
Initiation 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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Genetic Network Analyzer (GNA)
Qualitative 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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Qualitative simulation of sporulation
Predictions obtained through qualitative simulation consistent with observed behavior of B. subtilis cells under starvation
Decision 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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Evaluation of PL differential equations
Pro: captures important dynamical phenomena of network, by suitable approximation of nonlinearities
Pro: qualitative analysis of dynamics of complex systems possible, due to favorable mathematical properties
Pro: powerful techniques for induction of model of network from gene expression data
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Conclusions
Several kinds of mathematical model of genetic regulatory networks
Nonlinear models give reasonably accurate description of regulatory interactions, but difficult to apply in practice
Linear models have favorable mathematical and computational properties, but can only give rough picture of regulatory structure
Piecewise-linear models are compromise between nonlinear and linear models, satisfying biological applicability and computational feasibility
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Beyond modeling and simulation
Integration of modeling and simulation with other computational and experimental tools:
Biological knowledge and databases
Selection of discriminatory experiments
Validation of model predictions with experimental data
chooseexperiments
simulate
compare
perform
experiments
constructand revise
models
predictions observationsexperimental
conditions
observations
fit of models
models
experimental
conditions
biological
system
biological
knowledge
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Beyond genetic regulatory networks
Integration of genetic networks with metabolic and signal transduction networks
Virtual cell or whole-cell simulation Tomita et al. (1999), Bioinformatics, 15(1):72-84