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: Hidde.de-Jong@inrialpes.fr

2

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

3

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

4

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

5

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

6

Organism as biochemical system

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

7

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

8

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

9

Genetic regulatory networks

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

10

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)

11

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

12

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

13

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

14

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

15

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

.

.

16

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

17

Experimental test of model

Experimental test of mathematical model (bistability and hysteresis)

Gardner et al. (2000), Nature, 403(6786): 339-342

18

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

19

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

20

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

21

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

22

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

23

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.

24

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

.

.

25

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

26

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

27

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

28

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

29

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

30

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

31

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

32

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

33

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

34

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

35

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

36

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

37

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

38

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

39

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

40

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