Mathematical Modeling ofGenetic Regulatory Networks

Hidde de JongProjet HELIX

INRIA Rhône-Alpes655, avenue de l’Europe

Montbonnot, 38334 Saint Ismier CEDEX

Email: Hidde.de-Jong@inrialpes.fr

2

Overview

1. Genetic regulatory networks

2. Modeling and simulation of genetic regulatory networks

3. Modeling and simulation approaches:

l differential equations

l stochastic equations

4. Conclusions

3

Genes and proteinsv Genes code for proteins that are essential for development

and functioning of organism: gene expression

DNA

RNA

transcription

protein

translation

protein andmodifier molecule

post-translational modification

4

Regulation of gene expression

v Regulation of gene expression on several levels

v Gene expression controlled by proteins produced by othergenes: regulatory interactions

transcriptional regulation

translational regulation

regulation ofpost-translational modification

5

repressors

repressor complex

activator

gene 1

gene 3

gene 2

Genetic regulatory network

v Genetic regulatory network consists of set of genes, proteins,small molecules, and their mutual regulatory interactions

v Development and functioning of organisms cell emerges frominteractions in genetic regulatory networks

6

Bacteriophage λ infection of E. coli

v Response of E. coli to phage λinfection involves decisionbetween alternativedevelopmental pathways:lytic cycle and lysogenyPtashne, 1992

7

Genetic regulatory network phage λ

v Choice between alternative developmental pathways controlledby network of genes, proteins, and mutual regulatoryinteractions

McAdams & Shapiro, 1995

8

v Most genetic regulatory networks are large and complexCells have many components that can interact in complex ways

v Dynamics of large and complex genetic regulatory processeshard to understand by intuitive approaches alone

v Mathematical methods for modeling and simulation arerequired:

l precise and unambiguous description of network of interactions

l systematical derivation of behavioral predictions

v Practical application of mathematical methods requires user-friendly computer tools

Computational approaches

9

Mathematical modeling approachesv Mathematical modeling has developed since the 1960s and is

currently attracting much attentionBower and Bolouri, 2001; Hasty et al., 2001; McAdams and Arkin, 1998;Smolen et al., 2000; de Jong, 2002

v Two approaches to computer modeling and simulationdiscussed in this session:

l differential equations

l stochastic equations

v Jean-Luc Gouzé will discuss class of piecewise-lineardifferential equations central to this project in more detail

10

Differential equation models

v Cellular concentration of proteins, mRNAs, and other moleculesat time-point t represented by continuous variable xi(t) ∈ R≥0

v Regulatory interactions modeled by kinetic equations

where fi(x) is rate law

v Rate of change of variable xi is function of other concentrationvariables x = [x1,…, xn]´

v Differential equations are major modeling formalism inmathematical biology

Segel, 1984; Kaplan and Glass, 1995; Murray, 2002

xi = fi(x), 1 ≤ i ≤ n,.

11

Negative feedback system

v Gene encodes a protein inhibiting its own expression:negative feedback

v Negative feedback important for homeostasis, maintenance ofsystem near a desired state

Thomas and d’Ari, 1990

gene

mRNA

protein

-

12

Model of negative feedback systemgene

mRNA

protein

–

x1 = mRNA concentration

x2 = protein concentration

x1 = κ1 f (x2) - γ1 x1

x2 = κ2 x1 - γ2 x2

κ1 , κ2 > 0, production rate constants

γ1, γ2 > 0, degradation rate constants

.

.

f (x2) = , θ > 0 thresholdθ n

θ n + x2n

x2

f (x2 )

θ0

13

Steady state analysis

v No analytical solution of nonlinear differential equationsdescribing feedback system

v System has single steady state at x = 0

v Steady state is stable, that is, after perturbation system willreturn to steady state (homeostasis)

.

x2

x1

0

x2 = 0 .

x1 = 0 .

x1 = 0 : x1 = f (x2)κ1γ1

x2 = 0 : x1 = x2γ2κ2

.

.

14

Transient behavior after pertubation

v Numerical simulation of differential equations shows transientbehavior towards steady state after perturbation

Initial values x1 (0), x2 (0) correspond to perturbation

x1

x2

t0

x2

x1

0

x2 = 0 .

x1 = 0 .

15

Positive feedback system

v Gene encodes a protein activating its own expression:positive feedback

v Positive feedback important for differentiation, evolutiontowards one of two alternative states of system

gene

mRNA

protein

+

16

Model of positive feedback systemgene

mRNA

protein

+

x1 = mRNA concentration

x2 = protein concentration

x1 = κ1 f (x2) - γ1 x1

x2 = κ2 x1 - γ2 x2

κ1 , κ2 > 0, production rate constants

γ1, γ2 > 0, degradation rate constants

.

.

f (x2) =x2

θ n + x2n

n

x2

f (x2 )

θ0

17

Steady state analysis

v No analytical solution of nonlinear differential equationsdescribing feedback system

v System has three steady states

v Two stable and one unstable steady state. System will tend toone of two stable steady states (differentiation)

x2

x1

0

x2 = 0 .

x1 = 0 .

x1 = 0 : x1 = f (x2)κ1γ1

x2 = 0 : x1 = x2γ2κ2

.

.

18

Transient behavior after pertubation

v Depending on strength of perturbation, transient behaviortowards different steady states

x1

x2

t0

x2

x1

0

x2 = 0 .

x1 = 0 .

x1

x2

t0

19

Model of time-delay feedback system

v Time to complete transcription and translation introduces time-delay in differential equations

v Time-delay feedback systems may exhibit oscillatory behavior

gene

mRNA

protein

-

x1 = mRNA concentration

x2 = protein concentration

x1 = κ1 f (x2) - γ1 x1

x2 = κ2 x1 - γ2 x2

.

.

x1(t ) = x1(t - τ1) , τ1 > 0 time-delay

x2(t ) = x2(t - τ2) , τ2 > 0 time-delay

τ

τ

τ

τ

20

More complex feedback systems

v Gene encodes a protein activating synthesis of another proteininhibiting expression of gene: positive and negative feedback

v Interlocking feedback loops give rise to models with complexdynamics: numerical simulation techniques necessary

gene a

mRNA b

protein A

-

protein B

gene b

+

mRNA a

21

Application of differential equations

v Differential equations have been used to model a variety ofgenetic regulatory networks:

l circadian rhythms in Drosophila (Leloup and Goldbeter, 1998)

l λ phage infection of E. coli (McAdams and Shapiro, 1998)

l segmentation of early embryo of Drosophila (Reinitz and Sharp, 1996)

l cell division in Xenopus (Novak and Tyson, 1993)

l Trp synthesis in E. coli (Santillán and Mackey, 2001)

l induction of lac operon in E. coli (Carrier and Keasling, 1999)

l developmental cycle of bacteriophage T7 (Endy et al., 2000)

l ...

22

Simulaton of phage ? infection

v Kinetic model of the phage ? network underlying decisionbetween lytic cycle and lysogenyMcAdams & Shapiro, 1995

23

Simulaton of phage ? infection

v Time evolution of promoter activity and protein concentrationsin (a) lysogenic and (b) lytic pathways

McAdams & Shapiro, 1995

24

Evaluation of differential equations

v Pro: general formalism for which powerful analysis andsimulation techniques exist

v Contra: numerical techniques are often not appropriate due tolack of quantitative knowledge

value of parameters and evolution of concentrations are not known

v Contra: implicit assumptions of continuous and deterministicchange of concentrations may not be valid on molecular level

25

Gene expression is discrete process

v Gene expression is result of large number of discrete events:chemical reactions

0 1 2 3 4 n-1 n0 1 2 3 4 n-1 nDNA

0 1 2 3 4 n-1 n

RNA polymerase

DNA + RNAP → DNA0 • RNAP

0 1 2 3 4 n-1 nDNA

DNAi • RNAP → DNAi+1 • RNAP

26

Gene expression is stochastic process

v Gene expression is stochastic process: random timeintervals τ between occurrence of reactions

v Time interval τ has probability distribution

P(τ )

τ

0 1 2 3 4 n-1 n0 1 2 3 4 n-1 nDNA

RNA polymerase

27

Differential equations are abstractions

v Differential equation models make continuous anddeterministic abstraction of discrete and stochastic process

l xi(t) ∈ R≥0 is continuous variable

l xi = fi(x) determines change in xi at t

v Abstraction may not be warranted when modeling generegulation on molecular level: low number of molecules

v Therefore, more realistic stochastic models of gene regulation

.

.

28

Stochastic variables

v Stochastic variables Xi describe number of molecules ofproteins, mRNAs, etc.

l Xi(t) ∈ N≥0 is discrete variable

l P(Xi(t)) is probability distribution describing probability that at time-point t cell contains Xi molecules of i

.

P(Xi(t))

Xi(t)

29

Stochastic master equations

v Stochastic master equations describe evolution of state X =[X1,…, Xn]´ of regulatory system

l m is the number of reactions that can occur in the system

l αj ∆t is the probability that reaction j will occur in [t, t +∆t] given thatthe system is in state X at t

l βj ∆t is the probability that reaction j will bring the system in state Xfrom another state in [t, t +∆t]

van Kampen, 1997

P(X (t +∆t)) = P(X (t )) (1 - ∑ αj ∆t ) + ∑ βj ∆tj = 1 j = 1

mm

30

Stochastic simulation

v For ∆t → 0 we obtain

v Analytical solution of master equations is not possible

v Stochastic simulation by predicting a sequence of reactionschanging the state of the system, starting from initial state X0

Stochastic simulation uses stochastic variables τ and ρτ = time interval until occurrence of next reaction

ρ = type of reaction

Gillespie, 1977

P(X (t )) = ∑ (βj - αj P(X (t )))j = 1

m∂∂t

31

Reactions in gene expression

v Five possible reactions in gene expression are considered

0 1 2 3 4 n-1 n

RNA polymerase

0 1 2 3 4 n-1 nDNA + RNAP → DNA0 • RNAP

DNAi • RNAP → DNAi+1 • RNAP0 1 2 3 4 n-1 n 0 1 2 3 4 n-1 n

DNAn • RNAP → DNA + RNAP0 1 2 3 4 n-1 n 0 1 2 3 4 n-1 n

DNA + R → DNA • R0 1 2 3 4 n-1 n 0 1 2 3 4 n-1 n

repressor

DNA • R → DNA + R0 1 2 3 4 n-1 n 0 1 2 3 4 n-1 n

1

3

2

5

4

32

Simulation of gene expression

v Stochastic simulation from initial state

0 1 2 3 4 n-1 n 0 1 2 3 4 n-1 nDNA + RNAP → DNA0 • RNAP

DNAi • RNAP → DNAi+1 • RNAP0 1 2 3 4 n-1 n 0 1 2 3 4 n-1 n

DNAi • RNAP → DNAi+1 • RNAP0 1 2 3 4 n-1 n 0 1 2 3 4 n-1 n

reaction 1 chosen

reaction 2 chosen

reaction 2 chosen

33

Stochastic outcome of simulation

v Simulation starting from same initial state will generally lead todifferent results

0 1 2 3 4 n-1 n 0 1 2 3 4 n-1 nDNA + R → DNA • R

DNA • R → DNA + R0 1 2 3 4 n-1 n 0 1 2 3 4 n-1 n

DNA + RNAP → DNA0 • RNAP0 1 2 3 4 n-1 n 0 1 2 3 4 n-1 n

reaction 4 chosen

reaction 5 chosen

reaction 1 chosen

34

Stochastic simulation and master equation

v Repeating stochastic simulations allows approximation ofP(X (t )) in master equation to be given

X1

t

P(X1(t1))

X1(t1)

t1

35

Application of stochastic equations

v Stochastic equations have been used to model genetic andother regulatory systems:

l λ phage infection of E. coli (Arkin et al., 1998)

l chemotactic signalling in E. coli (Morton-Firth and Bray, 1998)

l ...

36

Stochastic analysis of phage ? infection

v Stochastic model of ?lysis-lysogenydecision networkArkin et al., 1998

37

Stochastic analysis of phage ? infection

v Time evolution of Cro and CIdimer concentrations

v Due to stochastic fluctuations,under identical conditions cellsfollow one or other pathway withsome probability

Arkin et al., 1998

38

Comparison with deterministic approach

v Deterministic models can beseen as predicting averagebehavior of cell populationGillespie, 2000

v However, analysis of averagebehavior may obscure thatone part of populationchooses one pathway ratherthan another

Arkin et al., 1998

39

Evaluation of stochastic equations

v Pro: more realistic models of gene regulation

v Contra: required information on regulatory mechanisms onmolecular level usually not available

reaction schemas and values of parameters τ and ρ are not orincompletely known

v Contra: stochastic simulation is computationally expensive

large networks cannot currently be handled

40

Conclusions

v Computer tools for modeling and simulation will be necessaryto understand genetic regulatory processes

v Variety of approaches available, representing geneticregulatory systems on different levels of abstraction

v Choice of approach depends on aim of analysis and onavailable information:

l knowledge on reaction mechanisms

l quantitative data on model parameters and gene expression levels

v Serious applications are beginning to emerge

41

Literature

A. Arkin et al., Stochastic kinetic analysis of developmental pathway

bifurcation in phage λ-infected Escherichia coli cells, Genetics,149:1633-1648, 1998

J.M. Bower and H. Bolouri, Computational Modeling of Genetic andBiochemical Networks, MIT Press, 2001

T.A. Carrier and J.D. Keasling, Investigating autocatalytic gene expressionsystems through mechanistic modeling, J. Theor. Biol., 201:25-36, 1999

J.L. Cherry and F.R. Adler, How to make a biological switch, J. Theor. Biol.,203:117-133, 2000

42

Literature

D. Endy et al., Computation, prediction, and experimental tests of fitness forbacteriophage T7 mutants with permuted genomes, P. Nat. Acad. Sc.USA, 97(10):5375-5380, 2000

J. Hasty et al., Computational studies of gene regulatory networks: innumero molecular biology, Nat. Rev. Genet., 2(4):268-279, 2001

H. de Jong, Modeling and simulation of genetic regulatory systems: Aliterature review, J. Comput. Biol., 9(1): 69-105, 2002

N. van Kampen, Stochastic Processes in Physics and Chemistry, Elsevier,1997

43

Literature

D. Kaplan and L. Glass, Understanding Nonlinear Dynamics, SpringerVerlag, 1995

A.D. Keller, Model genetic circuits encoding autoregulatory transcriptionfactors, J. Theor. Biol., 172:169-185, 1995

J.-C. Leloup and A. Goldbeter, A model for circadian rhythms in Drosophilaincorporating the formation of a complex between the PER and TIMproteins, J. Biol. Rhythms, 13(1):70-87, 1998

B. Lewin, Genes VI, Oxford University Press, 199

44

Literature

H. McAdams and A. Arkin, Simulation of prokaryotic genetic circuits, Annu.Rev. Biophys. Biomol. Struct., 27:199-224, 1998

C. Morton-Firth and D. Bray, Predicting temporal fluctuations in anintracellular signalling pathway, J. Theor. Biol., 192:117-128, 1998

H. McAdams and L. Shapiro, Circuit simulation of genetic networks,Science, 269:650-656

B. Novak and J. Tyson, Modeling the cell-division cycle: M-phase trigger,oscillations and size control, J. Theor. Biol., 165:101-134, 1993

M. Ptashne, A Genetic Switch: Phage ? and Higher Organisms,

J. Reinitz and D. Sharp, Gene circuits and their uses, in: J. Collado-Videset al. (eds), Integrative Approaches to Molecular Biology, MIT Press,253-272,1996

45

Literature

M. Santillán and M.C. Mackey, Dynamic regulation of the tryptophanoperon: A modeling study and comparison with experimental data,P. Nat. Acad. Sc. USA, 98(4):1364-1369, 2001

L. Segel, Modeling Dynamic Phenomena in Molecular and Cellular Biology,Cambridge University Press, 1984

P. Smolen et al., Modeling transcription control in gene networks: Methods,recent results, and future directions, Bull. Math. Biol., 62:247-292, 2000

S.H. Strogatz, Nonlinear Dynamics and Chaos: With Applications toPhysics, Biology, Chemistry, and Engineering, Perseus Books, 1994

R. Thomas and R. d’Ari, Biological Feedback, CRC Press, 1990