Transcript Qualitative Simulation of Large and Complex Genetic

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 l’Europe
Montbonnot, 38334 Saint Ismier CEDEX
Email: [email protected]
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
2
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
3
Genes and proteins
 Genes code for proteins that are essential for development
and functioning of organism: gene expression
DNA
transcription
RNA
translation
protein
protein and
modifier molecule
post-translational
modification
4
Molecular interactions
 Cellular processes involve interactions between proteins,
genes, metabolites, and other molecules:

cell structure

metabolism
membrane
enzyme
metabolite

gene regulation
gene
transcription factor
kinase

signal transduction
phosphorylated
regulatory protein
5
Organism as biochemical system
 Organism can be viewed as biochemical system, structured
by network of interactions between its molecular components
6
Systems biology
 Challenge of systems biology: understand how global
behavior of organism emerges from local interactions between
its molecular components
"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
 Elements of systems biology:

High-throughput experimental techniques

Advanced computational techniques and powerful computers

Integrated application of experimental and computational tools
7
Model-driven analysis of biological systems
 Model-driven analysis: integrated application of experimental
and computational tools
biological
system
experimental
conditions
choose
experiments
experimental
simulate
observations
predictions
conditions
perform
experiments
compare
models
biological
knowledge
construct
and revise
models
fit of models
experimental
observations
data
 Model composition versus model induction (reverse
engineering)
8
Genetic regulatory networks
 Genetic regulatory network is part of biochemical network
consisting (mainly) of genes and their regulatory interactions
9
Experimental tools
 Study of large and complex genetic regulatory networks
requires powerful experimental tools
High-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)
10
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 biology
Regulation of lac operon (Jacob and Monod)
Goodwin (1963), Temporal Organization in Cells
 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
11
Hierarchy of modeling formalisms
Graphs
Boolean equations
abstraction
precision
Ordinary differential equations
feasibility
Stochastic master equations
 Differential equations are major formalism for modeling
genetic regulatory networks :
nonlinear, linear, piecewise-linear differential equations
12
Nonlinear differential equation models
 Cellular concentration of proteins, mRNAs, and other molecules
at time-point t represented by continuous variable xi(t)  R0
 Regulatory interactions modeled by differential equations
dx
.
 x  f (x),
dt
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
13
Model of cross-inhibition network

gene 1

gene 2
x1 = 1 f (x2)  1 x1
.
x2 = 2 f (x1)  2 x2
.
x1 = concentration protein 1
x2 = concentration protein 2
1, 2 > 0, production rate constants
1, 2 > 0, degradation rate constants
f (x )
1
f (x) =
0


n
 +x
n
n
, 
> 0 threshold
x
14
Phase-plane analysis
 Analysis of steady states in phase plane
.
x1
x1 = 0
.
x1 = 0 : x1 =
.
.
x2 = 0
0
x2 = 0 : x2 =
1
1 f (x2)
2
f (x1)
2
x2
 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
15
Construction of cross inhibition network
 Construction of cross inhibition network in vivo
Gardner et al. (2000), Nature, 403(6786): 339-342
 Differential equation model of network
.
u=
α1
1+vβ
–u
.
v=
α2
1+u
–v
16
Experimental test of model
 Experimental test of mathematical model (bistability and
hysteresis)
Gardner et al. (2000), Nature, 403(6786): 339-342
17
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
18
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
19
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
20
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
21
Linear differential equation models
 Cellular concentration of proteins, mRNAs, and other molecules
at time-point t represented by continuous variable xi(t)  R0
 Regulatory interactions modeled by differential equations
dx
.
 x  f (x)  Ax  b,
dt
where x  [x1,…, xn]´ and f (x) is linear rate law
 Analytical solution exists for linear differential equations:
t
x(t)  eAt x0   eA(t-τ) dτ  b
0
22
Model of cross-inhibition network

gene 1

gene 2
x1 = 1 f (x2)  1 x1
.
x2 = 2 f (x1)  2 x2
.
1
1, 2 > 0, production rate constants
1, 2 > 0, degradation rate constants
f (x) = 1  x / (2 ) , 
f (x )
0
x1 = concentration protein 1
x2 = concentration protein 2
2
x
> 0,
x  2
x1 =  1 x1  11 x2  1
.
x2 =  22 x1  2 x2  2
.
23
Phase-plane analysis
 Analysis of steady states in phase plane
.
x1
x1 = 0
.
x1 = 0 : x1 =
.
.
x2 = 0
0
x2 = 0 : x2 =
1
1 f (x2)
2
f (x1)
2
x2
 Single unstable steady state.
 Linear differential equations too simple to capture dynamic
phenomena of interest: no bistability and no hysteresis
24
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
.
x  Ax  b  ξ,
with noise term ξ, fits expression data
 Powerful techniques for induction of linear model from
experimental data
Ljung (1995), System Identification, Prentice Hall, 1999
25
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
26
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
Gardner et al. (2003), Science,
301(5629): 102-105
 Method robust to measurement noise and upscalable
27
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
28
Piecewise-linear differential equation models
 Cellular concentration of proteins, mRNAs, and other molecules
at time-point t represented by continuous variable xi(t)  R0
 Regulatory interactions modeled by differential equations
AD1 x
dx
.
 x  f (x) 
dt
n

b D1 ,
D1  R0

bDm,
Dm  R0

ADm x
n
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
29
Model of cross-inhibition network

gene 1

gene 2
x1 = 1 f (x2)  1 x1
.
x2 = 2 f (x1)  2 x2
.
x1 = concentration protein 1
x2 = concentration protein 2
1, 2 > 0, production rate constants
1, 2 > 0, degradation rate constants
f (x )
1
0
f (x) = s( x, ) =

1,
x<
0,
x>
x
30
Phase-plane analysis
 Analysis of dynamics in phase plane
x1 = 1 s (x2, 2)  1 x1
.
x2 = 2 s (x1, 1)  2 x2
.
x1
.
x1 = 0
.
1
x2 = 0
D1 D 2
0 x. 1 = 0 2
.
x2 = 0
x2
in D21 :
x1 = 11 x11 x1 , x1 = 0 : x1 =c
= 01  1
.
.
,
x
x2 = 2  2 x2
2 = 0 : x2 = 2  2
.
.
 In every region Dj , model simplifies to system of piecewiseaffine differential equations
All solutions, while being in Dj , converge towards target steady state
 Different regions have different target steady states
31
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
Gouzé, Sari (2003), Dyn. Syst., 17(4):299-316
.
x1
x1 = 0
1
.
x2 = 0
0
2
x2
 Piecewise-linear model good approximation of nonlinear
model, retaining properties of bistability and hysteresis
32
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
33
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
34
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
35
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
36
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
37
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
38
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
biological
system
experimental
conditions
choose
experiments
experimental
simulate
observations
predictions
perform
experiments
conditions
compare
models
biological
knowledge
construct
and revise
models
fit of models
observations
39
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
40