Qualitative Modeling and Simulation of Genetic Regulatory Networks using Piecewise-Linear
Differential Equations
Hidde de Jong and Delphine Ropers
INRIA Rhône-Alpes655 avenue de l’Europe, Montbonnot
38334 Saint Ismier CedexFrance
Email: {Hidde.de-Jong,Delphine.Ropers}@inrialpes.fr
2
Overview
1. Genetic regulatory networks
2. Models of genetic regulatory networks
nonlinear differential equations
linear differential equations
piecewise-linear differential equations
3. Qualitative modeling, simulation, and validation using
piecewise-linear differential equations
4. Genetic Network Analyzer (GNA)
3
Escherichia coli: model organism
Enteric bacterium Escherichia coli has been most-studied organism in biology
« All cell biologists have two cells of interest: the one they are studying and
Escherichia coli »
2 μm 4300 genes107 bacteria
Schaechter and Neidhardt (1996), Escherichia coli and Salmonella, ASM Press, 4
4
Bacterial cell and proteins
Proteins are building blocks of cell
Cell membrane, enzymes, gene expression, …
5
Variation in protein levels
Protein levels in cell are adjusted to specific environmental conditions
Peng, Shimizu (2003), App. Microbiol. Biotechnol., 61:163-178
Ali Azam et al. (1999), J. Bacteriol., 181(20):6361-6370
2D gels
Western blots
DNA microarrays
6
Synthesis and degradation of proteins
DNA
mRNA
protein
modified protein
transcription
translation
post-translational modification
effector molecule
degradation protease
RNA polymerase
ribosome
7
Regulation of synthesis and degradation
RBS
mRNA
ribosome
modified protein
kinase
protease
RNA polymerasetranscription
factor
DNA
small RNA
response regulator
8
Example: σS in E. coli σS (RpoS) is sigma factor in E. coli and other bacteria
Subunit of RNA polymerase which recognizes specific promoters
σS is regulated on different levels: Transcription: repression by CRP·cAMP
Translation: increase in efficiency by binding of small RNAs DsrA, RprA
Activity: increase in promoter affinity of RNAP with σS by binding of Crl
Degradation: RssB targets σS for degradation by ClpXP
Adapted from: Hengge-Aronis (2002), Microbiol. Mol. Biol. Rev., 66(3):373-395
9
Genetic regulatory networks
Control of protein synthesis and degradation gives rise to genetic regulatory networks
Networks of genes, RNAs, proteins, metabolites, and their interactions
Activation Stress signal
CRP
crp
cya
CYA
fis
FIS
Supercoiling
TopA
topA
GyrAB
P1-P4P1 P2
P2P1-P’1
rrnP1 P2
P
gyrABP
tRNArRNA
GyrI
gyrIP
rpoSP1 P2nlpD
σS
RssB
rssAPA PB rssB
P5
Carbon starvation network in E. coli
10
Modeling of genetic regulatory networks
Abundant knowledge on components and interactions of genetic regulatory networks
Currently no understanding of how global dynamics emerges from local interactions between components
Shift from structure to behavior of genetic regulatory networks
« functional genomics », « integrative biology », « systems biology », …
Mathematical methods supported by computer tools allow modeling and simulation of genetic regulatory networks:
precise and unambiguous description of network
understanding through computer experiments
new predictions
11
Model formalisms
Many formalisms to model genetic regulatory networks
ODEs with implicit assumptions and additional simplifications: Continuous and deterministic dynamics
Lumping together protein synthesis and degradation in single step
Graphs
Differential equations
Stochastic master equations
precision abstraction
Boolean equations
de Jong (2002), J. Comput. Biol., 9(1): 69-105
12
Cross-inhibition network
Cross-inhibition network consists of two genes, each coding for transcription regulator inhibiting expression of other gene
Cross-inhibition network is example of positive feedback, important for differentiation
Thomas and d’Ari (1990), Biological Feedback
gene b
protein B
gene a
protein A
promoter a promoter b
13
Nonlinear model of cross-inhibition network
xa = concentration protein A
xb = concentration protein B
xa = a f (xb) a xa
xb = b f (xa) b xb
a, b > 0, production rate constants a, b > 0, degradation rate
constants
.
.
f (x) = , > 0 threshold
n
n + x n
x
f (x )
0
1
b
B
a
A
14
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
System displays hysteresis effect: transient perturbation may cause irreversible switch to another steady state
xb
xa
0
xb = 0 .
xa = 0 .
xa = 0 : xa = f (xb)a
a
xb = 0 : xb = f (xa)b
b
.
.
15
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
16
Experimental test of model
Experimental test of mathematical model (bistability and hysteresis)
Gardner et al. (2000), Nature, 403(6786): 339-342
17
Bifurcation analysis
Analysis of bifurcations caused by changes in control parameter
Change in control parameter may cause an irreversible switch to another steady state
xb
xa
0
xb = 0 .
xa = 0 .
xb
xa
0
xb = 0 .
xa = 0 .
xb
xa
0
xb = 0 .
xa = 0 .
value of b
18
Bacteriophage infection of E. coli
Response of E. coli to phage infection involves decision between alternative developmental pathways: lysis and lysogeny
Ptashne, A Genetic Switch, Cell Press,1992
19
Control of phage fate decision
Cross-inhibition feedback plays key role in establishment of lysis or lysogeny, as well as in induction of lysis after DNA damage
Santillán, Mackey (2004), Biophys. J., 86(1): 75-84
20
Simple model of phage fate decision
Differential equation model of cross-inhibition feedback network involved in phage fate decision
mRNA and protein, delays, thermodynamic description of gene regulation
Santillán, Mackey (2004), Biophys. J., 86(1): 75-84
21
Analysis of phage model
Bistability (lysis and lysogeny) only occurs for certain parameter values
Switch from lysis to lysogeny involves bifurcation from one monostable regime to another, due to change in degradation constant
Santillán, Mackey (2004), Biophys. J., 86(1): 75-84
22
Extended model of phage infection
Differential equation model of the extended network underlying decision between lysis and lysogeny
McAdams, Shapiro (1995), Science, 269(5524): 650-656
23
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
24
Linear model of cross-inhibition network
xa = concentration protein A
xb = concentration protein B
a, b > 0, production rate constants a, b > 0, degradation rate
constants xa = a f (xb) a xa
xb = b f (xa) b xb
.
.
f (x) = 1 x / (2 ) , > 0, x 2
x
f (x )
0 2
1
b
B
a
A
25
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
xb
xa
0
xa = 0 .
xb = 0 .
xa = 0 : xa = f (xb)a
a
xb = 0 : xb = f (xa)b
b
.
.
26
Evaluation of linear differential equations
Pro: analytical solution exists, thus facilitating analysis of complex systems
Contra: too simple to capture important dynamical phenomena of regulatory network, due to neglect of nonlinear character of interactions
27
Piecewise-linear model of cross-inhibition
f (x) = s(x, ) =1, x <
0, x >
x
f (x )
0
1
Glass and Kauffman (1973), J. Theor. Biol., 39(1):103-129
xa = concentration protein A
xb = concentration protein B
a, b > 0, production rate constants a, b > 0, degradation rate
constants xa = a f (xb) a xa
xb = b f (xa) b xb
.
.
b
B
a
A
28
PL models and gene regulatory logic
Step function expressions correspond to Boolean functions used to express gene regulatory logic
ba
A
B
xa a s-(xb , b ) – a xa.
xb b s-(xa , a) – b xb .
Thomas and d’Ari (1990), Biological Feedback
condition gene a: (xb < b )condition gene b: (xa < a )
xa a s-(xa , a2) s-(xb , b ) – a xa .
xb b s-(xa , a1) – b xb .
b
B
a
A
condition gene a: (xa < a2) (xb < b )condition gene b: (xa < a1)
29
Phase-plane analysis Analysis of dynamics of PL models in phase space
xb
xa
0 b
a
xa a – a xa .
xb b – b xb .
κa/γa
κb/γb
M1
xa a s-(xb , b ) – a xa.
xb b s-(xa , a) – b xb .
M1:
xb
xa
0 b
a
κb/γb
M3
xa – a xa .xb b – b xb ...M3:
30
Phase-plane analysis Analysis of dynamics of PL models in phase space
Extension of PL differential equations to differential inclusions using Filippov approach
xb
xa
0 b
a
κa/γa
κb/γb
xa a s-(xb , b ) – a xa.
xb b s-(xa , a) – b xb .
M2
Gouzé, Sari (2002), Dyn. Syst., 17(4):299-316
xb
xa
0 b
a
κb/γb
M5
κa/γa
31
Phase-plane analysis
Global phase-plane analysis by combining analyses in local regions of phase plane
Piecewise-linear model good approximation of nonlinear model, retaining properties of bistability and hysteresis
xb
xa
0
xb = 0 .
xa = 0 .
b
a
xb
xa
0
xb = 0 .
xa = 0 .
32
Hyperrectangular phase space partition: unique derivative sign pattern in regions
Qualitative abstraction yields state transition graph
Shift from continuous to discrete picture of network dynamics
Qualitative analysis using PL models
xb
xa
0 b
a
D1 D2
D3
D4
D5
D11 D12
D13
D14
D15
D16
D19
D23
D18
D21
D24
D25
D10
D6 D7 D8
D9
D17
D20
D22
D6
D22
D19
D10
D16
D1 D2 D3 D4 D5
D15
D25
D11 D12 D13 D14
D7
D8
D9
D17
D20
D23
D18
D21
D24
.
.xa > 0xb > 0D1:
.
.xa > 0xb < 0D17:
.
.xa = 0xb = 0D19:
de Jong, Gouzé et al. (2004), Bull. Math. Biol., 66(2):301-340
33
Qualitative analysis using PL models
Paths in state transition graph represent possible qualitative behaviors
.
.xa > 0xb > 0D1:
.
.xa > 0xb < 0D17:
.
.xa = 0xb = 0D19:
D6
D22
D19
D10
D16
D1 D2 D3 D4 D5
D15
D25
D11 D12 D13 D14
D7
D8
D9
D17
D20
D23
D18
D21
D24
D1 D11 D17 D19
a
κa/γa
D1 D11 D17 D19
b
κb/γb
34
State transition graph invariant for parameter constraints
Qualitative analysis using PL models
D1 D3
D11 D12 0 < a < a/a
0 < b < b/b
xb
xa
0 b
a
κa/γa
κb/γb
D1
D11 D12
D3
35
State transition graph invariant for parameter constraints
Qualitative analysis using PL models
D1 D3
D11 D12 0 < a < a/a
0 < b < b/b
xb
xa
0 b
a
κa/γa
κb/γb
D1
D11 D12
D3
36
State transition graph invariant for parameter constraints
Qualitative analysis using PL models
D1 D3
D11 D12 0 < a < a/a
0 < b < b/b
xa
0
a
κa/γa
κb/γb
D1
D11 D12
D3
xb0 b
a
κa/γa
κb/γb
D1
D11
D1
D11
0 < b/b < b
0 < a < a/a
37
Predictions well adapted to comparison with available experimental data: changes of derivative sign patterns
Model validation: comparison of derivative sign patterns in observed and predicted behaviors
Need for automated and efficient tools for model validation
D6
D22
D19
D10
D16
D1 D2 D3 D4 D5
D15
D25
D11 D12 D13 D14
D7
D9
D17
D20
D23
D18
D21
D24
Validation of qualitative models
Concistency?
0
xb
time
time0
xa
xa > 0.xb > 0.
xb > 0.xa < 0.
D8
38
Predictions well adapted to comparison with available experimental data: changes of derivative sign patterns
Model validation: comparison of derivative sign patterns in observed and predicted behaviors
Need for automated and efficient tools for model validation
Validation of qualitative models
Concistency?
Yes
0
xb
time
time0
xa
xa > 0.xb > 0.
xb > 0.xa < 0.
.
.xa > 0xb > 0D1:
.
.xa > 0xb < 0D17:
.
.xa = 0xb = 0D19:
D6
D22
D19
D10
D16
D1 D2 D3 D4 D5
D15
D25
D11 D12 D13 D14
D7
D8
D9
D17
D20
D23
D18
D21
D24
39
Model-checking approach
Dynamic properties of system can be expressed in temporal logic (CTL)
Model checking is automated technique for verifying that state transition graph satisfies temporal-logic statements
Computer tools are available to perform efficient and reliable model checking (NuSMV, CADP, …)
There Exists a Future state where xa > 0 and xb > 0 and starting from that state,
there Exists a Future state where xa < 0 and xb > 0
. .
. .
EF(xa > 0 xb > 0 EF(xa < 0 xb > 0) ). . . . 0
xb
time
time0
xa
xa > 0.xb > 0.
xb > 0.xa < 0.
40
Validation using model checking
Compute state transition graph using qualitative simulation
Use of model checkers to verify whether experimental data and predictions are consistent
Concistency?0
xb
time
time0
xa
xa > 0.xb > 0.
xb > 0.xa < 0.
Batt et al. (2005), Bioinformatics, 21(supp. 1): i19-i28
D6
D22
D19
D10
D16
D1 D2 D3 D4 D5
D15
D25
D11 D12 D13 D14
D7
D9
D17
D20
D23
D18
D21
D24
D8
41
Validation using model checking
Compute state transition graph using qualitative simulation
Use of model checkers to verify whether experimental data and predictions are consistent
Yes
Concistency?
Model corroborated
EF(xa > 0 xb > 0 EF(xa < 0 xb > 0) ). . . .
Batt et al. (2005), Bioinformatics, 21(supp. 1): i19-i28
D6
D22
D19
D10
D16
D1 D2 D3 D4 D5
D15
D25
D11 D12 D13 D14
D7
D8
D9
D17
D20
D23
D18
D21
D24
D19
D1
D11
D17
42
Analysis of attractors of PL systems
Search of steady states of PL systems in phase space
xb
xa
0 b
a
D6
D22
D19
D10
D16
D1 D2 D3 D4 D5
D15
D25
D11 D12 D13 D14
D7
D9
D17
D20
D23
D18
D21
D24
D8
43
Analysis of stability of steady states, using local properties of state transition graph
Definition of stability of equilibrium points on surfaces of discontinuity
Analysis of attractors of PL systems
Search of steady states of PL systems in phase space
Casey et al. (2006), J. Math Biol., 52(1):27-56
xb
xa
0 b
a
D6
D22
D19
D10
D16
D1 D2 D3 D4 D5
D15
D25
D11 D12 D13 D14
D7
D8
D9
D17
D20
D23
D18
D21
D24
44
Genetic Network Analyzer (GNA)
http://www-helix.inrialpes.fr/gna
Qualitative simulation method implemented in Java: Genetic Network Analyzer (GNA)
de Jong et al. (2003), Bioinformatics, 19(3):336-344
Distribution by Genostar SA
Batt et al. (2005), Bioinformatics, 21(supp. 1): i19-i28
45
Applications of GNA Qualitative simulation method used to analyze various
bacterial regulatory networks: initiation of sporulation in Bacillus subtilis
quorum sensing in Pseudomonas aeruginosa
carbon starvation response in Escherichia coli
onset of virulence in Erwinia chrysanthemi
de Jong, Geiselmann et al. (2004), Bull. Math. Biol., 66(2):261-300
Viretta and Fussenegger, Biotechnol. Prog., 2004, 20(3):670-678
Ropers et al., Biosystems, 2006, 84(2):124-152
Sepulchre et al., J. Theor. Biol., 2006, in press
46
Evaluation of PL differential equations
Pro: captures important dynamical phenomena of network, by suitable approximation of nonlinearities
Pro: qualitative analysis of dynamics possible, due to favorable mathematical properties
Contra: restricted class of models, not directly applicable to type of functions found in, for example, metabolism
47
Contributors and sponsorsGrégory Batt, Boston University, USA
Hidde de Jong, INRIA Rhône-Alpes, France
Hans Geiselmann, Université Joseph Fourier, Grenoble, France
Jean-Luc Gouzé, INRIA Sophia-Antipolis, France
Radu Mateescu, INRIA Rhône-Alpes, France
Michel Page, INRIA Rhône-Alpes/Université Pierre Mendès France, Grenoble, France
Corinne Pinel, Université Joseph Fourier, Grenoble, France
Delphine Ropers, INRIA Rhône-Alpes, France
Tewfik Sari, Université de Haute Alsace, Mulhouse, France
Dominique Schneider, Université Joseph Fourier, Grenoble, France
Ministère de la Recherche,
IMPBIO program European Commission,
FP6, NEST program INRIA, ARC program Agence Nationale de la
Recherche, BioSys program
