Grgory Batt

1
Validation of Hybrid Models of Genetic
Regulatory NetworksNutritional Stress Response
in E. coli

• Grégory Batt
• INRIA Rhône-Alpes, Grenoble
• Université Joseph Fourier, Grenoble
• (Now at Boston University)
• Email batt_at_bu.edu

2
Genetic regulatory networks

• Organism can be viewed as biochemical system,
  structured by networks of interactions between
  its molecular components
• Challenge of systems biology understand how
  global behavior emerges from local interactions
  between molecular components
• Genetic regulatory network is part of biochemical
  network consisting (mainly) of genes and their
  regulatory interactions

cross-inhibition network
3
Genetic regulatory networks

• Genetic regulatory networks underlie functioning
  and development of living organisms

4
Genetic regulatory networks

• Genetic regulatory networks underlie functioning
  and development of living organisms

Network controlling nutritional stress response
in E. coli
5
Genetic regulatory networks

• Genetic regulatory networks underlie functioning
  and development of living organisms
• Due to switch-like character of gene regulations,
  hybrid models have been proposed for network
  analysis
• Antoniotti et al.,
  Theor. Comput. Sci., 04 Belta et al., HSCC, 04
  de Jong et al., HSCC,
  03 Ghosh and Tomlin, Syst. Biol., 04
• Current constraints on network analysis
• lack of quantitative information on kinetic
  constants and molecular concentrations
• size of networks and complexity of dynamics
• Qualitative method tailored to analysis of
  genetic networks using coarse-grained, hybrid
  models

de Jong et al., HSCC, 03
6
Model validation

• Available information on structure of networks is
  incomplete
• Model is working hypothesis and needs to be
  tested
• Model validation is prerequisite for use of model
  as predictive and explanatory tool
• Check consistency between model predictions and
  experimental data

7
Model validation

• Available information on structure of networks is
  incomplete
• Model is working hypothesis and needs to be
  tested
• Model validation is prerequisite for use of model
  as predictive and explanatory tool
• Check consistency between model predictions and
  experimental data
• Current constraints on model validation
• predictions suitable for comparison with
  available experimental data
• model validation must be automatic and efficient
• Approach extend existing qualitative modeling
  and simulation method and combine with
  model-checking techniques

8
Overview

• Introduction
• Method for model validation
• Piecewise-affine (PA) differential equation
  models
• Symbolic analysis using qualitative abstraction
• Verification of properties by means
  model-checking techniques
• Genetic Network Analyzer 6.0
• Validation of model of nutritional stress
  response in E. coli
• Discussion and conclusions

9
Overview

• Introduction
• Method for model validation
• Piecewise-affine (PA) differential equation
  models
• Symbolic analysis using qualitative abstraction
• Verification of properties by means
  model-checking techniques
• Genetic Network Analyzer 6.0
• Validation of model of nutritional stress
  response in E. coli
• Discussion and conclusions

10
PA models of genetic regulatory networks

• Genetic networks modeled by class of differential
  equations using step functions to describe
  regulatory interactions

11
PA models of genetic regulatory networks

• Genetic networks modeled by class of differential
  equations using step functions to describe
  regulatory interactions

A
B
b
12
PA models of genetic regulatory networks

• Genetic networks modeled by class of differential
  equations using step functions to describe
  regulatory interactions

A
B
a
13
PA models of genetic regulatory networks

• Genetic networks modeled by class of differential
  equations using step functions to describe
  regulatory interactions

14
PA models of genetic regulatory networks

• Genetic networks modeled by class of differential
  equations using step functions to describe
  regulatory interactions

• Differential equation models of regulatory
  networks are piecewise-affine (PA)

Glass and Kauffman, J. Theor. Biol., 73
15
Qualitative analysis of network dynamics
.
x h (x), x ? ? \?

• Analysis of the dynamics in state space
• Partition of state space into mode domains

16
Qualitative analysis of network dynamics
.
x h (x), x ? ? \?

• Analysis of the dynamics in state space
• Partition of state space into mode domains

maxb
?b
M1
0
maxa
?a1
?a2
17
Qualitative analysis of network dynamics
.
x h (x), x ? ? \?

• Analysis of the dynamics in state space
• Partition of state space into mode domains

maxb
.
M11
xa ? ?a xa
.
xb ? ?b ?b xb
?b
0
maxa
?a1
?a2
18
Qualitative analysis of network dynamics
.
x h (x), x ? ? \?

• Analysis of the dynamics in state space
• Partition of state space into mode domains

maxb
?b
0
maxa
?a1
?a2
19
Qualitative analysis of network dynamics
.
x h (x), x ? ? \?

• Analysis of the dynamics in state space
• Partition of state space into mode domains
• Extension of PA differential equations to
  differential inclusions
• Filippov-like approach

maxb
maxb
kb/gb
?b
?b
M3
M4
M3
M1
M2
M5
0
0
maxa
maxa
?a1
ka/ga
?a1
ka/ga
?a2
?a2
.
x ? H (x), x ? ?
Gouzé and Sari, Dyn. Syst., 02
20
Qualitative analysis of network dynamics
.
x ? H (x), x ? ?

• Analysis of the dynamics in state space
• Partition of state space into mode domains
• Extension of PA differential equations to
  differential inclusions
• In mode domain M, the system either converges
  monotonically towards focal set, or
  instantaneously traverses M

maxb
?b
0
?a1
maxa
?a2
de Jong et al., Bull. Math. Biol., 04
Gouzé and Sari, Dyn. Syst., 02
.
21
Problem for model validation

• Model validation using gene expression data
  observation of changes in derivative signs
• Predictions obtained using mode domain partition
  not adapted to comparison with available
  experimental data
• Partition does not preserve unicity of derivative
  sign of solutions

22
Qualitative analysis of network dynamics

• Finer partition of state space flow domains
• Repartitioning mode domains by means of nullcline
  planes
• In every domain D, the system either converges
  monotonically towards focal set, or
  instantaneously traverses D
• In every domain D, derivative signs are identical
  everywhere

maxb
?b
0
?a1
maxa
?a2
23
Qualitative analysis of network dynamics

• Finer partition of state space flow domains
• Repartitioning mode domains by means of nullcline
  planes
• In every domain D, the system either converges
  monotonically towards focal set, or
  instantaneously traverses D
• In every domain D, derivative signs are identical
  everywhere
• Non-unicity of solutions of differential
  inclusion derivative sign pattern

maxb
?b
0
?a1
maxa
?a2
.
24
Continuous transition system

• PA system, ? (?,?,H), associated with
  continuous PA transition system, ?-TS (?,?,),
  where
• ? continuous state space

25
Continuous transition system

• PA system, ? (?,?,H), associated with
  continuous PA transition system, ?-TS (?,?,),
  where
• ? continuous state space
• ? transition relation

x1 ? x2,
x1 ? x3,
x3 ? x4
x2 ? x3,
26
Continuous transition system

• PA system, ? (?,?,H), associated with
  continuous PA transition system, ?-TS (?,?,),
  where
• ? continuous state space
• ? transition relation
• satisfaction relation
• ? and ?-TS have equivalent reachability properties

maxb
maxb
x5
kb/gb
x4
x3
?b
?b
x1
x2
0
0
?a1
?a1
?a2
?a2
maxa
maxa
27
Discrete abstraction

• Qualitative PA transition system, ?-QTS (D,
  ??,?), where
• D finite set of domains

D D1, , D27
28
Discrete abstraction

• Qualitative PA transition system, ?-QTS (D,
  ??,?), where
• D finite set of domains
• ?? quotient transition relation

29
Discrete abstraction

• Qualitative PA transition system, ?-QTS (D,
  ??,?), where
• D finite set of domains
• ?? quotient transition relation
• ? quotient satisfaction relation

D? p iff there exists x?D such that x p
30
Discrete abstraction

• Qualitative PA transition system, ?-QTS (D,
  ??,?), where
• D finite set of domains
• ?? quotient transition relation
• ? quotient satisfaction relation

maxb
maxb
kb/gb
?b
?b
0
0
?a1
?a1
?a2
?a2
maxa
maxa
31
Discrete abstraction

• ?-QTS simulates ?-TS conservative approximation
• Every solution of ? corresponds to a path in
  ?-QTS
• ?-QTS provides finite description of dynamics in
  phase space
• Well-adapted to model validation changes in
  derivative signs over time

Alur et al., Proc. IEEE, 00
32
Discrete abstraction

• ?-QTS simulates ?-TS conservative approximation
• Every solution of ? corresponds to a path in
  ?-QTS
• ?-QTS provides finite description of dynamics in
  phase space
• Well-adapted to model validation changes in
  derivative signs over time
• ?-QTS is invariant for all parameters ?, ?, and ?
  satisfying a set of inequality constraints

Alur et al., Proc. IEEE, 00
33
Discrete abstraction

• ?-QTS simulates ?-TS conservative approximation
• Every solution of ? corresponds to a path in
  ?-QTS
• ?-QTS provides finite description of dynamics in
  phase space
• Well-adapted to model validation changes in
  derivative signs over time
• ?-QTS is invariant for all parameters ?, ?, and ?
  satisfying a set of inequality constraints
• ?-QTS can be computed symbolically using
  parameter inequality constraints qualitative
  simulation
• Use of?-QTS to verify dynamical properties of
  original system ?

Alur et al., Proc. IEEE, 00
Batt et al., HSCC, 05
Problem need for efficient method to check
properties of ?-QTS
34
Model-checking approach

• Model checking is automated technique for
  verifying that discrete transition system
  satisfies certain temporal properties
• Computation tree logic model-checking framework
• set of atomic propositions AP
• discrete transition system is Kripke structure KS
  ? S, R, L ?,
• where S set of states, R transition relation, L
  labeling function over AP
• temporal properties expressed in Computation Tree
  Logic (CTL)
• p, f1, f1?f2, f1?f2, f1?f2, EXf1, AXf1, EFf1,
  AFf1, EGf1, AGf1, Ef1Uf2, Af1Uf2, where p?AP and
  f1, f2 CTL formulas
• Computer tools are available to perform efficient
  and reliable model checking (e.g., NuSMV, SPIN,
  CADP)

35
Validation using model checking

• Atomic propositions
• AP xa 0, xa lt qa1, ... , xb lt maxb, xa lt 0,
  xa 0, ... , xb gt 0
• Observed property expressed in CTL

36
Validation using model checking
37
Validation using model checking

• Discrete transition system computed using
  qualitative simulation

38
Validation using model checking

• Discrete transition system computed using
  qualitative simulation
• Use of model checkers to check consistency
  between experimental data and predictions

Consistency?
Yes
Batt et al., IJCAI, 05
39
Validation using model checking

• Discrete transition system computed using
  qualitative simulation
• Use of model checkers to check consistency
  between experimental data and predictions
• Fairness constraints used to exclude spurious
  behaviors

Consistency?
Yes
Batt et al., IJCAI, 05
40
Genetic Network Analyzer 6.0

• Model validation approach supported by new
  version of GNA, freely available for academic
  research

Batt et al., Bioinformatics, 05
41
Overview

• Introduction
• Method for model validation
• Piecewise-affine (PA) differential equation
  models
• Symbolic analysis using qualitative abstraction
• Verification of properties by means
  model-checking techniques
• Genetic Network Analyzer 6.0
• Validation of model of nutritional stress
  response in E. coli
• Discussion and conclusions

42
Overview

• Introduction
• Method for model validation
• Piecewise-affine (PA) differential equation
  models
• Symbolic analysis using qualitative abstraction
• Verification of properties by means
  model-checking techniques
• Genetic Network Analyzer 6.0
• Validation of model of nutritional stress
  response in E. coli
• Discussion and conclusions

43
Nutritional stress response in E. coli

• In case of nutritional stress, E. coli population
  abandons growth and enters stationary phase
• Decision to abandon or continue growth controlled
  by complex network
• Model 7 PADEs, 40 parameters and 54 inequality
  constraints

Ropers et al., Biosystems, 06
44
Validation of stress response model

• Qualitative simulation of carbon starvation
• 66 reachable domains (lt 1s.)
• single attractor domain (asymptotically stable
  equilibrium point)
• Experimental data on Fis
• CTL formulation
• Model checking with NuSMV property true (lt 1s.)

45
Validation of stress response model

• Other properties
• cya transcription is negatively regulated by the
  complex cAMP-CRP
• DNA supercoiling decreases during transition to
  stationary phase
• Inconsistency between observation and prediction
  calls for model revision or model extension
• Nutritional stress response model extended with
  global regulator RpoS

Kawamukai et al., J. Bacteriol., 85
True
Balke and Gralla, J. Bacteriol., 87
False
46
Overview

• Introduction
• Method for model validation
• Piecewise-affine (PA) differential equation
  models
• Symbolic analysis using qualitative abstraction
• Verification of properties by means
  model-checking techniques
• Genetic Network Analyzer 6.0
• Validation of model of nutritional stress
  response in E. coli
• Discussion and conclusions

47
Overview

• Introduction
• Method for model validation
• Piecewise-affine (PA) differential equation
  models
• Symbolic analysis using qualitative abstraction
• Verification of properties by means
  model-checking techniques
• Genetic Network Analyzer 6.0
• Validation of model of nutritional stress
  response in E. coli
• Discussion and conclusions

48
Conclusions

• Automated and efficient method for testing
  whether predictions from qualitative models of
  genetic regulatory networks are consistent with
  experimental data on system dynamics
• Approach adapted to current knowledge and
  available experimental data
• Combination of tailored symbolic analysis and
  model checking for verification of dynamical
  properties of hybrid models of large and complex
  networks
• Method implemented in new version of GNA
• Biological relevance demonstrated on validation
  of model of network of biological interest

49
Conclusions

• Discrete or qualitative abstraction for analysis
  of networks
• hybrid automata
  Belta et al., HSCC, 04 Ghosh et al., HSCC, 03
• qualitative differential equations Heidtke and
  Schulze-Kremer, Bioinformatics, 98
• Model-checking for verification of network
  properties
• generalized logical models Bernot et
  al., J. Theor. Biol., 04
• concurrent systems Chabrier et al., Theor.
  Comput. Sci., 04 Eker et al., PSB, 02
• simulation trace semantics Antoniotti
  et al., Theor. Comput. Sci., 04
• Further work
• integration of tailored model checker into GNA
• exploit network modularity with compositional
  model checking (scalability)
• use model validation method for model revision (?
  Calins talk)

50
Acknowledgements

• Thank you for your attention!

• Hidde de Jong (INRIA Rhône-Alpes, France)
• Johannes Geiselmann (Université Joseph Fourier,
  Grenoble, France)
• Jean-Luc Gouzé (INRIA Sophia-Antipolis, France)
• Radu Mateescu (INRIA Rhône-Alpes, France)
• Michel Page (Université Pierre Mendès-France,
  Grenoble, France)
• Delphine Ropers (INRIA Rhône-Alpes, France)
• Tewfik Sari (Université de Haute Alsace,
  Mulhouse, France)
• Dominique Schneider (Université Joseph Fourier,
  Grenoble, France)