Validation of Qualitative Models of Genetic Regulatory Networks A Method Based on Formal Verificatio

1
Validation of Qualitative Models of Genetic
Regulatory Networks A Method Based on Formal
Verification Techniques

• Grégory Batt
• Ph.D. defense
• --
• under supervision of Hidde de Jong,
• Helix research group
• INRIA Rhône-Alpes
• --
• Ecole doctorale
• Mathématiques, Sciences et technologies de
  linformation, Informatique
• Université Joseph Fourier

2
Stress response in Escherichia coli

• Bacteria capable of adapting to a variety of
  changing environmental conditions
• Stress response in E. coli has been much studied
• Model for understanding adaptation of pathogenic
  bacteria to their host

3
Nutritional stress response in E. coli

• Response of E. coli to nutritional stress
  conditions transition from exponential phase to
  stationary phase
• Important developmental decision profound
  changes of morphology, metabolism, gene
  expression,...

log (pop. size)
gt 4 h
time
4
Network controlling stress response

• Response of E. coli to nutritional stress
  conditions controlled by genetic regulatory
  network
• Despite abundant knowledge on network
  components, no global view of functioning of
  network available

5
Modeling and simulation

• Genetic regulatory network controlling E. coli
  stress response is large and complex
• Modeling and simulation indispensable for
  dynamical analysis of genetic regulatory networks
• Systematic prediction of possible network
  behaviors
• Current constraints on modeling and simulation
• knowledge on molecular mechanisms rare
• quantitative information on kinetic parameters
  and molecular concentrations absent
• Qualitative methods developed for analysis of
  genetic networks using coarse-grained models

6
Model validation

• Available information on structure of network
  controlling E. coli stress response 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 network
  controlling E. coli stress response 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
• available experimental data essentially
  qualitative in nature
• model validation must be automatic and efficient

8
Objectives and approach of thesis

• Objective of thesis
• Development of automated and efficient method
  for testing whether predictions from qualitative
  models of genetic regulatory networks are
  consistent with experimental data on dynamics of
  system
• Approach based on formal verification of hybrid
  systems
• qualitative analysis of piecewise-linear models
  of genetic networks
• model checking for testing consistency between
  predictions and data
• Expected contributions
• scalable method with sound theoretical basis
• implementation of method in user-friendly
  computer tool
• applications to validation of models of networks
  of biological interest

9
Overview

• Introduction
• Method for model validation
• Piecewise-linear (PL) 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
Overview

• Introduction
• Method for model validation
• Piecewise-linear (PL) 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

11
PL differential equation models

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

• Hybrid, piecewise-linear (PL) models of genetic
  regulatory networks

Glass and Kauffman, J. Theor. Biol., 73
12
Qualitative analysis of network dynamics
.
x h (x), x ? ? \?

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

maxb
?b
0
?a1
maxa
?a2
13
Qualitative analysis of network dynamics
.
x h (x), x ? ? \?

• Analysis of the dynamics in phase space

maxb
?b
M1
0
maxa
?a1
?a2
14
Qualitative analysis of network dynamics
.
x h (x), x ? ? \?

• Analysis of the dynamics in phase space

maxb
.
M11
xa ? ?a xa
.
xb ? ?b ?b xb
?b
0
maxa
?a1
?a2
15
Qualitative analysis of network dynamics
.
x h (x), x ? ? \?

• Analysis of the dynamics in phase space

maxb
?b
0
maxa
?a1
?a2
16
Qualitative analysis of network dynamics
.
x h (x), x ? ? \?

• Analysis of the dynamics in phase space
• Extension of PL differential equations to
  differential inclusions using Filippov 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
17
Qualitative analysis of network dynamics
.

• Analysis of the dynamics in phase space
• In every mode domain M, the system either
  converges monotonically towards focal set, or
  instantaneously traverses M

x ? H (x), x ? ?
maxb
?b
0
?a1
maxa
?a2
de Jong et al., Bull. Math. Biol., 04
Gouzé and Sari, Dyn. Syst., 02
18
Problem for model validation

• Partition does not preserve unicity of derivative
  sign
• Predictions not adapted to comparison with
  available experimental data temporal evolution
  of direction of change of protein concentrations

19
Qualitative analysis of network dynamics

• Finer partition of phase space flow domains
• 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
20
Continuous transition system

• PL system, ? (?,?,H), associated with
  continuous PL transition system, ?-TS (?,?,),
  where
• ? continuous phase space

21
Continuous transition system

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

x1 ? x2,
x1 ? x3,
x3 ? x4
x2 ? x3,
22
Continuous transition system

• PL system, ? (?,?,H), associated with
  continuous PL transition system, ?-TS (?,?,),
  where
• ? continuous phase 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
23
Discrete abstraction

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

D D1, , D27
24
Discrete abstraction

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

25
Discrete abstraction

• Qualitative PL 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
26
Discrete abstraction

• Qualitative PL transition system, ?-QTS (D,
  ??,?), where
• D finite set of domains
• ?? quotient transition relation
• ? quotient satisfaction relation
• Quotient transition system ?-QTS is a simulation
  of ?-TS (but not a bisimulation)

maxb
maxb
kb/gb
?b
?b
0
0
?a1
?a1
?a2
?a2
maxa
maxa
Alur et al., Proc. IEEE, 00
27
Discrete abstraction

• Important properties of ?-QTS
• ?-QTS provides finite and qualitative description
  of the dynamics of system ? in phase space
• ?-QTS is a conservative approximation of ?
  every solution of ? corresponds to a path in
  ?-QTS
• ?-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 ?
• Need for automatic and efficient method to
  verify properties of ?-QTS

Batt et al., HSCC, 05
28
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)

29
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

30
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
31
Overview

• Introduction
• Method for model validation
• Piecewise-linear (PL) 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

32
Genetic Network Analyzer

• Model validation approach implemented in version
  6.0 of GNA, freely available for academic
  research

Batt et al., Bioinformatics, 05
33
Genetic Network Analyzer

• GNA implemented in Java 1.4
• gt 17000 lines of code in 6 packages
• 35 of lines modified with respect to version 5.5
• (up to 60 in kernel)

34
Genetic Network Analyzer

• Rules for symbolic computation of refined
  partition and corresponding transition relation
  and domain properties
• Tailored algorithms and implementation favor
  upscalability
• Export functionalities to model checkers (NuSMV,
  CADP)

35
Overview

• Introduction
• Method for model validation
• Piecewise-linear (PL) 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

36
Nutritional stress response in E. coli

• Entry into stationary phase is an important
  developmental decision

exponential phase
stationary phase
?
signal of nutritional deprivation
How does lack of nutrients induce decision to
stop growth?
37
Model of nutritional stress response

• Carbon starvation network modeled by PL model
• 7 PL differential equations, 40 parameters and 54
  inequality constraints

Ropers et al., Biosystems, in press
How does response emerge from network of
interactions?
38
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.)

39
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 (lt1s)
Balke and Gralla, J. Bacteriol., 87
False (lt1s)
40
Novel prediction of stress response model

• Qualitative simulation of carbon upshift
  response
• 1143 reachable domains (lt 2s)
• several strongly connected components
• Are some strongly connected components
  attractors?
• Attractor corresponds to damped oscillations
  towards stable equilibrium point unexpected
  prediction
• Experimental verification of model predictions
• Time-series measurements of protein
  concentrations in parallel and at high sampling
  rate using gene reporter system

AG(statesInSCCi ? AG statesInSCCi)
True (lt1s, i3)
Grognard et al., in preparation
41
Overview

• Introduction
• Method for model validation
• Piecewise-linear (PL) 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
Summary

• Development of automated and efficient method for
  testing whether predictions from qualitative
  models of genetic regulatory networks are
  consistent with experimental data on system
  dynamics
• Use of discrete abstraction that yields
  predictions well-adapted to comparison with
  available experimental data
• Combination of tailored symbolic analysis and
  model checking for verification of dynamical
  properties of hybrid models of large and complex
  networks
• Biological relevance demonstrated on validation
  of models of networks of biological interest

Batt et al., HSCC, 05
Batt et al., IJCAI, 05
Batt et al., Bioinformatics, 05
43
Discussion

• Discrete abstractions used for analysis of
  continuous and hybrid models
• symbolic reachability analysis of hybrid automata
  models
• more precise analysis of system dynamics
• need for complex decision procedures
• no treatment of discontinuities in vector field
• qualitative simulation using qualitative
  differential equations
• more general class of model
• methods are not scalable
• Model checking used for analysis of discrete
  models
• verification of properties of logical models
• intuitive connection between underlying
  continuous dynamics
• and discrete representation
• no explicit representation of dynamical phenomena
  at threshold concentrations

Ghosh and Tomlin, Systems Biology, 04
Heidtke and Schulze-Kremer, Bioinformatics, 98
Bernot et al., J. Theor. Biol., 04
44
Perspectives

• Further integration of model-checking task into
  GNA
• Property specification, verification,
  interpretation of diagnostics
• Exploitation of advanced model-checking
  techniques
• Partial order reduction, graph minimization,
  modular model checking, ...
• Extensions of model validation
• model inference complete partially-specified
  models
• model revision modify inconsistent models
• network design find model satisfying set of
  design constraints

45

• Thanks for your attention!