validation of qualitative models of genetic regulatory networks a method based on formal...
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Validation of Qualitative Models of Genetic Regulatory Networks
A Method Based on Formal Verification Techniques
Grégory BattPh.D. defense
--
under supervision of Hidde de Jong,
Helix research group
INRIA Rhône-Alpes
--
Ecole doctorale
Mathématiques, Sciences et technologies de l’information, Informatique
Université Joseph Fourier
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
Nutritional stress
Osmotic stress
Heat shock
Cold shock
…
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)
time
> 4 h
Network controlling stress response Response of E. coli to nutritional stress conditions controlled by
genetic regulatory networkDespite abundant knowledge on network components, no global view of
functioning of network available
rrnP1 P2
CRP
crp
cya
CYA
cAMP•CRP
FIS
TopA
topA
GyrAB
P1-P4P1 P2
P2P1-P’1
P
gyrABP
Signal (carbon starvation)
DNA supercoiling
fis
stable RNAs
protein
gene
promoter
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
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
consistency?experimental datanetwork predictions
x = f (x) .
model
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
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
Overview
I. Introduction
II. Method for model validation
1. Piecewise-linear (PL) differential equation models
2. Symbolic analysis using qualitative abstraction
3. Verification of properties by means model-checking techniques
III. Genetic Network Analyzer 6.0
IV. Validation of model of nutritional stress response in E. coli
V. Discussion and conclusions
Overview
I. Introduction
II. Method for model validation
1. Piecewise-linear (PL) differential equation models
2. Symbolic analysis using qualitative abstraction
3. Verification of properties by means model-checking techniques
III. Genetic Network Analyzer 6.0
IV. Validation of model of nutritional stress response in E. coli
V. Discussion and conclusions
PL differential equation models
Genetic networks modeled by class of differential equations using step functions to describe switch-like regulatory interactions
xa a s-(xa , a2) s-(xb , b ) – a xa .
xb b s-(xa , a1) – b xb .
x : protein concentration
, : rate constants : threshold concentration
x
s-(x, θ)
0
1
Hybrid, piecewise-linear (PL) models of genetic regulatory networks Glass and Kauffman, J. Theor. Biol., 73
b
B
a
A
Analysis of the dynamics in phase space:
Partition of phase space into mode domains
a10
maxb
a2
b
maxa
Qualitative analysis of network dynamics
a10
maxb
a2
b
maxa a10
maxb
a2
b
maxaa10
maxb
a2
b
maxa
M1 M2 M3 M4 M5
M10
M15M14M13M12M11
M6 M7 M8 M9
x = h (x), x \ .
Analysis of the dynamics in phase space:
a10
maxb
a2
b
maxa
Qualitative analysis of network dynamics
xa a s-(xa , a2) s-(xb , b ) – a xa.
xb b s-(xa , a1) – b xb .
a10
maxb
a2
b
maxa
xa a – a xa .
xb b – b xb .
aa
bb
0 < a1 < a2 < a/a < maxa
0 < b < b/b < maxb
0 a – a xa 0 b – b xb
.
M1
x = h (x), x \
Analysis of the dynamics in phase space:
a10
maxb
a2
b
maxa
Qualitative analysis of network dynamics
a10
maxb
a2
b
maxa
xa – a xa .
xb b – b xb .
M11
0 < a1 < a2 < a/a < maxa
0 < b < b/b < maxb
. x = h (x), x \
bb
Analysis of the dynamics in phase space:
a10
maxb
a2
b
maxa
Qualitative analysis of network dynamics
a10
maxb
a2
b
maxa
M2
aa
bb
M3M1
. x = h (x), x \
Analysis of the dynamics in phase space:
Extension of PL differential equations to differential inclusions using Filippov approach:
a10
maxb
a2
b
maxa
Qualitative analysis of network dynamics
a10
maxb
a2
b
maxaaa
bb
M3
a10
maxb
a2
b
maxaa10
maxb
a2
b
maxaaa
M3M1M5
Gouzé and Sari, Dyn. Syst., 02
.
M2 M4
. x = h (x), x \
x H (x), x
Analysis of the dynamics in phase space:
In every mode domain M, the system either converges monotonically towards focal set, or instantaneously traverses M
a10
maxb
a2
b
maxa
Qualitative analysis of network dynamics
a10
maxb
a2
b
maxa a10
maxb
a2
b
maxaa10
maxb
a2
b
maxa
M1 M2 M3 M4 M5
M10
M15M14M13M12M11
M6 M7 M8 M9
. x H (x), x
de Jong et al., Bull. Math. Biol., 04Gouzé and Sari, Dyn. Syst., 02
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
Problem for model validation
a10
maxb
a2
b
maxaa10
maxb
a2
b
maxa a10
maxb
a2
b
maxaa10
maxb
a2
b
maxa
M1 M2 M3 M4 M5
M10
M15M14M13M12M11
M6 M7 M8 M9
xa < 0, xb ?x M11:. .
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
a10
maxb
a2
b
maxa
Qualitative analysis of network dynamics
a10
maxb
a2
b
maxa a10
maxb
a2
b
maxaa10
maxb
a2
b
maxa
bb
D12 D22 D23 D24
D17 D18
D21 D20
D1 D3 D5 D7 D9
D15
D27 D26 D25
D11 D13 D14
D2 D4 D6 D8
D10 D16
D19
xa < 0, xb > 0x D17:. .
Continuous transition system
PL system, = (,,H), associated with continuous PL transition system, -TS = (,→,╞), where
continuous phase space
Continuous transition system
PL system, = (,,H), associated with continuous PL transition system, -TS = (,→,╞), where
continuous phase space
→ transition relation
and x and x’ in same or in adjacent domain
: transition from x to x’ iff a solution reaches x’ from x
maxamaxa
a10
maxb
a2
b
a10
maxb
a2
b
bb x1 → x2, x1 → x3,
x3 → x4x2 → x3,x1 x2
x3x4
x5
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
: describes derivative sign of solutions at x
maxamaxa
a10
maxb
a2
b
a10
maxb
a2
b
bb
x1 x2
x3x4
. x1╞ xa > 0,x5 . x1╞ xb
> 0,
. x4╞ xa < 0, . x4╞ xb
> 0,
Discrete abstraction
Qualitative PL transition system, -QTS = (D, →,╞), where
D finite set of domains :D = {D1, …, D27}
D1 D ;
maxamaxa
a10
maxb
a2
b
a10
maxb
a2
b
bb
D12 D22 D23 D24
D17 D18
D21 D20
D1 D3 D5 D7 D9
D15
D27 D26 D25
D11 D13 D14
D2 D4 D6 D8
D10 D16
D19
Discrete abstraction
Qualitative PL transition system, -QTS = (D, →,╞), where
D finite set of domains → quotient transition relation : transition from D to D’ iff there exist
xD, x’D’ such that x → x’
D1 D ; D1 →~ D1, D1 →~ D11, D11 →~ D17,
maxamaxa
a10
maxb
a2
b
a10
maxb
a2
b
bb
x1
D17
D1
D11
x1 x2
x3x4
x5
D1 D1
D11
D17
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 xD such that x╞ p
maxamaxa
a10
maxb
a2
b
a10
maxb
a2
b
bb
x1
D17
D1
D11
x1 x2
x3x4
x5
D1 D ; D1 →~ D1, D1 →~ D11, D11 →~ D17, D1╞ xa>0, D1╞ xb>0, D4╞ xa < 0. . .
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)
D1 D3 D5 D7 D9
D15
D27D26D25
D11 D12 D13 D14
D2 D4 D6
D8
D10
D16D17
D18
D20
D19
D21
D22
D23
D24
maxamaxa
a10
maxb
a2
b
a10
maxb
a2
b
bb
D12 D22 D23 D24
D17 D18
D21 D20
D1 D3 D5 D7 D9
D15
D27 D26 D25
D11 D13 D14
D2 D4 D6 D8
D10 D16
D19
D1
D11
D17
D18
Alur et al., Proc. IEEE, 00
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
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, f1f2, f1f2, f1→f2, EXf1, AXf1, EFf1, AFf1, EGf1, AGf1, Ef1Uf2, Af1Uf2,
where pAP and f1, f2 CTL formulas
Computer tools are available to perform efficient and reliable model checking (e.g., NuSMV, SPIN, CADP)
Validation using model checking
Atomic propositionsAP = {xa = 0, xa < a
1, ... , xb < maxb, xa < 0, xa= 0, ... , xb > 0}
Observed property expressed in CTL
There Exists a Future state where xa > 0 and xb > 0
and 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.
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
Yes
Consistency?0
xb
time
time0
xa
xa > 0.xb > 0.
xb > 0.xa < 0.
EF(xa > 0 xb > 0 EF(xa < 0 xb > 0) ). . . .
D1 D3 D5 D7 D9
D15
D27D26D25
D11 D12 D13 D14
D2 D4 D6
D10
D16D17
D18
D20
D19
D21
D22
D23
D24
D8
Batt et al., IJCAI, 05
Overview
I. Introduction
II. Method for model validation
1. Piecewise-linear (PL) differential equation models
2. Symbolic analysis using qualitative abstraction
3. Verification of properties by means model-checking techniques
III. Genetic Network Analyzer 6.0
IV. Validation of model of nutritional stress response in E. coli
V. Discussion and conclusions
Genetic Network Analyzer
Model validation approach implemented in version 6.0 of GNA, freely available for academic research Batt et al., Bioinformatics, 05
Integration into environmentfor explorative genomics atGenostar SA
structure into packages
class diagram of kernel
Genetic Network Analyzer
GNA implemented in Java 1.4
> 17000 lines of code in 6 packages
35% of lines modified with respect to version 5.5
(up to 60% in kernel)
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)
Overview
I. Introduction
II. Method for model validation
1. Piecewise-linear (PL) differential equation models
2. Symbolic analysis using qualitative abstraction
3. Verification of properties by means model-checking techniques
III. Genetic Network Analyzer 6.0
IV. Validation of model of nutritional stress response in E. coli
V. Discussion and conclusions
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?
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
Signal (carbon starvation)
CRP
crp
cya
CYA
cAMP•CRP
fis
Fis
Supercoiling
TopA
topA
GyrAB
P1-P4 P1 P2
P2P1-P’1
rrnP1 P2
stable RNAs
PgyrABP
How does response emerge from network of interactions?
Validation of stress response model
Qualitative simulation of carbon starvation:
66 reachable domains (< 1s.)
single attractor domain (asymptotically stable equilibrium point)
Experimental data on Fis:
CTL formulation:
Model checking with NuSMV: property true (< 1s.)
“Fis concentration decreases and becomes steady in stationary phase”
Ali Azam et al., J. Bacteriol., 99
EF(xfis < 0 EF(xfis = 0 xrrn < rrn) ). .
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
True (<1s)
Kawamukai et al., J. Bacteriol., 85
Balke and Gralla, J. Bacteriol., 87
False (<1s)
AG(xcrp > 3crp xcya > 3
cya xs > s → EF xcya < 0).
EF( (xgyrAB < 0 xtopA > 0) xrrn < rrn). .
Novel prediction of stress response model
Qualitative simulation of carbon upshift response: 1143 reachable domains (< 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 (<1s, i=3)
Grognard et al., in preparation
Overview
I. Introduction
II. Method for model validation
1. Piecewise-linear (PL) differential equation models
2. Symbolic analysis using qualitative abstraction
3. Verification of properties by means model-checking techniques
III. Genetic Network Analyzer 6.0
IV. Validation of model of nutritional stress response in E. coli
V. Discussion and conclusions
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., Bioinformatics, 05
Batt et al., IJCAI, 05
Batt et al., HSCC, 05
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
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
Thanks for your attention!