formal verification of hybrid models of genetic regulatory networks grégory batt center for...

Formal Verification of Hybrid Models of Genetic Regulatory Networks

Grégory Batt

Center for Information and Systems Engineering

and Center for BioDynamics

at Boston University

Email: [email protected]

Overview

1. Introduction to genetic regulatory networks

2. Hybrid models of genetic regulatory networks

3. Formal verification of piecewise affine models

1. Symbolic reachability analysis

2. Discrete abstraction

3. Model checking

4. Application to model validation: nutritional stress response in E. coli

4. Discussion and conclusions

Genetic regulatory networks

Organism can be viewed as biochemical system, structured by network of interactions between its molecular components

Genetic regulatory network is part of biochemical network consisting (mainly) of genes and their regulatory interactions

Genetic regulatory networks underlie functioning and development of living organisms

protein

gene

promoter

a

A

B

b

cross-inhibition network

Gardner et al., Nature, 00

Analysis of genetic regulatory networks

Constraints for network analysis: presence of non-linear phenomena and of feed-back loops

large number of genes involved in most biologically-interesting networks

knowledge on molecular mechanisms rare

quantitative information on parameters and concentrations still scarce

Need for approaches dealing specifically with these constraints

Ropers et al., Biosystems, 06

Signal (lack of nutrients)

CRP

crp

cya

CYA

cAMP•CRP

fis

FIS

Supercoiling

TopA

topA

GyrAB

P1-P4 P1 P2

P2P1-P’1

rrnP1 P2

stable RNAs

PgyrABP

nutritional

stress response

network in E.

coli

Differential equation models

Genetic networks modeled by differential equations reflecting switch-like character of regulatory interactions

xa a f (xa , a2) f (xb , b ) – a xa .

xb b f (xa , a1) – b xb .

x : protein concentration

, : rate constants : threshold concentrationb

B

a

A

x

h-(x, θ, n)

0

1

Yagil and Yagil, Biophys. J., 71

Hill function

Glass and Kauffman, J. Theor. Biol., 73

x

s-(x, θ)

0

1

step function

x

r-(x, θ, h)

0

1

h

Mestl et. al., J. Theor. Biol., 95

ramp function

Hybrid models

Use of step functions results in piecewise affine models

xa

xb

Hybrid models


xa

xb


In every rectangular region, the system converges monotonically towards a focal set

Gouzé and Sari, Dyn. Syst., 02

de Jong et al., Bull. Math. Biol., 04

xa

xb

Hybrid models


Use of ramp functions results in piecewise multiaffine models

xa

xb





xa

xb

Hybrid models



In every rectangular region, the flow is a convex combination of its values at the vertices Belta and Habets, Trans. Automatic Control, 06

xa

xb





Hybrid models



Key properties used to deal with uncertainties on parameters: properties proven for every parameter satisfying qualitative constraints:

symbolic analysis for PA systems

properties proven for polyhedral sets of parameters: parametric analysis for

PMA systems

xa xa

xbxb

Analysis of the dynamics in phase space:

a10

maxb

a2

b

maxa

Symbolic analysis of PA models

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

.

D1

x = h (x), x \


a10

maxb

a2

b

maxa


a10

maxb

a2

b

maxa

D3

xa a – a xa .

xb – b xb .

aa

D5 0 < a1 < a2 < a/a < maxa

0 < b < b/b < maxb

. x = h (x), x \


a10

maxb

a2

b

maxa


a10

maxb

a2

b

maxa

D3

aa

bb

D5D1

. x = h (x), x \


Extension of PA differential equations to differential inclusions using Filippov approach:

a10

maxb

a2

b

maxa


a10

maxb

a2

b

maxaaa

bb

D5

a10

maxb

a2

b

maxaa10

maxb

a2

b

maxaaa

D5D1D9


.

D3 D7

. x = h (x), x \

x H (x), x

Partition of phase space into domains

In every domain D D, the system either converges monotonically towards focal set, or instantaneously traverses D

In every domain D D, derivative signs are identical everywhereDomains are regions having qualitatively-identical dynamics

a10

maxb

a2

b

maxa


a10

maxb

a2

b

maxa

bb

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

D1

xa > 0, xb > 0x D1:. .

Continuous transition system

PA system, = (,,H), associated with continuous PA transition system, -TS = (,→,╞), where

continuous phase space




→ transition relation

maxamaxa

a10

maxb

a2

b

a10

maxb

a2

b

bb x1 → x2, x1 → x3,

x3 → x4x2 → x3,x1 x2

x3x4

x5




→ transition relation ╞ satisfaction relation

and -TS have equivalent reachability properties

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 PA transition system, -QTS = (D, →,╞), where

D finite set of domains

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



D finite set of domains → quotient transition relation

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 D ;




╞ quotient satisfaction relation

D1 D ; D1 →~ D1, D1 →~ D11, D11 →~ D17, D1╞ xa>0, D1╞ xb>0, D4╞ xa < 0. . .

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,




╞ 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, 05de Jong et al., Bull. Math. Biol., 04

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)

Clarke et al., MIT Press, 01

Verification using model checking

Atomic propositionsAP = {xa = 0, xa < a

1, ... , xb < maxb, xa < 0, xa= 0, ... , xb > 0}

Expected 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.

Verification using model checking

Discrete transition system computed using qualitative simulation

Use of model checkers to check whether predictions satisfy expected properties

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

Genetic Network Analyzer

Model verification approach implemented in version 6.0 of GNABatt et al., Bioinformatics, 05

Integration into environmentfor explorative genomics atGenostar SA

de Jong et al., Bioinformatics, 03

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 is controlled by complex genetic regulatory network

Model: 7 PADEs, 40 parameters and 54 inequality constraints

?exponential phase

stationary phase

signal of nutritional

deprivation

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

Ropers et al., Biosystems, 06

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). .

Discussion

Related work: Discrete abstraction used for symbolic analysis of PA models of biological

networks

Model checking used for analysis of biological networks

Tailored combination of symbolic reachability analysis, discrete abstraction and model checking is effective for verification of dynamical properties of qualitative models of genetic networks

Ongoing work: Use similar ideas for the identification of parameter sets for which a model satisfies given specifications

Application to network design in synthetic biology

Ghosh and Tomlin, Systems Biology, 04

Bernot et al., J. Theor. Biol., 04

Chabrier et al., Theor. Comput. Sci., 04 Eker et al., PSB, 02

Acknowledgements

Contributors :

In France (PA models): In USA (PMA models):

Thanks for your attention!

Hidde de Jong (INRIA)

Johannes Geiselmann (UJF, Grenoble)

Jean-Luc Gouzé (INRIA)

Radu Mateescu (INRIA)

Michel Page (UPMF, Grenoble)

Delphine Ropers (INRIA)

Tewfik Sari (UHA, Mulhouse)

Dominique Schneider (UJF, Grenoble)

Calin Belta (Boston University)

Marius Kloetzer (Boston University)

Boyan Yordanov (Boston University)

Ron Weiss (Princeton University)

formal verification of hybrid models of genetic regulatory networks grégory batt center for...

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x h x

x b b f x

a1 b x

step function x r x

a2 f x b

coli slide

ramp function slide

threshold concentration