Constraints-based methods for the qualitative modeling of

biological networks

Eric FanchonTIMC-IMAG (Grenoble)

‘Molecular’ networksNon-linear interactionsFeedback loops

State of knowledge assumed :The molecular players, and the connectivity of the

system are known (structural knowledge)Some knowledge on behaviour[Parameters are unknown or partially known]

Develop a computer tool to:Infer model parameters from known behavioursRevise a modelDesign informative experiments...

Modeling context and objectives

Modeling objectives

• Inference of model parameters from behaviours

• Building / revision of qualitative models

Outline

1. Formalism: Multivalued asynchronous networks

1. Computational approach: Constraint Logic Programming

1. Revision of the model of Nutritional stress in E. coli

Part 1 : Multivalued asynchronous networks

• Thomas’ networks...

Piecewise-linear differential equations

= focal point(D0)x2

x111 max10

max2

12

21

22

D0

Thresholds → rectangular partition of state (concentration) spaceIn each regular domain D, the system tends monotonically to a focal point (D) :

Phase portrait : determined by the distribution of focal points

dxi/dt = fi (x) - γi xi with γi > 0

i (D) = fi (x) / γi

9 domains

Discrete abstraction

x2

x111 max1

max2

12

21

22

0 (0,0) (1,0) (2,0)

(0,1)

(2,2)

(0,0)

(0,0)

Transition S(D) S(D’) : if continuous trajectories going from D to D’.

Rectangular domain D discrete state Real concentrations xi integers

Concentration space Grid

(0,0)

Asynchronous updating

E. H. Snoussi and R. Thomas (1993)

Transition rule, Transition graph

State x’ is a successor of x if :

• There is exactly one component i such that x’i ≠ xi

• If φi(x) > xi : x’i = xi + 1

• If φi(x) < xi : x’i = xi – 1

Stationary state :

• ∀i, φi(x) = xi

• state which has no successor(0,0) (1,0) (2,0)

(0,1)

(0,0)

Family of models

R. Thomas & M. Kaufman,Chaos, 11, 180 (2001)+,2

+,1

-,1

x y

Black wall

x2

x1

11 max10

max2

12

21

22

(0,0) (1,0) (2,0)

(0,0)(1,0)

(1,0)(0,0)

Introduction of Singular states de Jong, Gouzé et al., Bull. Math. Biology, 66, 301 (2004)

Sliding mode / persistent state

(1,0) (0,0)

Black wall

x2

x1

11 max10

max2

12

21

22

(0,0) (1,0) (2,0)

(0,0)(1,0)

(1,0)(0,0)

(1,0) (0,0)

• Introduction of Singular states to take into account all stationary states (E. H. Snoussi and R. Thomas, Bull. Math. Biology, 55, 973, 1993)

• Rule to compute the successors of singular states (de Jong, Gouzé et al., Bull. Math. Biology, 66, 301, 2004)

Part 2 : computational approach

• Constraint Logic Programming (CLP)

CLP: Declarative programming

• Declarative modeling by constraints:

• Description of properties and relationships between partially known objects.

Problem = set of constraints (equations/inequations)

• Solvers satisfiability of the set of constraints

• Consistency: a single logical specification for diverse functionalities (diverse types of queries).

• Iterative modeling: add new constraints whenever new information become available from experiments. The model can be ‘refined’ progressively.

• Correct handling of finite and infinite, partial and full information Handling of incomplete knowledge.

• No unnecessary commitments: No need to set parameters to arbitrary values if parameter not determined by available knowledge. Keep all solutions.

• High-level, Expressive language

Advantages of CLP

Prolog implementation of Asynchronous Multivalued

Networks

• Implementation (Fabien Corblin) in SICStus Prolog of:• the 'regular' formalism• the extended formalism (with singular states)

• Main predicates :• Definition of the transition rules• Definition of a specific model (focal points

equations and inequalities between parameters) structural knowledge

• Behavioral observations

Regular states only

successor(M, State_i, State_s) is true iff

State_s is a possible successor

of State_i according to model M

successor(M, State_i, State_s) <=

focal_state(M, State_i, State _f)

successor_constraints(State _i, State _f, State _s).

Regular states only (2)

successor_constraints(State_i, State_f, State_s) <=

D = (State_i State _f)

at_most_one_jump(D, State _i, State _f, State_s).

......

Part 3 : application to the revision of the E. coli nutritional

stress model

Nutritional stress response in E. coli

• Response of E. coli to nutritional stress conditions: transition from exponential phase to stationary phase

log (pop. size)

time

> 4 h

Carbon starvation response

Ropers et al. (2006) BioSystems, 84, 124–152

rrnP1 P2

CRP

crp

cya

CYA

cAMP•CRP

FIS

TopA

topA

GyrAB

P1-P4P1 P2

P2P1-P’1

P

gyrABP

Signal (lack of carbon source)

DNA supercoiling

fis

tRNArRNA

protein

gene

promoter

Piecewise-Linear Diff. Eqs (PLDEs)

• Example of TopA : 2 influences• Fis

• Supercoiling (GyrAB and TopA)

D. Ropers et al. (2006) BioSystems, 84, 124–152.

Behavioral knowledge

• State corresponding to growth (Sgrowth) :

Fis at high level; supercoiling high; ...

• State corresponding to the stressed phase (Sstress)

Fis at low level; supercoiling low; ...

Supercoiling must be lower in Sstress than in Sgrowth

• The model must accept a path going from ‘Sgrowth &

signal=1’ to Sstress, and a path from ’Sstress &

signal=0’ to Sgrowth.

Results of qualitative simulation

Simulation of transition from exponential to stationary phase

CYA

FIS

GyrAB

Signal

TopA

rrn

CRP

D. Ropers et al. (2006) BioSystems, 84, 124–152.

Simulations done with GNA: H. de Jong et al. (2003) Bioinformatics, 19, 336-344.

Inconsistency Model revision

• Add new interaction/element(s) in the network ??

• Other possibilities should be considered :

• Parameter values different from those originally chosen

• Other ways of combining interactions

• Different order between thresholds

• Re-analysis of the data

Try to revise model (without adding new genes) with our declarative/parameterized approach.

A discrete model is constituted of :

• Focal point equations

depency relationships between variables

• A set of inequalities between parameters : sign of interactions, combination of interactions.

One influence on node x

• Two contexts for x : y on or off

2 parameters Kx1 and Kx

2

• φx(y) = Kx1 . c(y=0) + Kx

2 . c(y≥1)

• Sign of the interaction: + Kx2 > Kx

1

Observation : the production rate of x increases wheny is at high concentration (y≥1)

+,1y x

Two influences

+,1

+,1

x

y

Observations :• the production rate of x increases when y is at

high concentration (y=1)• the production rate of x increases when z is at

high concentration (z=1)

z

• 4 contexts : y on/off; z on/off

4 parameters Kx1, Kx

2, Kx3 and Kx

4

• φx(y,z) = Kx1 . c(y=0) c(z=0) + Kx

2 . c(y=1)

c(z=0)

+ Kx3 . c(y=0) c(z=1) + Kx

4 . c(y=1)

c(z=1)

Combination of 2 influences

• Observation: y and z together activate x

additivity constraints

Kx(y=1)(z=1) ≥ Kx

(y=0)(z=1)

Kx(y=1)(z=1) ≥ Kx

(y=1)(z=0)

Kx(y=1)(z=0) ≥ Kx

(y=0)(z=0)

Kx(y=0)(z=1) ≥ Kx

(y=0)(z=0)

• Two extreme cases :• y and z work independently (y or z)

Kx(y=0)(z=0) = 0 and Kx

(y=1)(z=0), Kx(y=0)(z=1), Kx

(y=1)(z=1) ≥ 1

• y and z need to be together to activate x (y and z)

Kx(y=0)(z=0) = Kx

(y=1)(z=0) = Kx(y=0)(z=1) = 0 and Kx

(y=1)(z=1) ≥ 1

Combination of 2 influences (2)

Other situation :• y alone activates x

• z alone activates x

• y and z together form a complex, and the complex does not activate x.

(Kx(y=1)(z=0) ≥ Kx

(y=0)(z=0)) and

(Kx(y=0)(z=1) ≥ Kx

(y=0)(z=0)) and

(Kx(y=1)(z=1) ≤ Kx

(y=0)(z=0))

Discrete (qualitative) description :

• More flexible than PLDE descriptions in that we do not need to choose an analytical form specifying how influences combine on a given node. The inequalities contain this information.

• From a discrete description, differential equations can be written, if needed.

• {Observations} ‘Thomas’ model (PLDE model)

Method

• ‘Discrete model first / PLDEs later’

• Work on a parameterized model

• Constraints between parameters deduced from the observations

Re-examination : the example of topA

Biological observations:

Proteins GyrAB et TopA influence the expression of the topA gene via DNA coiling: GyrAB favors TopA expression; TopA has an antagonistic influence.

Fis increases the expression rate of TopA.

New focal equation:

φtopA = K1topA (xfis < 3) (1 - [(xgyAB 2)(xtopA < 1)] ) +

K2topA

(xfis < 3) [(xgyAB 2)(xtopA < 1)] +

K3topA (xfis 3) (1 - [(xgyAB 2)(xtopA < 1)] ) +

K4topA (xfis 3) [(xgyAB 2)(xtopA < 1)]

Contraints on the Ks:

( (K1topA < K3

topA) (K2topA < K4

topA) ) ( (K1topA < K3

topA) (K2topA < K4

topA) )

K1topA K3

topA K1topA K

2topA K

2topA K4

topA K3topA K

4topA

where KitopA {0,1,2}

(Sébastien Tripodi)

Expression of TopA (2)

Global interaction graph

Parameterized_model_1

Re-analysis of biological data • No K parameters instanciated

• Two influences on TopA and GyrAB → 4 parameters each.

• 3 influences on Crp → 6 parameters.

Do not assume anything about how the influences combine on Crp.

Total : 20 discrete parameters

State S : [signal, crp, cya, fis, gyrAB, topA]

Expression in Prolog (Query 1) :

biomodel(Model_Stress_Coli),

S1 = [0,1,1,3, Xg, Yg],

S2 = [1,2,1,0, Xs, Ys],

Xs-Ys #=< Xg-Yg,

Path1 = [S1,S1],

Path2 = [S2,S2],

multival_asynch_model_tc(Model_Stress_Coli, Path1),

multival_asynch_model_tc(Model_Stress_Coli, Path2).

NO solution There exists no model having both observed stationary states...

Behavioral knowledge

Identify 'blocking' constraint

• The system is allowed to remove one of the 6 constraints on TopA parameters retry the same query.

Result:

• Only 1 constraint on TopA parameters is incompatible with the existence of the 2 stationary states.

(additivity constraint)

Identify 'blocking' constraint (2)

• Results : only 1 solution :

S1 = [0,1,1,3,1,0] S2 = [1,2,1,0,1,1]

(S = [signal, crp, cya, fis, gyrAB, topA])

Enumerate the K parameters 3 models

Parameterized_model_2

• Changes with respect to previous model:• No K parameters instantiated (same as before)

• Enforce additive constraints on Crp

• Remove the 'blocking' constraint on TopA

• Query 2 :

Existence of models possessing a path (L≤6) corresponding to the transition to stressed phase in presence of starvation signal, and the reverse path (transition to exponential phase when the starvation signal disappears).

• All 3 models have this property

New PLDE for TopA

• The suppression of the ‘additive’ constraint on topA translates as a new term in the topA equation of the original PLDE model.

d/dt xtopA = κtopA1 s-(xfis) + κtopA

2 s+(xgyr).s-(xtopA)

+ κtopA3 s+(xfis) s+(xgyr).s-(xtopA) - γtopA.

xtopA

With :

• (κtopA1 + κtopA

2)/ γtopA, κtopA1/ γtopA and κtopA

2/ γtopA in the same

interval

• κtopA3/ γtopA and (κtopA

2 + κtopA3) / γtopA in the same interval

Biological interpretation

• A low level of Fis (alone) is compatible with TopA expression

Fis acts as an inhibitor when it is alone (prediction).

(and as an activator in presence of supercoiling)

• Paper published recently dealing with oxydative stress!

«When Fis levels are low, hydrogen peroxide treatment results in topA activation»

(Weinstein-Fischer & Altuvia, Mol. Microbio., 2007)

Same behavior in nutritional stress ?

Taking into account singular states

• There are 9 singular stationary states along the path going from ‘exponential phase & stress signal’ to ‘stationary phase & NO stress signal’.

• Some of these states are asymptotically stable but all have at least one successor.

• It may be necessary to add constraints on real parameters to be sure the system does not get trapped in a stable singular state.

Summary

• Framework: multivalued asynchronous networks

• PLC implementation (‘regular’ and ‘singular’ versions)

• Constraints Systematic analysis (no trial and error)

Work with sets of models and stay close to biological data.

Summary (2)

• Method to build/revise models (‘discrete-first’ approach)

{Observations} → discrete/regular

→ discrete with singular states

→ PLDEs

• Automatic identification of blocking constraint

• Nutritional stress model: consistent models were found by changing two equations and some parameter values. Prediction of a new role for Fis.

Perspectives

• Play with threshold orders (ordering of θ’s)

• Automatic elimination of solution models whose transition graph contains ‘non-biological’ paths.

• Discovery of relationships between parameters that are obeyed by all solution models

proposition of experiments

Participants and collaborators

• Fabien Corblin • Sébastien Tripodi• Laurent Trilling

...from TIMC-IMAG, Grenoble

In collaboration with :

• Delphine Ropers (Helix, INRIA Rhône-Alpes)