Constraints-based methods for the qualitative modeling of biological networks Eric Fanchon TIMC-IMAG (Grenoble)

‘Molecular’ networks Non-linear interactions Feedback 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 behaviours Revise a model Design 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) x2x2 x1x1 1111 max 1 0 max 2 1212 2121 2222 D0 Thresholds → rectangular partition of state (concentration) space In each regular domain D, the system tends monotonically to a focal point  ( D ) : Phase portrait : determined by the distribution of focal points dx i /dt = f i (x) - γ i x i with γ i > 0  i (D) = f i (x) / γ i 9 domains

Discrete abstraction x2x2 x1x1 1111 max 1 max 2 1212 2121 22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 x i  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 ≠ x i If φ i (x) > x i : x’ i = x i + 1 If φ i (x) < x i : x’ i = x i – 1 Stationary state : ∀ i, φ i (x) = x i 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 x2x2 x1x1 1111 max 1 0 max 2 1212 2121 22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 x2x2 x1x1 1111 max 1 0 max 2 1212 2121 22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

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 (S growth ) : Fis at high level; supercoiling high;... State corresponding to the stressed phase (S stress ) Fis at low level; supercoiling low;...  Supercoiling must be lower in S stress than in S growth The model must accept a path going from ‘S growth & signal=1’ to S stress, and a path from ’S stress & signal=0’ to S growth.

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,

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 K x 1 and K x 2 φ x (y) = K x 1. c(y=0) + K x 2. c(y≥1) Sign of the interaction: +  K x 2 > K x 1 Observation : the production rate of x increases when y is at high concentration (y≥1) +,1 y x

Two influences +,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 K x 1, K x 2, K x 3 and K x 4 φ x (y,z) = K x 1. c(y=0) c(z=0) + K x 2. c(y=1) c(z=0) + K x 3. c(y=0) c(z=1) + K x 4. c(y=1) c(z=1)

Combination of 2 influences Observation: y and z together activate x  additivity constraints K x (y=1)(z=1) ≥ K x (y=0)(z=1) K x (y=1)(z=1) ≥ K x (y=1)(z=0) K x (y=1)(z=0) ≥ K x (y=0)(z=0) K x (y=0)(z=1) ≥ K x (y=0)(z=0) Two extreme cases : y and z work independently (y or z)  K x (y=0)(z=0) = 0 and K x (y=1)(z=0), K x (y=0)(z=1), K x (y=1)(z=1) ≥ 1 y and z need to be together to activate x (y and z)  K x (y=0)(z=0) = K x (y=1)(z=0) = K x (y=0)(z=1) = 0 and K x (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. (K x (y=1)(z=0) ≥ K x (y=0)(z=0) ) and (K x (y=0)(z=1) ≥ K x (y=0)(z=0) ) and (K x (y=1)(z=1) ≤ K x (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: φ topA = K 1 topA (x fis < 3) (1 - [(x gyAB  2)(x topA < 1)] ) + K 2 topA (x fis < 3) [(x gyAB  2)(x topA < 1)] + K 3 topA (x fis  3) (1 - [(x gyAB  2)(x topA < 1)] ) + K 4 topA (x fis  3) [(x gyAB  2)(x topA < 1)] Contraints on the Ks: ( (K 1 topA < K 3 topA )  (K 2 topA < K 4 topA ) )  ( (K 1 topA < K 3 topA )  (K 2 topA < K 4 topA ) ) K 1 topA  K 3 topA  K 1 topA  K 2 topA  K 2 topA  K 4 topA  K 3 topA  K 4 topA where K i topA  {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 x topA = κ topA 1 s - (x fis ) + κ topA 2 s + (x gyr ).s - (x topA ) + κ topA 3 s + (x fis ) s + (x gyr ).s - (x topA ) - γ topA. x topA With : (κ topA 1 + κ topA 2 )/ γ topA, κ topA 1 / γ topA and κ topA 2 / γ topA in the same interval κ topA 3 / γ topA and (κ topA 2 + κ topA 3 ) / γ topA 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)