1. François Fages MPRI Bio-info 2007 Formal Biology of the Cell Modeling, Computing and Reasoning with Constraints François Fages, Constraint Programming Group, INRIA Rocquencourt mailto:Francois.Fages@inria.fr http://contraintes.inria.fr/

2. François Fages MPRI Bio-info 2007 Overview of the Lectures 1.Introduction. Formal molecules and reactions in BIOCHAM. 2.Formal biological properties in temporal logic. Symbolic model-checking. 3.Continuous dynamics. Kinetics models. 4.Learning kinetic parameter values. Constraint-based model checking. 5.…

3. François Fages MPRI Bio-info 2007 Cell Cycle Control [Qu et al. 03] k1 for _=>Cyclin. k2*[Cyclin] for Cyclin=>_. k3*[Cyclin]*[Cdc2~{p1}] for Cyclin+Cdc2~{p1}=>Cdc2~{p1}-Cyclin~{p1}. k4p*[Cdc2~{p1}-Cyclin~{p1}] for Cdc2~{p1}-Cyclin~{p1}=>Cdc2-Cyclin~{p1}. k4*[Cdc2-Cyclin~{p1}]^2*[Cdc2~{p1}-Cyclin~{p1}] for Cdc2~{p1}-Cyclin~{p1}=[Cdc2-Cyclin~{p1}]=>Cdc2-Cyclin~{p1}. k5*[Cdc2-Cyclin~{p1}] for Cdc2-Cyclin~{p1}=>Cdc2~{p1}-Cyclin~{p1}. k6*[Cdc2-Cyclin~{p1}] for Cdc2-Cyclin~{p1}=>Cdc2+Cyclin~{p1}. k7*[Cyclin~{p1}] for Cyclin~{p1}=>_. k8*[Cdc2] for Cdc2=>Cdc2~{p1}. k9*[Cdc2~{p1}] for Cdc2~{p1}=>Cdc2. parameter(k1,0.015). parameter(k2,0.015). parameter(k3,200). parameter(k4p,0.018). parameter(k4,180). parameter(k5,0). parameter(k6,1). parameter(k7,0.6). parameter(k8,100). parameter(k9,100). present(Cdc2,1). make_absent_not_present.

4. François Fages MPRI Bio-info 2007 Cell Cycle Control [Qu et al. 2003]

5. François Fages MPRI Bio-info 2007 Linear Time Logic with Constraints Constraints over concentrations and derivatives as FOL formulae over the reals: [M] > 0.2 [M]+[P] > [Q] d([M])/dt < 0

6. François Fages MPRI Bio-info 2007 Linear Time Logic with Constraints Constraints over concentrations and derivatives as FOL formulae over the reals: [M] > 0.2 [M]+[P] > [Q] d([M])/dt < 0 Linear Time Logic LTL operators for time X, F, U, G F([M]>0.2) FG([M]>0.2) F ([M]>2 & F (d([M])/dt 0 & F(d([M])/dt<0)))) oscil(M,n) defined as at least n alternances of sign of the derivative

7. François Fages MPRI Bio-info 2007 Linear Time Logic with Constraints Constraints over concentrations and derivatives as FOL formulae over the reals: [M] > 0.2 [M]+[P] > [Q] d([M])/dt < 0 Linear Time Logic LTL operators for time X, F, U, G F([M]>0.2) FG([M]>0.2) F ([M]>2 & F (d([M])/dt 0 & F(d([M])/dt<0)))) oscil(M,n) defined as at least n alternances of sign of the derivative Period(A,75)=  t  v F(T = t & [A] = v & d([A])/dt > 0 & X(d([A])/dt < 0) & F(T = t + 75 & [A] = v & d([A])/dt > 0 & X(d([A])/dt < 0)))

8. François Fages MPRI Bio-info 2007 How to Evaluate a Constraint LTL Formula ? Consider the ODE’s of the concentration semantics dX/dt = f(X)

9. François Fages MPRI Bio-info 2007 How to Evaluate a Constraint LTL Formula ? Consider the ODE’s of the concentration semantics dX/dt = f(X) Numerical integration methods produce a (clever) discretization of time (adaptive step size Runge-Kutta or Rosenbrock method for stiff syst.)

10. François Fages MPRI Bio-info 2007 How to Evaluate a Constraint LTL Formula ? Consider the ODE’s of the concentration semantics dX/dt = f(X) Numerical integration methods produce a (clever) discretization of time (adaptive step size Runge-Kutta or Rosenbrock method for stiff syst.) The trace is a linear Kripke structure: (t 0,X 0,dX 0 /dt), (t 1,X 1,dX 1 /dt), …, (t n,X n,dX n /dt). over concentrations and their derivatives at discrete time points Evaluate the formula on that Kripke structure with a model checking alg.

11. François Fages MPRI Bio-info 2007 1. Simulation based Forward Model Checking Hypothesis 1: the initial state is completely known Hypothesis 2: the formula can be checked over a finite period of time [0,T] 1.Run the numerical integration from 0 to T producing values at a finite sequence of time points 2.Iteratively label the time points with the sub-formulae of  that are true: Add  to the time points where a FOL formula  is true, Add F  (X  ) to the (immediate) previous time points labeled by  Add  U  to the predecessor time points of  while they satisfy  (Add G  to the states satisfying  until T) Model checker and numerical integration methods implemented in Prolog

12. François Fages MPRI Bio-info 2007 2. Learning Parameters from Temporal Properties biocham: learn_parameter([k3,k4],[(0,200),(0,200)],20, oscil(Cdc2-Cyclin~{p1},3),150).

13. François Fages MPRI Bio-info 2007 Learning Parameters from Temporal Properties biocham: learn_parameter([k3,k4],[(0,200),(0,200)],20, oscil(Cdc2-Cyclin~{p1},3),150). First values found : parameter(k3,10). parameter(k4,70).

14. François Fages MPRI Bio-info 2007 Learning Parameters from Temporal Properties biocham: learn_parameter([k3,k4],[(0,200),(0,200)],20, oscil(Cdc2-Cyclin~{p1},3) & F([Cdc2-Cyclin~{p1}]>0.15), 150). First values found : parameter(k3,10). parameter(k4,120).

15. François Fages MPRI Bio-info 2007 Learning Parameters from LTL Specification biocham: learn_parameter([k3,k4],[(0,200),(0,200)],20, period(Cdc2-Cyclin~{p1},35), 150). First values found: parameter(k3,10). parameter(k4,280).

16. François Fages MPRI Bio-info 2007 3. Backward Constraint-based Model Checking Reason backward from the set of states satisfying a formula to the set of initial states for which the formula is true. Makes it possible to reason with a partially know initial state. Approximate set of states with constraints: polyhedrons defined by linear constraints.

17. François Fages MPRI Bio-info 2007 Hybrid (Continuous-Discrete) Dynamics Gene X activates gene Y but above some threshold gene Y inhibits X. 0.1*[X] for _ =[X]=> Y. if [Y]<0.8 then 0.1 for _ => X. 0.2*[X] for X => _. absent(X). absent(Y).

18. François Fages MPRI Bio-info 2007 Translation to Constraint Logic Programs over Reals Hybrid Differential Equation System: dx/dt = 0.1 – 0.2*x if y < 0.8 dx/dt = – 0.2*x if y ≥ 0.8 dy/dt = 0.1*x (Concurrent) transition system of the trace using Euler’s method: y < 0.8  x’ = x + dt*(0.1-0.2*x), y’ = y + dt*0.1*x y ≥ 0.8  x’ = x + dt*(0.1-0.2*x), y’ = y + dt*0.1*x Initial condition: x=0, y=0. Translation into a Constraint Logic Program over the reals (dt=1): Init :- X=0, Y=0, p(X,Y). p(X,Y):- X>=0, Y>=0, Y<0.8, X1=X-02*X+01, Y1=Y+0.1*X, p(X1,Y1). p(X,Y):- X>=0, Y>=0, Y>=0.8, X1=X-02*X, Y1=Y+0.1*X, p(X1,Y1).

19. François Fages MPRI Bio-info 2007 Constraint-based CTL Backward Model Checking Theorem [Delzanno Podelski 99] EF(f)=lfp(T P  {p(x):-f} ), EG(f)=gfp(T P  f ). Safety property AG(  f) iff  EF(f) iff init  lfp(T P  {f} ) Liveness property AG(f1  AF(f2)) iff init  lfp(T P  f1  gfp(T P  {f2} ) ) Deductive Model Checking DMC system [Delzanno 00] Implemented in Sicstus-Prolog CLP(Herbrand,Real,Boolean) Fourier-Motzkin elimination and Simplex algorithm.

20. François Fages MPRI Bio-info 2007 Constraint-based Backward Reasoning in DMC r(init, p(s_s,A,B), {A=0,B=0}). r(p(s_s,A,B), p(s_s,C,D), {A>=0,B>=0.8,C=A-02*A,D=B+01*A}). r(p(s_s,A,B), p(s_s,C,D), {A>=0,B>=0,B<0.8, C=A-02*A+01,D=B+01*A}). ? prop(P,S). P = unsafe, S = p:s*(x>=0.6) ? ti. Property satisfied. Execution time 0 ? ls. s(0, p(s_s,A,_), {A>=0.6}, 1, (0,0)).

21. François Fages MPRI Bio-info 2007 Constraint-based Backward Simulation in DMC ? prop(P,S). P = unsafe, S = p:s*(x>=0.2) ? ? ti. Property NOT satisfied. Execution time 1.5 ? ls. s(0, p(s_s,A,_), {A>=0.2}, 1, (0,0)). s(1, p(s_s,A,B), {B =-0,A>=0.19387755102040816}, 2, (2,1)). … s(26, p(s_s,A,B), {B>=0,A>=0, B+0.1982676351105516*A<0.7741338175552753}, 27, (2,26)). s(27, init, {}, 28, (1,27)).