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François Fages MPRI Bio-info 2005 Formal Biology of the Cell Modeling, Computing and Reasoning with Constraints François Fages, Constraint Programming Group, INRIA Rocquencourt

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François Fages MPRI Bio-info 2005 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.Computational models of the cell cycle control [L. Calzone]. 5.Mixed models of the cell cycle and the circadian cycle [L. Calzone]. 6.Machine learning reaction rules from temporal properties. 7.Learning kinetic parameter values. Constraint-based model checking. 8.Constraint Logic Programming approach to protein structure prediction.

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François Fages MPRI Bio-info 2005 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.

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François Fages MPRI Bio-info 2005 Cell Cycle Control [Qu et al. 2003]

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François Fages MPRI Bio-info 2005 Constraint-Based Linear Time Logic Constraints over concentrations and derivatives as FOL formulae over the reals: [M] > 0.2 [M]+[P] > [Q] d([M])/dt < 0 LTL operators for time X, F, G, U (no non-determinism). F([M]>0.2) FG([M]>0.2) F ([M]>2 & F (d([M])/dt 0 & F(d([M])/dt<0))))

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François Fages MPRI Bio-info 2005 Traces from Numerical Simulation From a system of Ordinary Differential Equations dX/dt = f(X) Numerical integration (by Euler, Runge-Kutta, adaptive step size Runge-Kutta, Rosenbrock methods) produces a discretization of time The trace is a linear Kripke structure: (t 0,X 0 ), (t 1,X 1 ), …, (t n,X n ). the derivatives can be added to the trace (t 0,X 0,dX 0 /dt), (t 1,X 1,dX 1 /dt), …, (t n,X n,dX n /dt). Equality x=v true if x i ≤v & x i+1 ≥ v or if x i ≥ v & x i+1 ≤v (Rolle’s theorem!)

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François Fages MPRI Bio-info 2005 Constraint-Based LTL (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] Simple algorithm based on the trace of the numerical simulation: 1.Run the numerical simulation 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 (optimistic abstraction…)

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François Fages MPRI Bio-info 2005 Model-Checking Specific First-Order LTL Formulae Let us introduce the time variable t We can model-check a First-Order Logic LTL formula such as period(A,75) defined as 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)))

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François Fages MPRI Bio-info 2005 Example in Qu’s Model of the Cell Cycle K1=0, K5u=0

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François Fages MPRI Bio-info 2005 Learning Parameter Values from LTL Specification ? learn_parameter([k5u,k1],[(0,10),(0,500)], 20, oscil(CycB-CDK~{p1},2,10.0),300). parameter(k5u,0.5). parameter(k1,350).

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François Fages MPRI Bio-info 2005 Learning Parameter Values from LTL Specification ? learn_parameter([k5u,k1],[(0,10),(0,500)], 20, period(CycB-CDK~{p1},75), 300). parameter(k5u,2). parameter(k1,200).

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François Fages MPRI Bio-info 2005 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.

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François Fages MPRI Bio-info 2005 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).

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François Fages MPRI Bio-info 2005 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*( *x), y’ = y + dt*0.1*x y ≥ 0.8 x’ = x + dt*( *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+01*X, p(X1,Y1). p(X,Y):- X>=0, Y>=0, Y>=0.8, X1=X-02*X, Y1=Y+01*X, p(X1,Y1).

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François Fages MPRI Bio-info 2005 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.

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François Fages MPRI Bio-info 2005 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)).

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François Fages MPRI Bio-info 2005 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>= }, 2, (2,1)). … s(26, p(s_s,A,B), {B>=0,A>=0, B *A< }, 27, (2,26)). s(27, init, {}, 28, (1,27)).

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