ON THE APPROXIMATION OF STOCHASTIC CONCURRENT CONSTRAINT PROGRAMMING BY MASTER EQUATION

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1 ON THE APPROXIMATION OF STOCHASTIC CONCURRENT CONSTRAINT PROGRAMMING BY MASTER EQUATION Luca Bortolussi 1,2 1 Department of Mathematics and Informatics University of Trieste luca@dmi.units.it 2 Center for Biomolecular Medicine, Area Science Park, Trieste QAPL 28, Budapest, 3 th March 28

2 SEMANTICS FOR SPA

3 OUTLINE 1 INTRODUCTION 2 BASICS ON SCCP 3 APPROXIMATIONS AND MASTER EQUATION 4 EXAMPLES

4 STOCHASTIC CONCURRENT CONSTRAINT PROGRAMMING CCP = CONSTRAINTS + AGENTS Constraints are formulae over an interpreted first order language (i.e. X = 1, Y > X 3); they can be added to a "container", the constraint store, but can never be removed. Agents can perform two basic operations on this store (asynchronously): tell or ask a constraint. p :- A; π = [g u] λ ; M = π.a M + M A = M p; N = A A N rw(x):- [X > X = X 1] λ(x).rw(x) + [true X = X + 1] λ(x).rw(x) STOCHASTIC CCP Each ask and tell instruction has a rate (function) attached to it: λ : C R +. The semantics of the language is given in terms of a Continuous Markov Chain. L. Bortolussi, Stochastic Concurrent Constraint Programming, QAPL, 26

MODELING IN SCCP OREGONATOR MODELING BIOCHEMICAL REACTIONS R 1 +... + R n f (R,X;k) P 1 +... + P m f -reaction(r, X, P, k) :- tell f (R,X;k) (R = R 1 P = P + 1).

5 MODELING IN SCCP OREGONATOR MODELING BIOCHEMICAL REACTIONS R R n f (R,X;k) P P m f -reaction(r, X, P, k) :- tell f (R,X;k) (R = R 1 P = P + 1). f -reaction(r, X, P, k) B k1 A A + B k2 A k3 2A + C 2A k4 C k5 B ANALYSIS TOOLS Stochastic simulation (Gillespie algorithm) Stochastic model checking and CTMC analysis Approximation with ODE s and Hybrid Automata L. Bortolussi, A. Policriti. Modeling Biological systems in sccp, Constraints, 13(1), 28.

FROM SCCP TO ODE ν = X G 1 G 1 1 1 +1 +1 1 Gene(X) :- [ X = X

6 FROM SCCP TO ODE ν = X G 1 G Gene(X) :- [ X = X + 1] kp.gene(x) + [X 1 ] kb X.[ ] ku.gene(x) Degrade(X) :- [X X = X 1] kd X.Degrade(X) φ = k pg 1 k b XG 1 k ug k d X Ẋ = k pg 1 k d X Φ 1 = ν φ : G 1 = k ug k b XG 1 G = k b XG 1 k ug

7 CIRCADIAN CLOCK

8 CIRCADIAN CLOCK p_gate(α A, α A, γ A, θ A, M A, A) p_gate(α R, α R, γ R, θ R, M R, A) reaction(β A, [M A ], [A]) reaction(δ MA, [M A ], []) reaction(β R, [M R ], [R]) reaction(δ MR, [M R ], []) reaction(γ C, [A, R], [AR]) reaction(δ A, [AR], [R]) reaction(δ A, [A], []) reaction(δ R, [R], []) A R A R

9 CIRCADIAN CLOCK p_gate(α A, α A, γ A, θ A, M A, A) p_gate(α R, α R, γ R, θ R, M R, A) reaction(β A, [M A ], [A]) reaction(δ MA, [M A ], []) reaction(β R, [M R ], [R]) reaction(δ MR, [M R ], []) reaction(γ C, [A, R], [AR]) reaction(δ A, [AR], [R]) reaction(δ A, [A], []) reaction(δ R, [R], []) A R

10 MASTER EQUATION FOR SCCP The master equation is equivalent to the Kolmogorov Forward Equation: it is a PDE for the time-evolution of the probability density function P(X, t). INCREASE OF P(X, t) DUE TO t j IN dt P(Y ν j, t)φ j (Y ν j )dt DECREASE OF P(X, t) DUE TO t j IN dt P(Y, t)φ j (Y)dt P(Y, t) t = j ( φj (Y ν j )P(Y ν j, t) φ j (Y)P(Y, t) )

11 FIRST-ORDER APPROXIMATION DIFFERENTIAL EQUATION FOR THE AVERAGE OF SCCP d Y i t dt = Φ 1 i (Y) t TAYLOR EXPANSION OF Φ 1 i (Y) t Φ 1 (Y) t Φ 1 ( Y t ) Y h,k=1 2 hkφ 1 ( Y t ) Y h Y k t FIRST-ORDER EQUATION FOR THE AVERAGE d Y i t dt = Φ 1 i ( Y t )

12 SECOND-ORDER APPROXIMATION EXACT EQUATION FOR COVARIANCE d Y i Y k t dt = Φ 2 ik(y) + (Y i Y i t )Φ 1 k(y) + (Y k Y k t )Φ 1 i (Y) t t t SECOND-ORDER EQUATIONS FOR AVERAGE AND COVARIANCE d Y i t dt = Φ 1 ( Y t ) T (N) h,k=1 2 hkφ 1 ( Y t ) Y h Y k t d Y i Y k t dt Y = Φ 2 ik( Y t ) + h Φ 1 k( Y t ) Y i Y h t h=1 Y + h Φ 1 i ( Y t ) Y k Y h t h=1

13 RANDOM WALK RW X :- [ X = X + 1] k.rw X + [ X = X 1] k.rw X, Φ 1 (X) = Φ 2 (X) = 2k { X = Φ 1 ( X ) X 2 XX 2 Φ 1 ( X ) = X 2 = Φ 2 ( X ) + 2 X 2 X Φ 1 ( X ) = 2k X t = X X 2 t = 2kt + X 2

14 EFFECTS OF VARIANCE R 1 :- [ X = X + 1] k.r 1 ; R 2 :- [ Y = Y + 1] k.r 2 ; R 3 :- [X > X = X 1] α1 X.R 3 R 4 :- [Y > Y = Y 1] α2 Y.R 4 ; R 5 :- [X > Y > X = X 1 Y = Y + 1] ka X Y.R 5 R 1 R 2 R 3 R 4 R 5 Stochastic SO Average/Variance X Y

15 CIRCADIAN CLOCK

16 CIRCADIAN CLOCK Stochastic FO approximation A R A R

17 CIRCADIAN CLOCK Stochastic average FO approximation A R A R

18 CIRCADIAN CLOCK Stochastic average SO approximation A R

19 CIRCADIAN CLOCK Robustness of the system: increase translation rate of R from β R = 5 to β R = 5. Stochastic, β R = 5 FO approximation, β R = A R

20 CIRCADIAN CLOCK Robustness of the system: increase translation rate of R from β R = 5 to β R = 5. Stochastic average, β R = FO approximation, β R = A R

21 CIRCADIAN CLOCK Robustness of the system: increase translation rate of R from β R = 5 to β R = 5. Stochastic average, β R = SO approximation, β R = A R

22 CONCLUSIONS Many works in statistical mechanics deal with the relation between stochastic and deterministic description of systems. The Master Equation for a SPA is the key to use these methods also for the analysis of quantitative programming languages. SPA introduce many new challenges: the main one is synchronization, which introduces discontinuities in the expression of rates. Synchronization is discrete in nature: hybrid schemes of approximation should work better.

Modeling Biological Systems in Stochastic Concurrent Constraint Programming

Modeling Biological Systems in Stochastic Concurrent Constraint Programming Luca Bortolussi 1 Alberto Policriti 1 1 Department of Mathematics and Computer Science University of Udine, Italy. Workshop on