Biochemical simulation by stochastic concurrent constraint programming and hybrid systems

Transcription

1 Biochemical simulation by stochastic concurrent constraint programming and hybrid systems Luca Bortolussi 1 Alberto Policriti 2,3 1 Dipartimento di Matematica ed Informatica Università degli studi di Trieste luca@dmi.units.it 2 Dipartimento di Matematica ed Informatica Università degli studi di Udine 3 Istituto di Genomica Applicata Parco Scientifico Tecnologico, Udine. Siena, 20 th April 2007

2 A biological circuit: MAPKinase

3 A biological circuit: MAPKinase

4 Circadian Clock (J. M. G. Vilar, H. Yuan Kueh, N. Barkai, and S. Leibler. PNAS, 2002.)

5 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], [])

6 Modeling Biological Systems with Stochastic Process Algebras Pros Cons Simple Language Compositionality Hard to encode general information Lacking computational extensibility

7 How to Increase Expressiveness... Add a computational twist to π-calculus. Add a form of local storage. Keep separated the description of interactions and the description of memory and computations. Introduce more general format for rates. Constraints... why not?

8 Outline 1 Introduction 2 Stochastic Concurrent Constraint Programming 3 Modeling Biological Systems in sccp 4 sccp and Hybrid Automata sccp and ODEs sccp and Hybrid Automata

9 Concurrent Constraint Programming Constraint Store In this process algebra, the main object are constraints, which are formulae over an interpreted first order language (i.e. X = 10, Y > X 3). Constraints can be added to a "container", the constraint store, but can never be removed. Agents Agents can perform two basic operations on this store (asynchronously): Add a constraint (tell ask) Ask if a certain relation is entailed by the current configuration (ask instruction) Syntax of CCP Program = Decl.A D = ε Decl.Decl p(x) : A A = 0 tell(c).a ask(c 1 ).A 1 + ask(c 2 ).A 2 A 1 A 2 x A p(x) V. Saraswat, Concurrent Constraint Programming, MIT press, 1993

10 Syntax of sccp Syntax of Stochastic CCP Program = D.A D = ε D.D p( x ) : A A = 0 tell (c).a M xa A A M = π.g M + M π = tell λ (c) ask λ (c) G = 0 tell (c).g p( y ) M xg G G L. Bortolussi, Stochastic Concurrent Constraint Programming, QAPL, 2006 Stochastic Rates Rates are functions from the constraint store C to positive reals: λ : C R +. Rates can be thought as speed or duration of communications.

11 sccp technical details Operational Semantics There are two transition relations, one instantaneous (finite and confluent) and one stochastic. Show Details Traces are sequences of events with variable time delays among them. Discrete vs. Continuous Semantics Show Details The operational semantics is abstract w.r.t. the notion of time: we can map the labeled transition system into a discrete or a continuous time Markov Chain. Implementation We have an interpreter written in Prolog, using the CLP engine of SICStus to manage the constraint store. Efficiency issues. Stream Variables Quantities varying over time can be represented in sccp as unbounded lists. Hereafter: special meaning of X = X + 1.

12 Continuous Time Markov Chains A Continuous Time Markov Chain (CTMC) is a direct graph with edges labeled by a real number, called the rate of the transition (representing the speed or the frequency at which the transition occurs). In each state, we select the next state according to a probability distribution obtained normalizing rates (from S to S 1 r with prob. 1 ). r 1 +r 2 The time spent in a state is given by an exponentially distributed random variable, with rate given by the sum of outgoing transitions from the actual node (r 1 + r 2 ).

13 General Principles Measurable Entities Stream Variables Logical Entities Processes (Control Variables) Interactions Processes L. Bortolussi, A. Policriti Modeling Biological Systems in Stochastic Concurrent Constraint Programming, 2006

14 Biochemical Arrows to sccp processes R R n k P P m R R n k 1 k2 P P m S E K,V 0 P S E K,V 0,h P reaction(k, [R 1,..., R V n], [P 1,..., P m]) : ask ni=1 rma (k,r 1,...,Rn) (R i > 0). n i=i tell (R i = R i 1) m j=1 tell (P j = P j + 1). reaction(k, [R 1,..., R n], [P 1,..., P m]) reaction(k 1, [R 1,..., R n], [P 1,..., P m]) reaction(k 2, [P 1,..., P m], [R 1,..., R n]) mm_reaction(k, V 0, S, P) : ask rmm (K,V 0,S)(S > 0). (tell (S = S 1) tell (P = P + 1)). mm_reaction(k, V 0, S, P) hill_reaction(k, V 0, h, S, P) : ask rhill (K,V 0,h,S)(S > 0). (tell (S = S h) tell (P = P + h)). Hill_reaction(K, V 0, h, S, P) where r MA (k, X 1,..., X n) = k X 1 X n; r MM (K, V 0, S) = V 0S S + K ; r Hill (k, V 0, h, S) = V h 0S S h + K h

15 Enzymatic reaction S + E k 1 k 1 ES k2 P + E Mass Action Kinetics enz_reaction(k 1, k 1, k 2, S, E, ES, P) :- reaction(k 1, [S, E], [ES]) reaction(k 1, [ES], [E, S]) reaction(k 2, [ES], [E, P]) Michaelis-Menten Equations d[p] = V 0 S dt S+K V 0 = k 2 [E 0 ] K = k 2 +k 1 k 1 Mass Action Equations d[es] = k dt 1 [S][E] k 2 [ES] k 1 [ES] d[e] = k dt 1 [S][E] + k 2 [ES] + k 1 [ES] d[s] = k dt 1 [S][E] d[p] = k dt 2 [ES] Michaelis-Menten Kinetics mm_reaction! k 2 + k 1, k 2 E, S, P k 1

16 Enzymatic reaction S + E k 1 k 1 ES k2 P + E Mass Action Kinetics enz_reaction(k 1, k 1, k 2, S, E, ES, P) :- reaction(k 1, [S, E], [ES]) reaction(k 1, [ES], [E, S]) reaction(k 2, [ES], [E, P]) Michaelis-Menten Kinetics! k 2 + k 1 mm_reaction, k 2 E, S, P k 1

17 MAP-Kinase cascade enz_reaction(k a, k d, k r, KKK, E1, KKKE1, KKKS) enz_reaction(k a, k d, k r, KKKS, E2, KKKSE2, KKK ) enz_reaction(k a, k d, k r, KK, KKKS, KKKKKS, KKP) enz_reaction(k a, k d, k r, KKP, KKP1, KKPKKP1, KK ) enz_reaction(k a, k d, k r, KKP, KKKS, KKPKKKS, KKPP) enz_reaction(k a, k d, k r, KP, KP1, KPKP1, K ) enz_reaction(k a, k d, k r, K, KKPP, KKKPP, KP) enz_reaction(k a, k d, k r, KKPP, KKP1, KKPPKKP1, KKP) enz_reaction(k a, k d, k r, KP, KKPP, KPKKPP, KPP) enz_reaction(k a, k d, k r, KPP, KP1, KPPKP1, KP)

18 The gene machine

19 The instruction set null_gate(k p, X) : tell kp (X = X + 1).null_gate(k p, X) pos_gate(k p, k e, k f, X, Y ) : tell kp (X = X + 1).pos_gate(k p, k e, k f, X, Y ) +ask r(ke,y ) (true).tell ke (X = X + 1).pos_gate(k p, k e, k f, X, Y ) neg_gate(k p, k i, k d, X, Y ) : tell kp (X = X + 1).neg_gate(k p, k i, k d, X, Y ) +ask r(ki,y ) (true).ask k d (true).neg_gate(k p, k i, k d, X, Y ) where r(k, Y ) = k Y. L. Cardelli, A. Phillips, 2005.

20 Repressilator neg_gate(0.1, 1, , A, C) reaction(0.0001, [A], []) neg_gate(0.1, 1, , B, A) reaction(0.0001, [B], []) neg_gate(0.1, 1, , C, B) reaction(0.0001, [C], [])

21 Miscellanea Model Checking Show Details We defined a translation procedure that maps (a restricted version of) sccp into PRISM, a probabilistic symbolic model checker. Hence, we can check if sccp programs satisfy properties specified as Continuous Stochastic Logic formulae.

22 Connecting SPA and ODE models

23 From sccp to ODE What? We want to associate a set of ODE to an sccp program (written with a restricted syntax). Why? ODE can be numerically simulated faster than stochastic processes. On the market... Show Details How things can go wrong There are (syntactic) methods to write set of ODEs for PEPA and stochastic π-calculus, looking at the speed of creation and destruction of terms (We did the same for sccp). However, the ODE can show a behavior different from that of SPA models. J. Hillston, Fluid Flow Approximation of PEPA models, QEST, L. Cardelli, From Processes to ODEs by Chemistry, L. Bortolussi, A. Policriti. Connecting Process Algebras and Differential Equations for systems biology, 2006.

24 The strange beast of repressilator Repressilator with gene gates Repressilator in sccp Repressilator: ODE from sccp Repressilator: average of sccp model Return

25 Hybrid Automata - Intuitively Intuitively, a hybrid automaton is a finite state automaton H with continuous variables Z such that: H evolves from Z to Z in time T according to differential equations Ż = f (v, Z ) Z must always satisfy invariant conditions in each state H can cross e only when guards on Z are true when H crosses e, some variables may be reset.

26 From sccp to Hybrid Automata What? We want to associate an hybrid automata to a sccp network. Why? The mixed discrete/continuous dynamics of HA is more natural, as it can preserve the logical structure of sccp models. Hybrid automata are equipped with well developed analysis methods. How? The separation between constraint store and logical description of agents makes easy to identify (discrete) modes of the automata Activation conditions need to look at the temporal semantics of stochastic actions. L. Bortolussi, A. Policriti. Hybrid approximation of Stochastic Concurrent Constraint Programming, 2007.

27 From sccp to Hybrid Automata: example A :- tell k+ (X = X + 1).A + ask k X (X > 0). tell (X = X 1).A + ask k0 (true).b B :- tell k (X = X + 1).B + ask k+ X (X > 0). tell (X = X 1).B + ask k0 (true).a

28 Hybrid Repressilator Repressilator with gene gates Repressilator in sccp Repressilator: ODE from sccp Hybrid Repressilator from sccp Analysis of Hybrid Repressilator

29 Conclusions We introduced a stochastic version of Concurrent Constraint Programming. We applied it in the field of biological modeling for a wide spectrum of problems: biochemical reactions, genetic networks, but also protein folding. The analysis techniques comprehend: simulation, model checking, solution of related ODEs. A new promising analysis method consists in associating hybrid automata to sccp models. We are currently investigating the use of sccp to simulate implicitly complex protein networks, in order to tame combinatorial explosion

30 Biology, Computation and Information 4 th International School on Biology, Computation, and Information (BCI 2007). 2-7 July, Area Science Park, Trieste.

31 sccp technical details Further biological applications sccp and ODE Operational Semantic: Instantaneous Transition Instantaneous Transition (IR1) tell (c), d, V 0, d c, V (IR2) D p( E D x ), d, V A[ x / E y ], d, V if p( y ) : A (IR3) x A, d, V A[y/x], d, V {y} with y V 2 \ V (IR4) (IR5) A 1, d, V A 1, d, V A 1.A 2, d, V A 1.A 2, d, V A 1, d, V A 1, d, V A 1 A 2, d, V A 1 A 2, d, V Theorem The instantaneous transition is confluent and can be applied only a finite number of steps to each configuration C. Return

32 sccp technical details Further biological applications sccp and ODE Operational Semantic: Stochastic Transition Stochastic Transition (SR1) tell λ (c), d, V = (1,λ(d)) 0, d c, V if d c false (SR2) ask λ (c), d, V = (1,λ(d)) 0, d, V if d c (SR3) π, d, V = (p,λ) 0, d, V π.a, d, V = (p,λ) A, d, V with π = ask or π = tell (SR4) (SR5) (SR6) A 1, d, V = (p,λ) A 1, d, V A 1.A 2, d, V = (p,λ) A 1.A 2, d, V M 1, d, V = (p,λ) A 1, d, V M 1 + M 2, d, V = (p,λ ) A 1, d, V with p pλ = λ+rate( M 2,d,V ) and λ = λ + rate( M 2, d, V ) A 1, d, V = (p,λ) A 1, d, V A 1 A 2, d, V = (p,λ ) A 1 A 2, d, V with p pλ = λ+rate( A 2,d,V ) and λ = λ + rate( A 2, d, V ) rate returns the sum of rates of all active agents.

33 sccp technical details Further biological applications sccp and ODE Operational Semantic: Stochastic Transition Theorem Let A, d, V C be the current configuration. Then the next stochastic transition executes one of the agents prefixed by a guard belonging to the set exec( A, d, V ), call it A. Moreover, the probability of the transition (i.e. the first label in = ) is rate( A, d, V ) rate (exec( A, d, V )), and the rate associated to the transition (the second label in = ) is rate (exec( A, d, V )). Return

34 sccp technical details Further biological applications sccp and ODE Rates Rates can be interpreted as priorities or as frequencies. Rates as Priorities A rate can represent the priority of execution of a process. There is a global scheduler choosing probabilistically between active processes, according to their priority. Discrete time evolution. Rates as Frequencies A rate can represent the frequency or speed of a process. The higher the speed, the higher the probability of seeing a certain process executed. Continuous time evolution. Return

35 sccp technical details Further biological applications sccp and ODE Discrete and Continuous Time Discrete time Discrete time transition can be recovered from stochastic transition = (p,λ) by dropping the second label. Hence we leave only the probability associated to transitions, obtaining a Discrete time Markov Chain. Continuous Time Continuous time transition can be recovered from stochastic transition = (p,λ) by multiplying the two labels. Hence we consider the rate associated to the transition, obtaining a Continuous time Markov Chain. Return

36 sccp technical details Further biological applications sccp and ODE Discrete and Continuous Time Observables Discrete time I/O observables O d ( A, d ) = { (d, p) p = Prob ( A, d 0, d )}. Continuous time I/O observables O c ( A, d )(t) = { (d, p) p = Prob ( A, d 0, d ) (t) }. Theorem lim O c( A, d )(t) = O d ( A, d ). t Return

37 sccp technical details Further biological applications sccp and ODE Continuous Stochastic Logic Return (state formulae) φ ::= true a φ φ φ P p [ψ] S p [φ] E c [φ] (path formulae) ψ ::= Xφ φu I φ φuφ, X: next. U: until. U I : bounded until, second formula must be true within interval I. P p [ψ]: path formula ψ must have probability p, with {<,,, >}. S p [φ]: states satisfying must have steady state probability p, with {<,,, >}. E c [φ]: expected reward of reaching a state satisfying φ is p, with {<,,, >}.

38 sccp technical details Further biological applications sccp and ODE PRISM: a probabilistic model checker Probabilistic Model Checking To model check a formula in CSL, we need essentially to solve linear systems in the size of the state space. Hence model checking is polynomial (but no more linear). PRISM PRISM is a symbolic model checking for CSL (and other probabilistic logics), using both Algebraic Decision Diagrams and sparse matrices to speed up the basic algorithms. Models can be described using a concurrent language having variable update and comparison as basic primitives. Model Checking sccp To model check (restricted) sccp, we defined an encoding of sccp programs into PRISM programs. Return

39 sccp technical details Further biological applications sccp and ODE Example: encoding S-System s Repressilator Ẋ 1 = α 1 X 1 3 β 1 X1 0.5, Ẋ 2 = α 2 X 1 1 β 2 X2 0.5, Ẋ 3 = α 3 X 1 2 β 3 X3 0.5, α i = 0.2, β i = 1. Return subs(x 1 ) subs(x 2 ) subs(x 3 ) ::= (tell(x 1 = X 1 + σ) [α1 X 1 3 ] +tell(x 1 = X 1 σ) [β1 X ] ).subs(x 1) ::= (tell(x 2 = X 2 + σ) [α2 X 1 1 ] +tell(x 2 = X 2 σ) [β2 X ] ).subs(x 2) ::= (tell(x 3 = X 3 + σ) [α3 X 1 2 ] +tell(x 3 = X 3 σ) [β3 X ] ).subs(x 3) σ = 1 Show more on behavioral equivalence

40 sccp technical details Further biological applications sccp and ODE Repressilator gone wild S-System s model of repressilator suffers from an high sensitivity from parameters, differently from the usual PA models. sccp model with variable rates has the same wild behaviour! β i = 0.01 β i = Return

41 sccp technical details Further biological applications sccp and ODE From sccp to ODE: example RW X :- tell 1 (X = X 1).RW X + tell 1 (X = X + 2).RW X ) + ask f (X) (true).( tell 1 (X = X 2).RW X + tell 1 (X = X + 1).RW X ) f (X) = 1 X 2 +1 X P P Ẋ = P 0 P 1 Ṗ 0 = 1 X 2 +1 P 0 + 2P 1 Ṗ 1 = 1 X 2 +1 P 0 2P 1 Return L. Bortolussi, A. Policriti. Stochastic Concurrent Constraint Programming and Differential Equations, 2007.

42 sccp technical details Further biological applications sccp and ODE Analysis of Hybrid Repressilator Determinism vs non-determinism Stability Return