Theory Bio-Modeling 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 Constraint Based Methods for Bioinformatics, Nantes, 25 th September 2006
Theory Bio-Modeling Modeling Biological Systems with Stochastic Process Algebras Pros Simple Language Compositionality Cons Hard to encode general information Lacking computational extensibility Constraints... why not?
Theory Bio-Modeling Outline Theory 1 Concurrent Constraint Programming Continuous Time Markov Chains Stochastic CCP Bio-Modeling 2 Modeling Biochemical Reactions Modeling Gene Regulatory Networks
Theory Bio-Modeling Outline Theory 1 Concurrent Constraint Programming Continuous Time Markov Chains Stochastic CCP Bio-Modeling 2 Modeling Biochemical Reactions Modeling Gene Regulatory Networks
Theory Bio-Modeling 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 "pot", called the constraint store, but can never be removed. Syntax of CCP Agents Agents can perform two basic operations on this Program = Decl . A store: D = ε | Decl . Decl | p ( x ) : − A Add a constraint ( tell ask) A = 0 | tell ( c ) . A Ask if a certain relation is entailed by the | ask ( c 1 ) . A 1 + ask ( c 2 ) . A 2 current configuration ( ask instruction) | A 1 � A 2 | ∃ x A | p ( x )
Theory Bio-Modeling 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 1 with prob. 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 ).
Theory Bio-Modeling Syntax of sCCP Syntax of Stochastic CCP Program = D . A D = ε | D . D | p ( x ) : − A π = tell λ ( c ) | ask λ ( c ) M = π. A | π. A . p ( y ) | M + M A = 0 | tell ∞ ( c ) . A | ∃ x A | M | ( A � A ) Stochastic Rates Each basic instruction (tell, ask, procedure call) has a rate attached to it. Rates are functions from the constraint store C to → R + . positive reals : λ : C −
Theory Bio-Modeling sCCP soup Operational Semantics There are two transition relations , one instantaneous (finite and confluent) and one stochastic. Traces are sequences of events with variable time delays among them. Implementation Stream Variables We have an interpreter Quantities varying over written in Prolog, using the time can be represented in CLP engine of SICStus to sCCP as unbounded lists. manage the constraint Hereafter: special meaning store. of X = X + 1. Efficiency issues.
Theory Bio-Modeling Outline Theory 1 Concurrent Constraint Programming Continuous Time Markov Chains Stochastic CCP Bio-Modeling 2 Modeling Biochemical Reactions Modeling Gene Regulatory Networks
Theory Bio-Modeling General Principles Measurable Entities ↔ Stream Variables Processes Logical Entities ↔ (Control Variables) Interactions ↔ Processes
Theory Bio-Modeling Biochemical Arrows to sCCP processes reaction ( k , [ R 1 , . . . , R n ] , [ P 1 , . . . , P m ]) : − �❱ n ask rMA ( k , R 1 ,..., Rn ) i = 1 ( R i > 0 ) ✁ . R 1 + . . . + R n → k P 1 + . . . + P m � 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 ]) R 1 + . . . + R n ⇋ k 1 reaction ( k 1 , [ R 1 , . . . , R n ] , [ P 1 , . . . , P m ]) � k 2 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 ) . S �→ E K , V 0 P ( 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 ) . S �→ E K , V 0 , h P ( tell ∞ ( S = S − h ) � tell ∞ ( P = P + h )) . Hill _ reaction ( K , V 0 , h , S , P ) V 0 S h V 0 S r MA ( k , X 1 , . . . , X n ) = k · X 1 · · · X n ; r MM ( K , V 0 , S ) = ; r Hill ( k , V 0 , h , S ) = where S h + K h S + K
Theory Bio-Modeling A simple reaction: H + Cl ⇌ HCl We have two reaction agents. The reagents and the products are stream variables of the constraint store (put down in the environment). Independent on the number of molecules . reaction ( 100 , [ H , CL ] , [ HCL ]) � reaction ( 10 , [ HCL ] , [ H , CL ])
Theory Bio-Modeling Another reaction: Na + Cl ⇌ Na + + Cl − reaction ( 100 , [ NA , CL ] , [ NA + , CL − ]) � reaction ( 10 , [ NA + , CL − ] , [ NA , CL ])
Theory Bio-Modeling Enzymatic reaction S + E ⇋ k 1 k − 1 ES → k 2 P + E Mass Action Equations Mass Action Kinetics d [ ES ] = k 1 [ S ][ E ] − k 2 [ ES ] − k − 1 [ ES ] enz_reaction( k 1 , k − 1 , k 2 , S , E , ES , P ) :- dt d [ E ] reaction( k 1 , [ S , E ] , [ ES ] ) � = − k 1 [ S ][ E ] + k 2 [ ES ] + k − 1 [ ES ] dt reaction( k − 1 , [ ES ] , [ E , S ] ) � d [ S ] = − k 1 [ S ][ E ] reaction( k 2 , [ ES ] , [ E , P ] ) dt d [ P ] = k 2 [ ES ] dt Michaelis-Menten Equations Michaelis-Menten Kinetics V 0 S ✥ ✦ d [ P ] k 2 + k − 1 = dt S + K mm _ reaction , k 2 · E , S , P V 0 = k 2 [ E 0 ] k 1 k 2 + k − 1 K = k 1
Theory Bio-Modeling Enzymatic reaction S + E ⇋ k 1 k − 1 ES → k 2 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
Theory Bio-Modeling MAP-Kinase cascade enz_reaction( k a , k d , k r , KKK , E 1 , KKKE 1 , KKKS ) � enz_reaction( k a , k d , k r , KKKS , E 2 , KKKSE 2 , KKK ) � enz_reaction( k a , k d , k r , KK , KKKS , KKKKKS , KKP ) � enz_reaction( k a , k d , k r , KKP , KKP 1 , KKPKKP 1 , KK ) � enz_reaction( k a , k d , k r , KKP , KKKS , KKPKKKS , KKPP ) � enz_reaction( k a , k d , k r , KP , KP 1 , KPKP 1 , K ) � enz_reaction( k a , k d , k r , K , KKPP , KKKPP , KP ) � enz_reaction( k a , k d , k r , KKPP , KKP 1 , KKPPKKP 1 , KKP ) � enz_reaction( k a , k d , k r , KP , KKPP , KPKKPP , KPP ) � enz_reaction( k a , k d , k r , KPP , KP 1 , KPPKP 1 , KP )
Theory Bio-Modeling The gene machine
Theory Bio-Modeling 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 kd ( true ) . neg _ gate ( k p , k i , k d , X , Y ) where r ( k , Y ) = k · Y . L. Cardelli, A. Phillips, 2005.
Theory Bio-Modeling Repressilator neg_gate(0 . 1 , 1 , 0 . 0001 , A , C ) � reaction(0 . 0001 , [ A ] , [] ) � neg_gate(0 . 1 , 1 , 0 . 0001 , B , A ) � reaction(0 . 0001 , [ B ] , [] ) � neg_gate(0 . 1 , 1 , 0 . 0001 , C , B ) � reaction(0 . 0001 , [ C ] , [] )
Theory Bio-Modeling Circadian Clock
Theory Bio-Modeling Circadian Clock pos_gate( α A , α ′ A , γ A , θ A , M A , A ) � pos_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 ] , [] )
Theory Bio-Modeling Conclusions We have introduced a stochastic version of CCP, with functional rates. We showed that sCCP may be used for modeling biological systems, defining libraries for biochemical reactions and gene regulatory networks. We showed that non-constant rates allow to use more complex chemical kinetics than mass action one.
Theory Bio-Modeling The End THANKS FOR THE ATTENTION! QUESTIONS?
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