S EMANTICS FOR SPA I NTRODUCTION B ASICS ON S CCP A PPROXIMATIONS AND - PowerPoint PPT Presentation

I NTRODUCTION B ASICS ON S CCP A PPROXIMATIONS AND M ASTER E QUATION E XAMPLES O N THE A PPROXIMATION OF S TOCHASTIC C ONCURRENT C ONSTRAINT P ROGRAMMING BY M ASTER E QUATION 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 2008, Budapest, 30 th March 2008

I NTRODUCTION B ASICS ON S CCP A PPROXIMATIONS AND M ASTER E QUATION E XAMPLES S EMANTICS FOR SPA

I NTRODUCTION B ASICS ON S CCP A PPROXIMATIONS AND M ASTER E QUATION E XAMPLES O UTLINE 1 I NTRODUCTION 2 B ASICS ON S CCP 3 A PPROXIMATIONS AND M ASTER E QUATION 4 E XAMPLES

I NTRODUCTION B ASICS ON S CCP A PPROXIMATIONS AND M ASTER E QUATION E XAMPLES S TOCHASTIC C ONCURRENT C ONSTRAINT P ROGRAMMING CCP = C ONSTRAINTS + A GENTS Constraints are formulae over an interpreted first order language (i.e. X = 10, 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. STOCHASTIC CCP p :- A ; π = [ g → u ] λ ; M = π. A | M + M Each ask and tell instruction has A = 0 | M | p ; N = A | A � N a rate (function) attached to it: → R + . λ : C − [ X > 0 → X ′ = X − 1 ] λ ( X ) . rw ( X ) The semantics of the language is rw ( X ) :- + [ true → X ′ = X + 1 ] λ ( X ) . rw ( X ) given in terms of a Continuous Time Markov Chain. L. Bortolussi, Stochastic Concurrent Constraint Programming , QAPL, 2006

I NTRODUCTION B ASICS ON S CCP A PPROXIMATIONS AND M ASTER E QUATION E XAMPLES M ODELING IN S CCP O REGONATOR M ODELING BIOCHEMICAL REACTIONS B → k 1 A R 1 + . . . + R n → f ( R , X ; k ) P 1 + . . . + P m A + B → k 2 ∅ A → k 3 2 A + C f -reaction ( R , X , P , k ) :- 2 A → k 4 ∅ tell f ( R , X ; k ) ( R ′ = R − 1 ∧ P ′ = P + 1 ) . C → k 5 B f -reaction ( R , X , P , k ) A NALYSIS 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), 2008.

I NTRODUCTION B ASICS ON S CCP A PPROXIMATIONS AND M ASTER E QUATION E XAMPLES F ROM S CCP TO ODE   X 1 0 0 − 1 − 1 ν = G 1 0 + 1 0   − 1 G 0 0 + 1 0 Gene ( X ) :- [ ∗ → X ′ = X + 1 ] kp .Gene ( X ) + [ X ≥ 1 → ∗ ] kbX . [ ∗ → ∗ ] ku .Gene ( X )   k p G 1 k b XG 1 Degrade ( X ) :-   φ =   k u G 0 [ X ≥ 0 → X ′ = X − 1 ] kd X .Degrade ( X )   k d X  ˙ X = k p G 1 − k d X  Φ 1 = ν · φ :  ˙ G 1 = k u G 0 − k b XG 1 ˙  G 0 = k b XG 1 − k u G 0 

I NTRODUCTION B ASICS ON S CCP A PPROXIMATIONS AND M ASTER E QUATION E XAMPLES C IRCADIAN C LOCK

I NTRODUCTION B ASICS ON S CCP A PPROXIMATIONS AND M ASTER E QUATION E XAMPLES C IRCADIAN C LOCK 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 ] , [] ) 2.200 1.800 2.100 1.700 2.000 1.600 1.900 1.800 1.500 1.700 1.400 1.600 1.300 1.500 1.200 1.400 1.300 1.100 Value 1.200 1.000 Value 1.100 900 1.000 800 900 800 700 700 600 600 500 500 400 400 300 300 200 200 100 100 0 0 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 0 100 0 0 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 0 100 Time Time A R A R

I NTRODUCTION B ASICS ON S CCP A PPROXIMATIONS AND M ASTER E QUATION E XAMPLES C IRCADIAN C LOCK 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 ] , [] ) 3.000 2.750 2.500 2.250 2.000 Value 1.750 1.500 1.250 1.000 750 500 250 0 0 2 5 5 0 7 5 100 125 150 175 200 225 250 Time A R

I NTRODUCTION B ASICS ON S CCP A PPROXIMATIONS AND M ASTER E QUATION E XAMPLES M ASTER E QUATION FOR S CCP 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 ) . I NCREASE OF P ( X , t ) DUE TO t j IN dt P ( Y − ν j , t ) φ j ( Y − ν j ) dt D ECREASE OF P ( X , t ) DUE TO t j IN dt P ( Y , t ) φ j ( Y ) dt ∂ P ( Y , t ) � � � = φ j ( Y − ν j ) P ( Y − ν j , t ) − φ j ( Y ) P ( Y , t ) ∂ t j

I NTRODUCTION B ASICS ON S CCP A PPROXIMATIONS AND M ASTER E QUATION E XAMPLES F IRST - ORDER APPROXIMATION D IFFERENTIAL E QUATION FOR THE A VERAGE OF S CCP d � Y i � t Φ 1 � � = i ( Y ) t dt Φ 1 � � i ( Y ) T AYLOR EXPANSION OF t | Y | t ≈ Φ 1 ( � Y � t ) + 1 Φ 1 ( Y ) � ∂ 2 hk Φ 1 ( � Y � t ) �� Y h Y k �� t � � 2 h , k = 1 F IRST - ORDER EQUATION FOR THE AVERAGE d � Y i � t = Φ 1 i ( � Y � t ) dt

I NTRODUCTION B ASICS ON S CCP A PPROXIMATIONS AND M ASTER E QUATION E XAMPLES S ECOND -O RDER A PPROXIMATION E XACT EQUATION FOR COVARIANCE d �� Y i Y k �� t � � � � � � Φ 2 ( Y i − � Y i � t )Φ 1 ( Y k − � Y k � t )Φ 1 = ik ( Y ) t + k ( Y ) t + i ( Y ) dt t S ECOND - ORDER EQUATIONS FOR AVERAGE AND COVARIANCE | T ( N ) | d � Y i � t = Φ 1 ( � Y � t ) + 1 � ∂ 2 hk Φ 1 ( � Y � t ) �� Y h Y k �� t dt 2 h , k = 1 | Y | d �� Y i Y k �� t Φ 2 � ∂ h Φ 1 = ik ( � Y � t ) + k ( � Y � t ) �� Y i Y h �� t dt h = 1 | Y | � ∂ h Φ 1 + i ( � Y � t ) �� Y k Y h �� t h = 1

I NTRODUCTION B ASICS ON S CCP A PPROXIMATIONS AND M ASTER E QUATION E XAMPLES R ANDOM WALK RW X :- [ ∗ → X ′ = X + 1 ] k .RW X + [ ∗ → X ′ = X − 1 ] k .RW X , � Φ 1 ( X ) = 0 ˙ � X � = Φ 1 ( � X � ) + 1 X 2 �� ∂ 2 XX Φ 1 ( � X � ) = 0 �� 2 ˙ Φ 2 ( X ) = 2 k �� X 2 �� = Φ 2 ( � X � ) + 2 �� X 2 �� ∂ X Φ 1 ( � X � ) = 2 k � X � t = X 0 X 2 �� X 2 �� �� �� t = 2 kt + 0

I NTRODUCTION B ASICS ON S CCP A PPROXIMATIONS AND M ASTER E QUATION E XAMPLES E FFECTS OF V ARIANCE R 1 :- [ ∗ → X ′ = X + 1 ] k .R 1 ; R 2 :- [ ∗ → Y ′ = Y + 1 ] k .R 2 ; R 3 :- [ X > 0 → X ′ = X − 1 ] α 1 · X .R 3 R 4 :- [ Y > 0 → Y ′ = Y − 1 ] α 2 · Y .R 4 ; R 5 :- [ X > 0 ∧ Y > 0 → 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 14.000 13.000 12.000 11.000 10.000 9.000 8.000 Value 7.000 6.000 5.000 4.000 3.000 2.000 1.000 0 0 5.000 10.000 15.000 20.000 25.000 30.000 35.000 40.000 Time X Y

I NTRODUCTION B ASICS ON S CCP A PPROXIMATIONS AND M ASTER E QUATION E XAMPLES C IRCADIAN C LOCK

I NTRODUCTION B ASICS ON S CCP A PPROXIMATIONS AND M ASTER E QUATION E XAMPLES C IRCADIAN C LOCK FO approximation Stochastic 1.800 2.200 2.100 1.700 2.000 1.600 1.900 1.500 1.800 1.400 1.700 1.600 1.300 1.500 1.200 1.400 1.100 1.300 1.000 Value 1.200 Value 1.100 900 1.000 800 900 700 800 600 700 600 500 500 400 400 300 300 200 200 100 100 0 0 0 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 0 100 0 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 0 100 Time Time A R A R

I NTRODUCTION B ASICS ON S CCP A PPROXIMATIONS AND M ASTER E QUATION E XAMPLES C IRCADIAN C LOCK Stochastic average FO approximation 1.800 1.800 1.700 1.700 1.600 1.600 1.500 1.500 1.400 1.400 1.300 1.300 1.200 1.200 1.100 1.100 1.000 1.000 Value Value 900 900 800 800 700 700 600 600 500 500 400 400 300 300 200 200 100 100 0 0 0 5 0 100 150 200 250 300 350 400 450 500 0 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 0 100 Time Time A R A R

I NTRODUCTION B ASICS ON S CCP A PPROXIMATIONS AND M ASTER E QUATION E XAMPLES C IRCADIAN C LOCK SO approximation Stochastic average 1.800 1.700 1.600 1.500 1.400 1.300 1.200 1.100 1.000 Value 900 800 700 600 500 400 300 200 100 0 0 5 0 100 150 200 250 300 350 400 450 500 Time A R

I NTRODUCTION B ASICS ON S CCP A PPROXIMATIONS AND M ASTER E QUATION E XAMPLES C IRCADIAN C LOCK Robustness of the system: increase translation rate of R from β R = 5 to β R = 50. Stochastic, β R = 50 FO approximation, β R = 50 3.000 2.750 2.500 2.250 2.000 Value 1.750 1.500 1.250 1.000 750 500 250 0 0 2 5 5 0 7 5 100 125 150 175 200 225 250 Time A R

I NTRODUCTION B ASICS ON S CCP A PPROXIMATIONS AND M ASTER E QUATION E XAMPLES C IRCADIAN C LOCK Robustness of the system: increase translation rate of R from β R = 5 to β R = 50. Stochastic average, β R = 50 FO approximation, β R = 50 3.250 3.000 2.750 2.500 2.250 2.000 Value 1.750 1.500 1.250 1.000 750 500 250 0 0 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 0 100 Time A R