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

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

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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

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INTRODUCTION BASICS ON SCCP APPROXIMATIONS AND MASTER EQUATION EXAMPLES

ON THE APPROXIMATION OF STOCHASTIC CONCURRENT CONSTRAINT PROGRAMMING BY MASTER EQUATION

Luca Bortolussi1,2

1Department of Mathematics and Informatics

University of Trieste luca@dmi.units.it

2Center for Biomolecular Medicine, Area Science Park, Trieste

QAPL 2008, Budapest, 30th March 2008

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INTRODUCTION BASICS ON SCCP APPROXIMATIONS AND MASTER EQUATION EXAMPLES

SEMANTICS FOR SPA

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INTRODUCTION BASICS ON SCCP APPROXIMATIONS AND MASTER EQUATION EXAMPLES

OUTLINE

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

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INTRODUCTION BASICS ON SCCP APPROXIMATIONS AND MASTER EQUATION EXAMPLES

STOCHASTIC CONCURRENT CONSTRAINT PROGRAMMING

CCP = CONSTRAINTS + AGENTS 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. p :- A; π = [g → u]λ; M = π.A | M + M A = 0 | M | p; N = A | A N rw(X):- [X > 0 → 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 Time Markov Chain.

  • L. Bortolussi, Stochastic Concurrent Constraint Programming, QAPL, 2006
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INTRODUCTION BASICS ON SCCP APPROXIMATIONS AND MASTER EQUATION EXAMPLES

MODELING IN SCCP

MODELING BIOCHEMICAL REACTIONS R1 + . . . + Rn →f(R,X;k) P1 + . . . + Pm f-reaction(R, X, P, k) :- tellf(R,X;k)(R′ = R − 1 ∧ P′ = P + 1). f-reaction(R, X, P, k) ANALYSIS TOOLS Stochastic simulation (Gillespie algorithm) Stochastic model checking and CTMC analysis Approximation with ODE’s and Hybrid Automata OREGONATOR B →k1 A A + B →k2 ∅ A →k3 2A + C 2A →k4 ∅ C →k5 B

  • L. Bortolussi, A. Policriti. Modeling

Biological systems in sCCP , Constraints, 13(1), 2008.

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INTRODUCTION BASICS ON SCCP APPROXIMATIONS AND MASTER EQUATION EXAMPLES

FROM SCCP TO ODE

Gene(X) :- [∗ → X′ = X + 1]kp .Gene(X) + [X ≥ 1 → ∗]kbX .[∗ → ∗]ku .Gene(X) Degrade(X) :- [X ≥ 0 → X′ = X − 1]kd X .Degrade(X)

ν = X G1 G0   1 −1 −1 +1 +1 −1   φ =     kpG1 kbXG1 kuG0 kdX     Φ1 = ν · φ :      ˙ X = kpG1 − kdX ˙ G1 = kuG0 − kbXG1 ˙ G0 = kbXG1 − kuG0

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INTRODUCTION BASICS ON SCCP APPROXIMATIONS AND MASTER EQUATION EXAMPLES

CIRCADIAN CLOCK

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INTRODUCTION BASICS ON SCCP APPROXIMATIONS AND MASTER EQUATION EXAMPLES

CIRCADIAN CLOCK

p_gate(αA, α′

A, γA, θA, MA, A)

p_gate(αR, α′

R, γR, θR, MR, A)

reaction(βA, [MA], [A]) reaction(δMA, [MA], []) reaction(βR, [MR], [R]) reaction(δMR, [MR], []) reaction(γC, [A, R], [AR]) reaction(δA, [AR], [R]) reaction(δA, [A], []) reaction(δR, [R], [])

A R 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 0 100 Time 100 200 300 400 500 600 700 800 900 1.000 1.100 1.200 1.300 1.400 1.500 1.600 1.700 1.800 1.900 2.000 2.100 2.200 Value A R 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 0 100 Time 100 200 300 400 500 600 700 800 900 1.000 1.100 1.200 1.300 1.400 1.500 1.600 1.700 1.800 Value

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INTRODUCTION BASICS ON SCCP APPROXIMATIONS AND MASTER EQUATION EXAMPLES

CIRCADIAN CLOCK

p_gate(αA, α′

A, γA, θA, MA, A)

p_gate(αR, α′

R, γR, θR, MR, A)

reaction(βA, [MA], [A]) reaction(δMA, [MA], []) reaction(βR, [MR], [R]) reaction(δMR, [MR], []) reaction(γC, [A, R], [AR]) reaction(δA, [AR], [R]) reaction(δA, [A], []) reaction(δR, [R], [])

A R 2 5 5 0 7 5 100 125 150 175 200 225 250 Time 250 500 750 1.000 1.250 1.500 1.750 2.000 2.250 2.500 2.750 3.000 Value

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INTRODUCTION BASICS ON SCCP APPROXIMATIONS AND MASTER EQUATION EXAMPLES

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 tj IN dt P(Y − νj, t)φj(Y − νj)dt DECREASE OF P(X, t) DUE TO tj IN dt P(Y, t)φj(Y)dt ∂P(Y, t) ∂t =

  • j
  • φj(Y − νj)P(Y − νj, t) − φj(Y)P(Y, t)
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INTRODUCTION BASICS ON SCCP APPROXIMATIONS AND MASTER EQUATION EXAMPLES

FIRST-ORDER APPROXIMATION

DIFFERENTIAL EQUATION FOR THE AVERAGE OF SCCP

d Yit dt =

  • Φ1

i (Y)

  • t

TAYLOR EXPANSION OF

  • Φ1

i (Y)

  • t
  • Φ1(Y)
  • t ≈ Φ1(Yt) + 1

2

|Y|

  • h,k=1

∂2

hkΦ1(Yt) YhYkt

FIRST-ORDER EQUATION FOR THE AVERAGE

d Yit dt = Φ1

i (Yt)

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INTRODUCTION BASICS ON SCCP APPROXIMATIONS AND MASTER EQUATION EXAMPLES

SECOND-ORDER APPROXIMATION

EXACT EQUATION FOR COVARIANCE

d YiYkt dt =

  • Φ2

ik(Y)

  • t +
  • (Yi − Yit)Φ1

k(Y)

  • t +
  • (Yk − Ykt)Φ1

i (Y)

  • t

SECOND-ORDER EQUATIONS FOR AVERAGE AND COVARIANCE

d Yit dt = Φ1(Yt) + 1 2

|T(N)|

  • h,k=1

∂2

hkΦ1(Yt) YhYkt

d YiYkt dt = Φ2

ik(Yt) + |Y|

  • h=1

∂hΦ1

k(Yt) YiYht

+

|Y|

  • h=1

∂hΦ1

i (Yt) YkYht

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INTRODUCTION BASICS ON SCCP APPROXIMATIONS AND MASTER EQUATION EXAMPLES

RANDOM WALK

RWX :- [∗ → X ′ = X + 1]k.RWX + [∗ → X ′ = X − 1]k.RWX,

Φ1(X) = 0 Φ2(X) = 2k

  • ˙

X = Φ1(X) + 1

2

  • X 2

∂2

XXΦ1(X) = 0

˙ X 2 = Φ2(X) + 2

  • X 2

∂XΦ1(X) = 2k

Xt = X0

  • X 2

t = 2kt +

  • X 2
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INTRODUCTION BASICS ON SCCP APPROXIMATIONS AND MASTER EQUATION EXAMPLES

EFFECTS OF VARIANCE

R1 :- [∗ → X′ = X + 1]k .R1; R2 :- [∗ → Y ′ = Y + 1]k .R2; R3 :- [X > 0 → X′ = X − 1]α1·X .R3 R4 :- [Y > 0 → Y ′ = Y − 1]α2·Y .R4; R5 :- [X > 0 ∧ Y > 0 → X′ = X − 1 ∧ Y ′ = Y + 1]ka·X·Y .R5 R1 R2 R3 R4 R5 Stochastic

X Y 5.000 10.000 15.000 20.000 25.000 30.000 35.000 40.000 Time 1.000 2.000 3.000 4.000 5.000 6.000 7.000 8.000 9.000 10.000 11.000 12.000 13.000 14.000 Value

SO Average/Variance

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INTRODUCTION BASICS ON SCCP APPROXIMATIONS AND MASTER EQUATION EXAMPLES

CIRCADIAN CLOCK

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INTRODUCTION BASICS ON SCCP APPROXIMATIONS AND MASTER EQUATION EXAMPLES

CIRCADIAN CLOCK

Stochastic

A R

1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 0 100

Time

100 200 300 400 500 600 700 800 900 1.000 1.100 1.200 1.300 1.400 1.500 1.600 1.700 1.800 1.900 2.000 2.100 2.200

Value

FO approximation

A R

1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 0 100

Time

100 200 300 400 500 600 700 800 900 1.000 1.100 1.200 1.300 1.400 1.500 1.600 1.700 1.800

Value

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INTRODUCTION BASICS ON SCCP APPROXIMATIONS AND MASTER EQUATION EXAMPLES

CIRCADIAN CLOCK

Stochastic average

A R

5 0 100 150 200 250 300 350 400 450 500

Time

100 200 300 400 500 600 700 800 900 1.000 1.100 1.200 1.300 1.400 1.500 1.600 1.700 1.800

Value

FO approximation

A R

1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 0 100

Time

100 200 300 400 500 600 700 800 900 1.000 1.100 1.200 1.300 1.400 1.500 1.600 1.700 1.800

Value

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INTRODUCTION BASICS ON SCCP APPROXIMATIONS AND MASTER EQUATION EXAMPLES

CIRCADIAN CLOCK

Stochastic average

A R

5 0 100 150 200 250 300 350 400 450 500

Time

100 200 300 400 500 600 700 800 900 1.000 1.100 1.200 1.300 1.400 1.500 1.600 1.700 1.800

Value

SO approximation

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INTRODUCTION BASICS ON SCCP APPROXIMATIONS AND MASTER EQUATION EXAMPLES

CIRCADIAN CLOCK

Robustness of the system: increase translation rate of R from βR = 5 to βR = 50.

Stochastic, βR = 50

A R

2 5 5 0 7 5 100 125 150 175 200 225 250

Time

250 500 750 1.000 1.250 1.500 1.750 2.000 2.250 2.500 2.750 3.000

Value

FO approximation, βR = 50

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INTRODUCTION BASICS ON SCCP APPROXIMATIONS AND MASTER EQUATION EXAMPLES

CIRCADIAN CLOCK

Robustness of the system: increase translation rate of R from βR = 5 to βR = 50.

Stochastic average, βR = 50

A R

1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 0 100

Time

250 500 750 1.000 1.250 1.500 1.750 2.000 2.250 2.500 2.750 3.000 3.250

Value

FO approximation, βR = 50

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INTRODUCTION BASICS ON SCCP APPROXIMATIONS AND MASTER EQUATION EXAMPLES

CIRCADIAN CLOCK

Robustness of the system: increase translation rate of R from βR = 5 to βR = 50.

Stochastic average, βR = 50

A R

1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 0 100

Time

250 500 750 1.000 1.250 1.500 1.750 2.000 2.250 2.500 2.750 3.000 3.250

Value

SO approximation, βR = 50

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INTRODUCTION BASICS ON SCCP APPROXIMATIONS AND MASTER EQUATION EXAMPLES

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.