Continuous and Spatial Extension of Stochastic Calculus

Introduction

Stochastic π Calculus

Continuous semantics

Spatial Extension

Continuous and Spatial Extension of Stochastic π Calculus Anton Stefanek Supervised by Dr. Maria Vigliotti Second marker Dr. Jeremy Bradley

August 16, 2009

Anton Stefanek

JSPiM

Introduction



Spatial Extension

Overview

1 2

Why Systems Biology and Stochastic Process Algebras? Theoretical contribution Continuous semantics Spatial extension

3

Practical contribution JSPiM

Anton Stefanek

JSPiM

Introduction



Spatial Extension

JSPiM

Systems Biology Experiment Differential Equations Stochastic simulation

Biological System

Model System 100 KKKst KKPP KPP

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Spatial Extension

Stochastic Process Algebras (SPAs)

From performance analysis of computer and communications sytems 1

Formal

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Compositional Different forms of analysis

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ordinary differential equations (ODEs) stochastic simulation model checking and others 4

Directly implementable – tools

Applied to biology – PEPA, BioPEPA, Stochastic π calculus, sCCP

Anton Stefanek

JSPiM

Introduction



Spatial Extension

JSPiM

SPAs in Systems Biology Experiment Description in Stochastic Process Algebra Biological System

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Stochastic π Calculus Based on π calculus Used to model biological systems – e.g. signalling pathways, chemical reactions. Growing community, accepted by Biologists (paper in Nature) State of the art tool SPiM, collection of models ⇒ Worth of further research Problem Does not provide continuous semantics. Our solution We will extend it with continuous semantics and provide a tool.

Anton Stefanek

JSPiM

Introduction



Spatial Extension

JSPiM

Stochastic π calculus – Syntax Processes communicating over channels Actions α On channel a: (rate ra )

output action !ahmi input action ?a(x)

silent action:

Processes Zero Summation Parallel composition Restriction Identifier instance

0 α1 .P1 + · · · + αn .Pm P|Q (new e@r )P Ahbi def

Environment a set of defining equations A(x) = P + top-level process ⇒ System Anton Stefanek

τ @r

Introduction



Spatial Extension

Stochastic π calculus – Discrete semantics Important Rules r

τ @r .P −→ P r

a !ahbi.P | ?a(x).Q −→ P | Q{x 7→ b}

r

a (new e)!ahei | ?a(x).Q −→ (new e)(P| Q{x 7→ e})

Exponential delays ⇒ Continuous Time Markov Chain r1

0|τ @r2 .0

r2

τ @r1 .0|τ @r2 .0

0|0 r2

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τ @r1 .0|0

r1

JSPiM

Introduction



Spatial Extension

Simulation Problem Each process a single entity Large populations – many states and transitions e.g. 5 transitions from P|P|P|P|P Our solution We use structural congruence to represent processes as multisets, e.g. P | Q | Q | P | P ≡ {|3 × P , 2 × Q |} We describe explicit enumeration of the transitions (mass action) ⇒ efficient stochastic simulation of CTMC (based on the Gillespie algorithm) SPiM uses abstract machine instead Anton Stefanek

JSPiM

Introduction



Spatial Extension

JSPiM

Example: SIR model (Kermack-McKendrick)

S I R System

{|200 × S , 2 × I |}

CTMC

Model def

=

def

=

def

= =

{|k × S , n × I , m × R |}

?i.I !i.I + τ @rrec .R

n × rrec

0 200 × S |2 × I

{|(k − 1) × S , (n + 1) × I , m × R |} {|k × S , (n − 1) × I , (m + 1) × R |} System

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k × n × ri

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Spatial Extension


Problem How to get a continuous, deterministic interpretation of the model? Our solution We define continuous semantics providing this interpretation in terms of a set of ODEs.

Anton Stefanek

JSPiM

Introduction



Spatial Extension

Continuous semantics – Intuition Each state of the form {|k × S , n × I , m × R |} ) S decrease ri when S | I −→ 2× I I increase ) I decrease rrec R when I −→ R increase Take real valued functions s(t), i(t), r (t) ⇒ system of coupled ODEs ds/dt = −s(t)i(t)ri di/dt = s(t)i(t)ri − i(t)rrec dr /dt = i(t)rrec Initial values s(0) = 200, i(0) = 2 Anton Stefanek

JSPiM

Introduction



Spatial Extension

JSPiM

Continuous semantics – General case

Define all the possible “species” that can arise Definition P is a prime process if there exist no Q, R s.t. P ≡ Q|R. Prime b processes of a system (S, E ) is P(S, E ) all P s.t. S −→∗ P| · · · We can represent reachable processes as a multisets of species {|k × , n × , m × Theorem Every process can be expressed as a parallel composition of prime processes in a unique way.

Anton Stefanek

|}

Introduction



Spatial Extension

R|T

Q τ @r

a

Enter

·|U

Exit

P τ @q

Definition Define Enter and Exit multisets for channels b Enterch,S,E (P) = {|n × (a, R, T ) | R, T ∈ P(S, E ), r

a R|T −→ {|n × P, . . . |}|}. ra b Exitch,S,E (P) = {|(a, U) | U ∈ P(S, E ), P|U −→ ·|}

Similar for silent actions. Anton Stefanek

JSPiM

Introduction



Spatial Extension

JSPiM

Definition b For each prime process P in P(S, E ) define a real valued function [P] expressing population of P over time. Using the enumeration: R|T

Q

r [Q]

a

τ @r

a

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ra [P][U]

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d[P]/dt

P τ @q

ra [R][T ]

·|U

q[P]

Anton Stefanek

Introduction



Spatial Extension

Definition b The set of ODEs for (S, E ) contains for each P ∈ P(S, E) d[P] = dt

X (r ,Q)∈Enterτ,S,E (P)

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r [Q] +

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ra [R][T ]

(a,R,T )∈Enterch,S,E (P)

X

q[P] −

q∈Exitτ,S,E (P)

(a,U)∈Exitch,S,E (P)

with initial conditions [P](0) = S#P.

Anton Stefanek

ra [P][U],

JSPiM

Introduction



Spatial Extension

Example: SIR model Prime processes { S , I , R }

Model S I R System

def

=

def

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def

= =

d[ S ]/dt = −ri [ S ][ I ] d[ I ]/dt = ri [ S ][ I ] − rrec [ I ] d[ R ]/dt = rrec [ I ]

?i.I !i.I + τ @rrec .R 0 200 × S |2 × I

Initial values [ S ](0) = 200, [ I ](0) = 2, [ R ](0) = 0 System

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Spatial Extension

Finiteness Problem b P(S, E ) can become infinite def

P = (new e)(!shei|?e) def

Q = ?s(x).F hxi def

F (x) = τ @r .(!e|F hxi) Then prime processes include (new e)(?e| !e | F hei ) , (new e)(?e| !e | !e | F hei ) , . . . (new e)(?e| !e | !e | · · · | !e | F hei )

Anton Stefanek

JSPiM

Introduction



Spatial Extension

Finiteness

Our solution Conditions for finiteness For Chemical Ground Form (CGF) always finite subset with no new operator and message passing can translate from Sπ to CGF (most of the models)

⇒ Implementation requires CGF

Anton Stefanek

JSPiM

Introduction



Spatial Extension

Spatial extension Problem Want to model system with important spatial features – compartments, tissues.

Our solution We extend stochastic π calculus. We allow reuse of models and keep continuous semantics.

Anton Stefanek

JSPiM

Introduction



Spatial Extension

JSPiM

Spatial modelling in Systems Biology Solution Spatial extension Description in Stochastic Process Algebra

Experiment Problem Spatial properties Biological System

Formal Model

Model Analysis

Anton Stefanek

Introduction



Spatial Extension

Spatial Extension – Syntax Lπ Our extension of stochastic π calculus Static compartments with constant volume Random movement between compartments (given by m) Processes embedded in Location Graphs Location graphs [l1 : P1 , . . . , ln : Pn ]v ,m li location names, v volume function m movement function Anton Stefanek

JSPiM

Introduction



Spatial Extension

Spatial Extension – Semantics Internal transitions from communication affected by v Processes move between locations by movement function New rules r

a If P −→ Q then

[. . . , li : P , . . . ]v ,m

τ @ra /v (li )

−→

[. . . , li : Q , . . . ]v ,m

If P is a summation or an identifier instance,m(li , lj , P ) 6= 0 [. . . , li : P |Q, lj : R, . . . ]v ,m

τ @m(li ,lj ,

−→

P)

[. . . , li : Q, lj : P |R, . . . ]v ,m

Exponential delays ⇒ still CTMC (aggregation applies) Anton Stefanek

JSPiM

Introduction



Spatial Extension

JSPiM

Example: SIR with quarantine Location graph [a : System, b : 0]1,m m(a, b, I ) = rdiagnose m(b, a, R ) = rdischarge

CTMC states pairs of multisets {|ka × S , na × I , ma × R |}, {|nb × I , mb × R |} Results SystemQ:a

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Spatial Extension

Example: plant tissue

Cell =?attack.(Life|6 × Reistance|4 × Warning )+?warn.RCell, RCell =?attack.(Life|20 × Reistance|4 × Warning )+?warn.RCell, Resistance =!defeat.!defeated + delay @expire, Life =?fight+?defeated.RCell, Virus =!attack.(!fight.(2 × Virus)+?defeat) c0,0 : Cell

c0,1 : Cell

c0,2 : Cell

c0,3 : Cell

c1,0 : Cell

c1,1 : Cell | Virus

c1,2 : Cell

c1,3 : Cell

c2,0 : Cell

c2,1 : Cell

c2,2 : Cell

c2,3 : Cell

c3,0 : Cell

c3,1 : Cell

c3,2 : Cell

c3,3 : Cell

Anton Stefanek

JSPiM

Introduction



Spatial Extension

Example: plant tissue

Cell

RCell

Anton Stefanek

Virus

JSPiM

Introduction



Spatial Extension

Continuous semantics Can extend continuous semantics – prime processes and real valued functions for each location [P]l R|T in l move from j

Q in l a

m(j, l, P)

τ @r

Enter

P in l τ @q

move to k ·|U in l

Anton Stefanek

Exit

JSPiM

Introduction



Spatial Extension

JSPiM

Definition The set of ODEs for location graph contains for each prime P and location l d[P]l = dt

X (r ,Q)∈Enterτ,S,E (P)

X

−

X

ra [R]l [T ]l /v (l)

(a,R,T )∈Enterch,S,E (P)

X

q[P]l −

q∈Exitτ,S,E (P)

+

X

r [Q]l +

ra [P]l [U]l /v (l)

(a,U)∈Exitch,S,E (P)

m(j, l, P)[P]j −

m(j,l,P)6=0

X m(l,k,P)6=0

with initial conditions [P]l (0) = Pl #P.

Anton Stefanek

m(l, k, P)[P]l

Introduction



Spatial Extension

JSPiM

Example: SIR model d[ S ]a /dt = −ri [ S ]a [ I ]a d[ I ]a /dt = ri [ S ]a [ I ]a − rrec [ I ]a − rdiag [ I ]a d[ R ]a /dt = rrec [ I ]a + rdis [ R ]b d[ I ]b /dt = rdiag [ I ]a − rrec [ I ]b d[ R ]b /dt = rrec [ I ]b − rdis [ R ]b SystemQ:a

SystemQ:b

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Spatial Extension

JSPiM

Tool bringing the formalisms to the community Features Editor with GUI to show results Simulation of Sπ and Lπ systems ODE solution of CGF systems and CGF subset of Lπ

Written in Java (∼ 6KLOC), ANTLR for parsing, JFreeChart Directly related to the theory – multisets from google-collections

Anton Stefanek

JSPiM

Introduction



JSPiM – Editor

Anton Stefanek

Spatial Extension

JSPiM

Introduction



JSPiM – Commands (Simulation,ODE)

Anton Stefanek

Spatial Extension

JSPiM

Introduction



JSPiM – Spatial Commands

Anton Stefanek

Spatial Extension

JSPiM

Introduction



Spatial Extension

Summary

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Theoretical contributions Multiset representation Continuous semantics Spatial extension Investigations into relationship of the two semantics

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Practical contributions JSPiM Collection of existing models

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Future work Formal relationship between the two semantics Spatial models Release and open source of JSPiM

Anton Stefanek

JSPiM

Introduction



Thank you. Questions?

Anton Stefanek

Spatial Extension

JSPiM

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