Simulation and Modelling

Tony Field and Jeremy Bradley

Room 372. Email: jb@doc.ic.ac.uk

Department of Computing, Imperial College London

Produced with prosper and LATEX

SM [11/08] – p. 1

Chemical Reactions and Simulation

With thanks to Daniel Gillespie!

Daniel T. Gillespie, Simulation Methods inSystems Biology. SFM 2008, LNCS 5016, pp.

125–167, 2008. Springer-Verlag.

Daniel T. Gillespie, Exact stochastic simulation ofcoupled chemical reactions. J. Phys. Chem.,

81(25), pp. 2340–2361, 1977.DOI: 10.1021/j100540a008

SM [11/08] – p. 2

Simulating Large Stochastic Models

Examples: Chemical reactions, biologicalsystems, epidemic models and parallel anddistributed systems

Underlying genuinely stochastic models,often reasonable to assume Markovianbehaviour, reactions between elementsconsitute synchronisation in a model

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Reaction: Mass action

Reaction between e.g. well-mixed fluids andgases

Molecules diffuse (Brownian motion)

Molecules can potentially react with any otherco-reagant molecule

Example reaction:

A + Bλ

−→ AB

Initially m A molecules, n B molecules

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Reaction: Mass action

A

B

SM [11/08] – p. 5

Reaction: Mass action

A

B

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SM [11/08] – p. 5

Reaction: Mass action

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SM [11/08] – p. 5

Reaction: Mass action

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Total number of possible interactions: mn

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Reaction: Mass action

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Total number of actual AB products: min(m, n)

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Reaction: Local action

Reaction between e.g. surface of two solids,two jellies, two very viscous fluids

No molecule diffusion

Molecules react with closest local neighbour

No reaction competition

Example reaction:

A + Bλ

−→ AB

Initially m A molecules, n B molecules

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Reaction: Local actionA B

-

-

-

Total number of possible reactions: min(m, n)

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Reaction: Local action

Total number of AB products: min(m, n)

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Reaction: Passive action

Reaction catalysed by one or more passivemolecules

Heavily spatially dependent on catalystshape/configuration

Example reaction:

A + Bλ

−→ A + B′

Initially 1 A molecule, n B molecules

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Reaction: Passive action

A

B

Total number of possible reactions: I(m > 0) n

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Reaction: Catalyst

A

B′

Total number of B′ products: I(m > 0) n

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Chemical Systems

Consider a system of molecules of Nchemical species S1, . . . , SN, which interactthrough M reaction channels R1, . . . , RM inreaction vessel of volume Ω

The state of a general chemical systemrequires giving the instantaneous position,velocity, and species of each molecule in thesystem

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Chemical Systems

Specifying the state of a well-stirred system ismuch easier – we need only specify thevector

~X(t) = (X1(t), . . . , XN(t))

where Xi(t) is the number of Si moleculescontained in a container at time t

The state-change vector ~νj = (ν1j, . . . , νNj)where νij is defined to be the change in the Si

molecular population caused by one Rj

reaction event

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Propensity

So now we can say that reaction Rj producesthe following change in the system state

~x → ~x + ~νj

where xi is the number Si molecules in aparticular state

Propensity function aj(~x) represents, for a givensystem state ~x, the propensity of reaction Rj

to occur

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Propensity

Fundamental premise of stochastic chemicalkinetics, for a given system state ~x:

aj(~x)δt = IP(for state ~x, reaction Rj will occurinside Ω in the next infinitesimaltime interval [t, t + δt) | ~X(t) = ~x)

Used to express Chemical Master Equation(CME) for finding P (~x, t), probability system isin state ~x at time t:

P (~x, t) = IP( ~X(t) = ~x | ~X(t0) = ~x0 for t0 ≤ t)

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Example Propensity Functions

Under reaction Rj happening with reaction rateconstant cj, there are 3 possibilities (assumingno n-way reactions for n > 2):

If S1cj

−→ products aj(~x) = cjx1

If S1 + S2cj

−→ products aj(~x) = cjx1x2

If 2S1cj

−→ products aj(~x) = cj12x1(x1 − 1)

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Chemical Master Equation

Consider the possible ways that a system canreach state ~x by time t + δt:

P (~x, t + δt) = P (~x, t) ×

(

1 −

M∑

j=1

(aj(~x)δt)

)

︸ ︷︷ ︸

IP(System does not undergo a reaction in δt)

+M∑

j=1

P (~x − ~νj, t) × (aj(~x − ~νj)δt)

︸ ︷︷ ︸

IP(System does undergo a reaction Rj in δt)

(∗)

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Chemical Master Equation

Subtracting P (~x, t) from both sides of (∗),dividing by δx and letting δx → 0, we get a partialderivative expression:

∂P (~x, t)

∂t=

M∑

j=1

[aj(~x − ~νj)P (~x − ~νj, t) − aj(~x)P (~x, t)]

This is the Chemical Master Equation (CME) for thesystem

In theory the CME completely defines thebehaviour of the system – in practice difficult tosolve analytically for all but the simplest of aj(·)

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Average Behaviour

If we define the mean behaviour of ~X to be:

IE(f( ~X)) =∑

~x

f(~x)P (~x, t)

We can sum CME over all possible states ~x toget a simpler expression defining the averageevolution of the system:

dIE( ~X)

dt=

M∑

j=1

~νjIE(aj( ~X))

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Reaction Rate Equation

If we assume that there is no variance in theindividual trajectories of Xi, we can rewrite theprevious equation as:

dIE( ~X)

dt=

M∑

j=1

~νj(aj(IE( ~X)))

This is the Reaction Rate Equation. It assumes thatthe mean trajectory dominates the system andany fluctations in ~X decay

This is the standard set of ODEs used to modelmany biochemical systems.

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Gillespie’s Algorithm

Instead of trying to solve the CME, Gillespie’sAlgorithm produces a simulated trace ofexecution of ~X(t)

Looking at the mean of many such traces willusually give a good approximation to IE( ~X)

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Gillespie’s SSA

1. Initialization: Initialize no. of molecules inthe system, reactions constants, and randomnumber generators

2. Monte Carlo Step: Generate random nos. to de-termine next reaction to occur as well as timeinterval

3. Update: Increase the time step by the ran-domly generated time in Step 2. Update themolecule count based on reaction that oc-curred

4. Iterate: Go back to Step 2 unless no. of reac-tants is zero or the simulation time is exceeded

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SSA in Action

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Voters (ODEs)Pollers (ODEs)

Counters (ODEs)Voters (SS)Pollers (SS)

Counters (SS)

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SSA in Action

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SSA in Action

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Counters (SS)

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