The Stochastic Simulation Algorithm

Computer Science Large Practical:
The Stochastic Simulation Algorithm (SSA)

Stephen Gilmore

School of Informatics

Friday 5th October, 2012

Stephen Gilmore (School of Informatics) Stochastic simulation Friday 5th October, 2012 1 / 25

Stochastic: Random processes

Fundamental to the principle of stochastic modelling is the idea that
molecular reactions are essentially random processes; it is impossible
to say with complete certainty the time at which the next reaction
within a volume will occur.


Stochastic: Predictability of macroscopic states

In macroscopic systems, with a large number of interacting molecules,
the randomness of this behaviour averages out so that the overall
macroscopic state of the system becomes highly predictable.
It is this property of large scale random systems that enables a
deterministic approach to be adopted; however, the validity of this
assumption becomes strained in in vivo conditions as we examine
small-scale cellular reaction environments with limited reactant
populations.


Stochastic: Propensity function

As explicitly derived by Gillespie, the stochastic model uses basic
Newtonian physics and thermodynamics to arrive at a form often termed
the propensity function that gives the probability aµ of reaction µ
occurring in time interval (t, t + dt).

aµ dt = hµ cµ dt

where the M reaction mechanisms are given an arbitrary index µ
(1 ≤ µ ≤ M), hµ denotes the number of possible combinations of reactant
molecules involved in reaction µ, and cµ is a stochastic rate constant.


Stochastic: Grand probability function

The stochastic formulation proceeds by considering the grand probability
function Pr(X; t) ≡ probability that there will be present in the volume V
at time t, Xi of species Si , where X ≡ (X1 , X2 , . . . XN ) is a vector of
molecular species populations.

Evidently, knowledge of this function provides a complete understanding of
the probability distribution of all possible states at all times.


Stochastic: Infinitesimal time interval

By considering a discrete infinitesimal time interval (t, t + dt) in which
either 0 or 1 reactions occur we see that there exist only M + 1 distinct
configurations at time t that can lead to the state X at time t + dt.

Pr(X; t + dt)
= Pr(X; t) Pr(no state change over dt)
M
+ µ=1 Pr(X − vµ ; t) Pr(state change to X over dt)

where vµ is a stoichiometric vector defining the result of reaction µ on
state vector X, i.e. X → X + vµ after an occurrence of reaction µ.


Stochastic: State change probabilities

Pr(no state change over dt)
M
1− aµ (X)dt
µ=1

Pr(state change to X over dt)
M
Pr(X − vµ ; t)aµ (X − vµ )dt
µ=1


Stochastic: Partial derivatives

We are considering the behaviour of the system in the limit as dt tends to
zero. This leads us to consider partial derivatives, which are deﬁned thus:

∂ Pr(X; t) Pr(X; t + dt) − Pr(X; t)
= lim
∂t dt→0 dt


Stochastic: Chemical Master Equation

Applying this, and re-arranging the former, leads us to an important
partial diﬀerential equation (PDE) known as the Chemical Master
Equation (CME).
M
∂ Pr(X; t)
= aµ (X − vµ ) Pr(X − vµ ; t) − aµ (X) Pr(X; t)
∂t
µ=1


The problem with the Chemical Master Equation

The CME is really a set of nearly as many coupled ordinary
diﬀerential equations as there are combinations of molecules that can
exist in the system!
The CME can be solved analytically for only a very few very simple
systems, and numerical solutions are usually prohibitively diﬃcult.

D. Gillespie and L. Petzold.
chapter Numerical Simulation for Biochemical Kinetics, in System Modelling
in Cellular Biology, editors Z. Szallasi, J. Stelling and V. Periwal.
MIT Press, 2006.


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Stochastic simulation algorithms


Gillespie’s Stochastic Simulation Algorithm (SSA) is essentially an exact
procedure for numerically simulating the time evolution of a well-stirred
chemically reacting system by taking proper account of the randomness
inherent in such a system.



Gillespie’s exact SSA (1977)

The algorithm takes time steps of variable length, based on the rate
constants and population size of each chemical species.
The probability of one reaction occurring relative to another is
dictated by their relative propensity functions.
According to the correct probability distribution derived from the
statistical thermodynamics theory, a random variable is then used to
choose which reaction will occur, and another random variable
determines how long the step will last.
The chemical populations are altered according to the stoichiometry
of the reaction and the process is repeated.



As described by in “Stochastic Simulation Algorithms for Chemical Reactions” by Ahn,
Cao and Watson, 2008

Suppose a biochemical system or pathway involves N molecular
species {S1 , . . . , SN }.
Xi (t) denotes the number of molecules of species Si at time t.
People would like to study the evolution of the state vector
X (t) = (X1 (t), . . . , XN (t)) given that the system was initially in the
state vector X (t0 ).

Example
The enzyme-substrate example had N = 4 molecular species, (E , S, C , P),
and the initial state vector X (t0 ) was (5, 5, 0, 0). If t = 200 we might ﬁnd
that X (t) was (5, 0, 0, 5).




Suppose the system is composed of M reaction channels
{R1 , . . . , RM }.
In a constant volume Ω, assume that the system is well-stirred and in
thermal equilibrium at some constant temperature.

Example
The enzyme-substrate example had M = 3 reaction channels, f , b and p.




There are two important quantities in reaction channels Rj :
the state change vector vj = (v1j , . . . , vNj ), and
propensity function aj .
vij is deﬁned as the change in the Si molecules’ population caused by
one Rj reaction,
aj (x)dt gives the probability that one Rj reaction will occur in the
next inﬁnitesimal time interval [t, t + dt).

Example
The reaction f: E + S -> C has state change vector (−1, −1, 1, 0).




The SSA simulates every reaction event.
With X (t) = x, p(τ, j | x, t)dτ is deﬁned as the probability that the
next reaction in the system will occur in the inﬁnitesimal time interval
[t + τ, t + τ + dτ ), and will be an Rj reaction.
M
By letting a0 (x) ≡ j=1 aj (x), the equation

p(τ, j | x, t) = aj (x) exp(−a0 (x)τ ),

can be obtained.




A Monte Carlo method is used to generate τ and j.
On each step of the SSA, two random numbers r1 and r2 are
generated from the uniform (0,1) distribution.
From probability theory, the time for the next reaction to occur is
given by t + τ , where
1 1
τ= ln( ).
a0 (x) r1




The next reaction index j is given by the smallest integer satisfying
j
aj (x) > r2 a0 (x).
j =1

After τ and j are obtained, the system states are updated by
X (t + τ ) := x + vj , and the time is updated by t := t + τ .
This simulation iteration proceeds until the time t reaches the ﬁnal
time.



Sampling from a probability distribution
In order to sample from a non-uniform probability distribution we can
think of an archer repeatedly blindly ﬁring random arrows at a patch of
painted ground. Because the arrows are uniformly randomly distributed
they are likely to hit the larger painted areas more often than the smaller
painted areas.

Archer
133 110 50 50 40 30

Note
We cannot predict beforehand where any particular arrow will land.



Sampling from a probability distribution
Here we interpret the picture as meaning that there are five reaction
channels (the red reaction, the blue reaction, the green reaction, the
yellow reaction and the black reaction). These have different propensities,
with the red reaction being the most likely to fire and the black reaction
being the least likely to fire.

Archer
133 110 50 50 40 30

Note
We know that the blue reaction fires because 110 + 50 > 133.



Gillespie’s SSA is a Monte Carlo Markov Chain simulation

The SSA is a Monte Carlo type method. With the SSA one may
approximate any variable of interest by generating many trajectories and
observing the statistics of the values of the variable. Since many
trajectories are needed to obtain a reasonable approximation, the eﬃciency
of the SSA is of critical importance.



Excellent introductory papers

T.E. Turner, S. Schnell, and K. Burrage.
Stochastic approaches for modelling in vivo reactions.
Computational Biology and Chemistry, 28:165–178, 2004.

D. Gillespie and L. Petzold.
System Modelling in Cellular Biology, chapter Numerical Simulation for
Biochemical Kinetics,.
MIT Press, 2006.


The Stochastic Simulation Algorithm

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