Biomedical Engineering Reference
In-Depth Information
is infinite. Therefore, the analytical solution is possible only in few cases and the
numerical solutions are usually very computationally intensive. To overcome this
limitation, Gillespie proposed the stochastic simulation algorithm (SSA) [
11
]. This
algorithm is a Monte Carlo strategy and provides exact numerical realizations of the
stochastic process defined by the CME.
6.5.1
Successive Approximations Lead to Reaction
Rate Equations
However, since the SSA simulates every single reaction event, it becomes computa-
tionally hard for systems in which a large number of reactions occur during its time
evolution. This situation determined the development of other approaches which are
no longer exact but reduce the computational cost of a simulation of the system.
The
tau leaping technique
[
4
] is one of these approximated approaches. This al-
gorithm advances the system firing more than one reaction during a pre-selected
time step
: the number of firings is obtained from a Poisson random variable
P.a
j
.
is computed
in order to satisfy the leaping condition, according to which the state change must
be sufficiently small that no propensity function changes its value by a significant
amount. Hence, the current system's state is calculated according to the following
formula:
x
/; /
of mean and variance equal to
a
j
.
x
/
d
t
.Thevalueof
X
D
x
C
X
.t C /
v
j
P
j
.a
j
.
X
.t //; /
j D1
For example, considering the pathway of Fig.
6.1
, the formula related to the species
s
2
is:
:
D x
2
C
v
2;3
P
3
.a
3
.
X
2
.t C /
X
.t //; / C
v
2;4
P
4
.a
4
.
X
.t //; /;
because the only null stoichiometric coefficients regarding
s
2
are
v
2;3
and
v
2;4
.
If the populations of all the reactant species are sufficiently large, each reaction
is expected to fire more than one in the next
. If this condition and the leaping
condition hold, the tau leaping procedure can be approximated obtaining a stochastic
differential equation, called the
chemical Langevin equation
[
12
]:
q
X
X
d
X
d
D
v
j
a
j
.
X
.t // C
v
j
a
j
.
X
.t //
j
.t /;
t
j D1
j D1
where
are statistically independent “Gaussian white noise” processes. Here,
the state of the system
X
j
.t /
.t / D
x
is a continuous random variable and is expressed
as the sum of two terms: a deterministic drift term and a fluctuating diffusion term.
This equation is derived approximating the Poisson random variable (integer valued)
with a normal random variable (real valued) with the same mean and variance.
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