Biology Reference
In-Depth Information
diffusion of the key signal transduction molecule in the cytoplasmic
space.
12
Other examples of reaction-diffusion signaling models
include a sporulation control network model
11
and plant shoot meris-
tem simulations.
135
8.2. Reaction-Diffusion in Particle Methods
Extending the simulation scheme outlined in Sec. 7 to reaction-diffusion
problems as governed by Eq. (35), all components of the concentration
vector
u
are represented on the same set of computational particles, sup-
porting property vectors
ω
p
.
Evaluating the reaction terms
f
i
amounts to a purely local exchange
of strength among species in the same particle. Reactions are thus evalu-
ated independently for each particle. The rate of exchange between dif-
ferent
u
i
is directly given by the reaction kinetics that are evaluated using
either a deterministic method based on kinetic ODEs or a stochastic
method such as Gillespie's SSA algorithm.
61,62
The latter is possible
because individual particles constitute homogeneous reaction spaces with
no spatial gradients present inside a single particle. The deterministic
solver uses the same time integrator as the diffusion part, and is thus
restricted by the time step stability limit; while the stochastic solver
directly operates on (scaled) molecule numbers, and is used outside of
the time integrator's right-hand side.
8.2.1. An example with moving reaction fronts
→
2
a
with rate
constant
k
. Both species
a
and
b
diffuse with the same isotropic diffusion
constant
D
. We use a combination of PSE
103
and SSA
61,62
to simulate the
example system on the surface of a sphere with diameter
l
. The initial
condition is such that one half of the sphere contains only
a
, and the
other half only
b
. Since
a
and
b
are initially unmixed, diffusion is required
in order to bring them together and allow the reaction to start. As
b
is
“eaten up” by
a
, the reaction front separating the two species moves into
the region where
b
initially was and thus forms a traveling Fisher wave,
which propagates in the direction orthogonal to the reaction front.
As an illustrative example, we consider the reaction
a
+
b
