Biology Reference
In-Depth Information
The wave stops as soon as the sphere contains only
a
and all of
b
has been
consumed. If we identify the concentration of
a
with [
a
]
u
and nor-
malize the total concentration to 1 everywhere, mass action kinetics gives
the reaction term
f
(
u
)
=
u
). Inserting this into Eq. (35) yields a
nonlinear PDE. Such systems can exhibit bifurcations (cf. Sec. 2), which
is the case in the present example as Fisher waves only exist for speeds
s
=
ku
(1
−
≥
s
*, no moving front exists.
For the stochastic simulations, we denote by
X
p
the total number
of molecules contained in particle
p
, and define an analog to
Avogadro's number, viz.
M
, the number of molecules per unit mass.
Gillespie introduced the product h
c
as the expected number of reac-
tions per unit time. For the binary reaction here, it is h =
s
*
=
2
√
f
′
(0)
=
2
√
k.
136,137
For
s
<
X
p
X
p
. The
rate constant
k
and the reaction parameter
c
are related as
k
MVc
.
61
We interpret
c
as the probability that two molecules of species
a
and
b
react, provided they meet in space and time. This probability is inde-
pendent of the particle volume.
c
Figure 10(a) shows the total mass, integrated over the surface of the
sphere, of
a
and
b
as they evolve in time. The front position is also shown,
defined as the location where [
a
]
=
0.5. As long as reactions occur,
[
a
] and [
b
] change and the wave travels at a more or less constant speed
s
, given by the slope of the dashed curve in Fig. 10(a). If the dimension-
less front speed is plotted against the dimensionless reaction parameter,
the curves for different diffusion constants
D
collapse as shown in Fig. 10(b).
We also observe that the theoretical scaling
136,137
is well approximated,
particularly if the reaction is significantly faster than the diffusion.
=
[
b
]
=
9. Conclusions
Spatiotemporal models and numerical computer simulations are of rap-
idly growing importance in almost all areas of biology. Besides computa-
tional data analysis, including image processing, gene and protein sequence
analysis, clustering, and machine learning, modeling and simulation are
c
The probability of an encounter to occur, on the other hand, does depend on the
particle volume. This is, however, already accounted for in h, as the number of mol-
ecules
X
p
per particle is an extensive property.
