Information Technology Reference
In-Depth Information
10
10*[ES]/[E0]
8
[S]
6
4
2
0
0
20
40
60
80
100
120
t
0.08
0.06
0.04
0.02
0.2
0.4
0.6
0.8
1
1.2
1.4
[S]
Fig. 2.
Simulation of one reaction using the CA. Cells marked by a 1 contain an enzyme
molecule, when marked with “1*” the enzyme is bound to a substrate molecule. The
other cells show the relative number of
P
molecules. On the left we show snapshots at
time
t
=50
∆t
and
t
= 110
∆t
, on the right the time evolution of [
S
] and [
ES
]
/
[
E
0
]
(the fraction of substrate-bound enzyme molecules), and right bottom the dependence
of
d
[
P
]
/dt
on [
S
] (for a larger 3-D system of 30
3
sites). The lines show the theoretical
predictions from the MM rate law.
age concentration. Here we use the microscopic procedure to ensure consistent
results.
5 CA Simulations
Figure 2 shows a simulation with such a cellular automaton for a system with
only one enzymatic reaction:
E
+
S → E
+
P
. We use the parameters
V
m
=
1
,K
m
=0
.
3
,c
=1
,∆t
=0
.
01. Here the time step is limited by the restriction
that at most one molecule can react at one time step (since the enzyme can only
bind one substrate molecule at a time),and we start the simulation with a high
substrate concentration of 10 molecules per site. We observe that the CA model
corresponds to the predictions from the macroscopic Michaelis-Menten rate law.
In this example diffusion of the metabolites is comparably fast,and the spatial
dimension does not have a measurable influence.
In a second test,we use this model to simulate a toy network of unidirectional
enzymatic reactions with four enzymes and four metabolites.
A
E
1
−→ B
E
2
−→ C
E
3
−→ D
E
4
−→ A
(6)
