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1
Km=3
0.8
0.6
Km = 0.3
0.4
0.2
Points: averaged over 100 trials
2
4
6
8
10
x/Km
Fig. 1.
Correct MM reaction rate (re-scaled by
K
m
, with
V
max
=1
,dt
= 1) (top curve)
compared to the average of 100 random realizations (dots) and predicted average from
Eq. (5) (two gray curves below the top curve).
3.2 Detailed Simulation of the Mechanism
A second possibility is to model the mechanism that leads to the MM rate law
in detail. For the simple unimolecular reaction,this mechanism is described by
Eq. (2). We can directly model this in the automaton,since at each lattice site
we have at most one enzyme molecule,and therefore we can keep track of the
state of this molecule: unbound or bound to
S
. The transitions between the
states are governed by simple linear rate laws,which we can model exactly in
the CA. We therefore have the following processes:
E
+
nS → ES
+(
n−
1)
S
with probability
p
1
(
n
);
ES
+
nS → E
+(
n
+1)
S
with probability
p
−
1
;
ES
+
nP → E
+(
n
+1)
P
with probability
p
2
,
where the probabilities can be calculated from the rate laws as
p
1
(
n
)=
∆t k
1
n
,
p
−
1
=
∆t k
−
1
,
p
2
=
∆t k
2
and
p
−
1
and
p
2
are independent of the metabolite
numbers present in the cell. One problem with this approach is that the rates
k
are usually not known. We can assume values by taking into account that
V
m
=
k
2
and
K
m
=(
k
−
1
+
k
2
)
/k
1
and that
k
2
is smaller than
k
1
and
k
−
1
.Ifwe
simply assume
k
−
1
=
ck
2
,we obtain
k
1
=(
c
+1)
V
m
/K
m
. One would assume
that
c
should be
>
1,but actually
c
= 1 works just as well. Note that now the
fastest processes are (
c
+ 1) times faster than before,which means that we need
to use a smaller time step. With such a model we can perform a number of time
steps and show that the average rate of transformation from
S
to
P
converges
to the rate given by the MM law for all values of
K
m
.
