Information Technology Reference
In-Depth Information
10
Well-mixed
B
8
6
A
4
D
2
C
50
100
150
200
time
10
Four compartments
8
B
6
4
D
A
C
2
50
100
150
200
time
Fig. 3.
Simulation of four-enzyme system with different spatial distributions of the en-
zymes. Top row: random distribution of enzymes (with equal numbers for all enzymes,
and probability of 0.2 for each site). Bottom row: distribution in four quadrants. The
concentration time series shown at right are compared with the predictions from the
differential equations using Michaelis-Menten kinetics. For the four quadrant model,
we compare with a compartmentalized ODE model, since the well-mixed ODE model,
which is the same in both cases, clearly gives wrong results. The initial conditions for
the metabolites are identical in both cases, and the number of enzyme molecules of
each type is also identical in both cases.
The parameters for the different enzymes are:
E
1
:
V
m
1
=1,
K
m
1
=0
.
3,
E
2
:
V
m
2
=2,
K
m
2
=1,
E
3
:
V
m
3
=3,
K
m
3
=0
.
3,
E
4
:
V
m
4
=3,
K
m
4
= 3,the initial
condition is [
B
] = 10 and [
A
]=[
C
]=[
D
]=0.
We simulate two different situations in Figure 3,one where the enzymes
are randomly distributed in space,and one where the enzymes are separated
in different compartments,and the metabolites must diffuse from one enzyme
to the other. In the first case,we compare the time evolution of the metabolite
concentrations with the solution of the ordinary differential equations,using the
Michaelis-Menten approximation for the construction of the ODEs. In the second
case (with identical parameters except for the spatial distribution of enzymes),
we observe a significantly different time evolution and also a very different steady
state. We can model this situation by constructing a system of ODEs where each
quadrant is treated as a different compartment,the reaction rates are adjusted
in the compartments (e.g.,in the compartment corresponding to the left lower
quadrant,only enzyme
E
1
is present,but in concentration four times higher than
