Biomedical Engineering Reference
In-Depth Information
f
2
(t)
f
1
(t)
V
max
(
q
1
+
K
M
)
K
12
q
1
q
2
K
21
K
10
K
20
FIGURE 8.11
A two-compartment model with a quasi-steady-state approximation transfer rate.
approximation is quite good, as is the upper bound for large values of
The accuracy of
the lower and upper bound approximations depends quite heavily on the closeness of
K
M
q
S
:
to
q
S
max
:
The farther apart the two are, the poorer the approximation.
8.3.2 Two-Compartment Model
Next, consider a two-compartment model with a quasi-steady-state approximation trans-
fer rate as shown in Figure 8.11. Such models are useful in describing some capacity-limited
biochemical reactions that are more complex than a one-compartment model. Note that the
quasi-steady-state approximation is for compartment 1 only, and the other transfer rates are
the usual constants. The equations describing the system are
0
@
1
A
q
1
V
max
q
1
þ
K
M
q
1
¼
K
þ
K
12
þ
þ
K
q
2
þ
f
ð
t
Þ
10
21
1
ð
Þ
ð
8
:
66
Þ
q
2
¼
K
12
q
1
K
20
þ
K
21
ð
Þ
q
2
þ
f
2
ð
t
Þ
EXAMPLE PROBLEM 8.3
Simulate the model in Eq. (8.66), given that
K
12
¼
1,
K
21
¼
2,
V
max
¼
3,
K
M
¼
0
:
5,
q
1
ð
0
Þ¼
0,
q
2
ð
0
Þ¼
0,
K
10
¼
0, and
K
20
ð
0
Þ¼
0
:
The inputs are
f
1
ð
t
Þ¼
u
ð
t
Þ
u
ð
t
4
Þ
, and
f
2
(
t
)
¼
0.
Solution
The equations describing this system are
3
q
1
þ
1
q
1
þ
q
1
¼
þ
2
q
2
þ
u
ð
t
Þ
u
ð
t
4
Þ
ð
0
:
5
Þ
q
2
¼
3
q
1
2
q
2
The SIMULINK model shown in Figure 8.12 was executed using the
ode23tb
integrator.
The solution is shown in Figure 8.13.
Continued
