Biomedical Engineering Reference
In-Depth Information
The equations that describe this system in Eqs. (8.114) and (8.115) are
q
S
¼
K
1
q
C
1
K
q
S
q
E
1
q
C
1
¼
K
q
S
q
E
K
1
ð
þ
K
Þ
q
C
1
1
2
ð
8
:
116
Þ
q
I
¼
K
3
q
C
2
K
2
q
I
q
E
q
C
2
¼
K
q
I
q
E
K
3
ð
þ
K
Þ
q
C
2
3
4
We eliminate
q
E
from Eq. (8.116) by using
q
E
¼
E
q
C
1
q
C
2
, giving
0
q
S
¼
K
ð
q
S
þ
K
1
Þ
q
C
1
þ
K
q
S
q
C
2
K
E
q
S
1
1
1
0
q
C
1
¼
K
1
E
0
q
S
K
1
q
S
þ
K
1
þ
K
2
ð
Þ
q
C
1
K
1
q
S
q
C
2
ð
8
:
117
Þ
q
I
¼
K
ð
q
I
þ
K
3
Þ
q
C
2
þ
K
q
I
q
C
1
K
E
q
I
3
3
3
0
q
C
2
¼
K
E
q
I
K
ð
q
I
þ
K
3
þ
K
Þ
q
C
2
K
q
I
q
C
1
3
0
3
4
3
with nonzero initial conditions of
Þ¼
E
0
:
The quasi-steady-state approximation is found from Eq. (8.117) with
q
S
ð
0
Þ¼
S
0
,
q
I
ð
0
Þ¼
I
0
and
q
E
ð
0
q
C
1
¼
q
C
2
¼
0,
yielding
0
¼
K
1
E
0
q
S
K
1
q
S
þ
K
1
þ
K
2
ð
Þ
q
C
1
K
1
q
S
q
C
2
ð
8
:
118
Þ
0
¼
K
3
E
0
q
I
K
3
q
I
þ
K
3
þ
K
4
ð
Þ
q
C
2
K
3
q
I
q
C
1
Next, we solve for
q
C
2
, which gives
q
C
2
¼
K
q
I
E
ð
q
C
1
Þ
3
0
ð
8
:
119
Þ
q
I
þ
K
3
þ
K
K
3
4
and then we solve for
q
C
1
, giving
E
q
S
0
q
C
1
¼
ð
8
:
120
Þ
þ
q
I
K
i
M
q
S
þ
K
s
M
1
K
s
M
¼
K
1
þ
K
2
K
i
M
¼
K
3
þ
K
4
where
and
:
The velocity of the reaction is given by
K
1
K
3
K
E
q
S
V
2
0
max
¼
V
¼
K
2
q
C
1
¼
ð
8
:
121
Þ
K
s
M
q
S
þ
q
I
K
i
M
þ
q
I
K
i
M
q
S
þ
K
s
M
1
1
þ
1
where
:
Comparing Eq. (8.121) with Eq. (8.47), we see that
V
¼
K
E
max
2
0
V
max
does not change with the inclu-
K
M
in Eq. (8.47) has been replaced by the
sion of an enzyme inhibitor. In this case the term
in Eq. (8.121), which reduces the reaction rate. The left side of Figure 8.27
shows a plot of the reaction rate versus the substrate with increasing quantities of the
þ
q
I
K
i
M
term
K
s
M
1
