Biomedical Engineering Reference
In-Depth Information
80
70
q
M
= 0
60
50
40
q
M
= 10
30
q
M
= 50
20
10
0
0 0 0 0 0 0 0 0 0 0 0
q
s
FIGURE 8.28
Velocity of product appearance for a substrate, enzyme, and allosteric modifier using Eq. (8.129).
The parameters used are:
K
¼
4,
K
1
¼
2, K
2
¼
5,
K
¼
1,
K
3
¼
10,
K
¼
0
:
3,
K
4
¼
0
:
03,
K
¼
2,
K
¼
2,
K
6
¼
4, and
1
3
4
5
6
E
¼
15
:
0
The equations describing the system are given as
q
S
¼
K
1
ð
q
C
1
þ
q
C
3
Þ
K
1
q
S
q
E
þ
q
C
2
ð
Þ
ð Þ
q
C
1
¼
K
3
q
C
3
þ
K
1
q
S
q
E
K
1
þ
K
2
þ
K
3
q
M
q
M
¼
K
3
ð
q
C
2
þ
q
C
3
Þ
K
3
q
M
q
E
þ
q
C
1
ð
Þ
q
C
1
ð
8
:
131
Þ
q
C
2
¼
K
q
ð Þ
q
C
2
q
C
3
¼
K
1
q
S
q
C
2
þ
K
3
q
M
q
C
1
K
1
þ
K
3
q
M
q
E
þ
K
1
q
C
3
K
3
þ
K
3
1
ð
Þ
q
C
3
q
P
¼
K
q
C
1
2
where the variables and initial conditions are given as before. As before,
q
E
is eliminated
from Eq. (8.131) by using
q
C
1
q
C
2
q
C
3
, and a steady-state analysis is used to
determine the reaction rate—that is, we let
q
E
¼
E
0
q
S
¼
q
M
¼
q
C
1
¼
q
C
2
¼
q
C
3
¼
0
:
Keener and Sneyd
give the reaction rate as
V
max
1
V
¼
ð
8
:
132
Þ
þ
K
3
q
M
K
3
þ
K
1
K
1
q
S
1
8.6.3 Cooperative Reactions
Our last topic in this chapter deals with cooperative reactions involving a sequence of
biochemical reactions. The enzyme is able to bind with more than one substrate, and with
each successive binding, the reaction of the next reaction is impacted. These reactions can
have a sigmoidal reaction rate, allowing for a more precise control of the reaction, either
