Biomedical Engineering Reference
In-Depth Information
equilibrium to either the products or the reactants (Chapter 9) pending on the step rate
were derived. In this case, it is intuitive to rewrite Eqns
(E10-2.54)
and
(E10-2.56)
as
r
P
1
max
½
M
1
r
P
1
¼
(E10-2.57)
K
M
1
þ K
f
P
1
½
P
1
þ½
M
1
r
P
1
max
½
M
1
r
P
2
max
½
M
1
r
M
1
¼
1
þ
(E10-2.58)
K
M
1
þ K
f
P
1
½
P
1
þ½
M
K
M
2
þ K
f
P
2
½
P
2
þ½
M
1
where [M
3
] has been related to [M
1
] via equilibrium, all the K's are rate constants.
This example shows that the metabolic pathways can be applied to derive at kinetics of
the metabolism.
Example 10-3. Fluxes in Sequential feedback control of branched pathways.
Derive a flux expression for the production of P
1
for sequential feedback as shown in
Fig. 10.15
c. Assume PSSH is applicable.
Solution:
In the previous example, we have shown how the kinetics can be derived from metabolic
pathway via reaction rate approach. In this example, we examine the flux approach to the
kinetics. To derive a reasonable flux expression from the simplified pathway given in
Fig. 10.15
c, we first identify all the nodes and connections as shown in
Fig. E10-3.1
.We
next identify all the fluxes:
J
1
0
¼ J
1
¼ k
1
½
M
1
½
E
2
k
1
½
M
E
2
(E10-3.1)
1
J
2
0
¼ J
2
¼ k
1c
½
M
E
2
(E10-3.2)
1
J
3
0
¼ J
3
¼ k
2
½
M
2
½
E
3
k
2
½
M
E
3
(E10-3.3)
2
J
4
0
¼ J
4
¼ k
2c
½
M
E
3
(E10-3.4)
2
J
5
¼ J
6
¼ k
f1
½
M
3
½
E
2
k
f1
½
M
E
2
(E10-3.5)
3
J
7
0
¼ J
7
¼ k
3
½
M
3
½
E
4
k
3
½
M
E
4
(E10-3.6)
3
J
8
0
¼ J
8
¼ k
3c
½
M
E
4
(E10-3.7)
3
J
9
¼ k
4
½
M
4
(E10-3.8)
P
1
E
4
P
1
E
4
11
11
10
10
E
4
E
4
M
3
E
4
M
3
E
4
M
4
M
4
P
1
P
1
2
2
3
3
1
1
4
4
8
8
9
9
7
7
M
1
M
1
M
1
E
2
M
1
E
2
M
2
M
2
M
2
E
3
M
2
E
3
M
3
M
3
E
2
E
2
E
3
E
3
12
12
13
13
14
14
M
3
E
5
M
3
E
5
P
2
P
2
M
5
M
5
6
6
5
5
E
5
E
5
15
15
M
3
E
2
M
3
E
2
16
16
P
2
E
5
P
2
E
5
FIGURE E10-3.1
Schematic of a sequential feedback control of branched pathways showing all the nodes and
flux connections.
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