Biomedical Engineering Reference
In-Depth Information
8.7.1. Pseudosteady-State Hypothesis
We next examine the approximation by PSSH based on the same set of parameters. The
PSSH assumes that the rate of change of intermediates be zero. In this case,
0 ¼ r
SE
¼ r
1
r
2
¼ k
1
ðC
E
C
S
C
SE
=K
S
Þk
c
ðC
SE
C
PE
K
S
=K
P
=K
C
Þ
(8.117)
0 ¼ r
PE
¼ r
2
r
3
¼ k
c
ðC
SE
C
PE
K
S
=K
P
=K
C
Þk
3
ðC
PE
=K
P
C
E
C
B
Þ
(8.118)
This is equivalent to saying that r
1
¼
r
3
. Referring to
Fig. 8.18
, the implication is that at
PSSH, the fluxes passing through all the nodes are equal. In this case, the overall reaction
rate or total flux is the same as any of individual ones (in series).
Equations
(8.117) and (8.118)
can be rearranged to yield
r
2
¼
C
SE
ðk
1
=K
S
þk
c
ÞþC
PE
k
c
K
S
=K
P
=K
C
þC
E
k
1
C
S
¼ 0
(8.119)
C
SE
k
c
C
PE
ðk
c
K
S
=K
P
=K
C
þk
3
=K
P
ÞþC
E
k
3
C
P
¼ 0
(8.120)
Equations
(8.119) and (8.120)
can be solved to give
C
SE
¼
ðK
C
k
1
3
þK
S
k
c
ÞC
S
þK
C
k
1
C
P
1
K
S
C
E
(8.121)
K
C
k
1
3
þK
S
k
1
þ k
1
1
c
C
PE
¼
ð
k
1
þ
K
S
k
c
Þ
C
P
þ
k
1
C
S
3
K
P
C
E
(8.122)
K
C
k
1
3
þK
S
k
1
þ k
1
1
c
Total enzyme balance leads to
C
E
0
¼ C
E
þC
SE
þC
PE
¼ C
E
þ
ðK
C
k
1
3
þK
S
k
1
ÞC
S
þK
C
k
1
C
P
c
1
K
S
C
E
K
C
k
1
3
þK
S
k
1
þ k
1
1
c
þ
ð
k
1
þ
K
S
k
c
Þ
C
P
þ
k
1
C
A
3
K
P
C
E
(8.123)
K
C
k
1
3
þK
S
k
1
þ k
1
1
c
Thus,
C
E
0
C
E
¼
(8.124)
1þ
ðK
S
K
C
k
1
3
þk
c
þk
1
K
P
ÞC
S
þðK
C
k
1
K
S
þ k
1
K
P
þK
S
k
c
K
P
ÞC
P
1
1
3
K
C
k
1
3
þK
S
k
1
þ k
1
1
c
Substituting Eqn
(8.124)
into Eqns
(8.121) and (8.122)
, we obtain the concentration of enzyme
that is associated with substrate or product.
The overall rate of reaction:
r ¼r
S
¼ r
1
¼ k
1
C
S
C
E
k
1
C
SE
(8.125)
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