Biomedical Engineering Reference
In-Depth Information
Figure 8.25
shows the comparison between the full solutions and those from PSSH treat-
ment for the particular case of k
1
C
E0
¼
10 k
c
. One can observe that solu-
tions based on PSSH are surprisingly close to the full solutions. Although we have used
the argument that all the intermediates remain at steady state (pseudosteady state) and
the solutions do not appear to be the case, the computed enzyme distributions (dashed lines)
agree quite well with the full solutions (solid lines) when k
c
t
10 k
c
and k
3
C
E0
¼
0.05.
To show the reasonable agreement with the solutions from the PSSH approximation to the
process,
Fig. 8.26
shows a comparison between the full solutions (solid lines) and those from
PSSH treatment for the case of k
1
C
E0
¼
>
10 k
c
(dashed lines). One can observe
that the agreement between the full solutions and the PSSH solutions is rather good at “long”
times when k
c
t
k
c
and k
3
C
E0
¼
>
0.3.
8.7.2. Fast Equilibrium Step Approximation
We next examine how Fast Equilibrium Step (FES) assumption is applied to approxi-
mate the enzyme reaction network. If the catalytic reaction is the rate-limiting step
(Michaelis
e
Menten), we have the overall rate for reaction (8.104):
r ¼ r
2
¼ k
c
C
SE
k
c
C
PE
(8.133)
The other two steps (8.103) and (8.105) are in equilibrium,
0 ¼ r
1
¼ k
1
C
S
C
E
k
1
C
SE
(8.134)
0 ¼ r
3
¼ k
3
C
PE
k
3
C
P
C
E
(8.135)
which can be rearranged to give
k
1
k
1
C
SE
¼
C
S
C
E
¼ K
S
C
S
C
E
(8.136)
and
k
3
k
3
C
PE
¼
C
B
C
E
¼ K
P
C
P
C
E
(8.137)
Total enzyme balance:
C
E
0
¼ C
E
þC
SE
þC
PE
¼ C
E
þK
S
C
S
C
E
þK
P
C
P
C
E
(8.138)
Thus,
C
E
0
1þK
S
C
S
þK
P
C
P
C
E
¼
(8.139)
Substituting (
8.135
) into
(8.136), (8.137)
, and then (
8.133
), we obtain
K
S
C
S
1þK
S
C
S
þK
P
C
P
C
SE
¼
(8.140)
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