Biomedical Engineering Reference
In-Depth Information
the amount of enzyme is very small in comparison to the substrate and product and that
K
K
1
:
From
q
ES
¼
0 and Eq. (8.34), we have
0
2
¼
K
q
S
q
E
K
1
ð
þ
K
Þ
q
ES
ð
8
:
36
Þ
1
2
Since
q
E
ð
0
Þ¼
q
E
þ
q
ES
, we substitute
q
E
¼
q
E
ð
0
Þ
q
ES
into Eq. (8.36) to give
0
¼
K
q
S
q
E
ð
ð
0
Þ
q
ES
Þ
K
1
ð
þ
K
Þ
q
ES
ð
8
:
37
Þ
1
2
and after rearranging,
Þ
K
1
q
S
þ
K
1
þ
K
2
K
q
S
q
E
ð
0
Þ
q
S
þ
K
1
þ
K
2
K
1
q
S
q
E
ð
0
Þ
q
S
þ
K
M
q
S
q
E
ð
0
q
E
ð
0
Þ
1
q
ES
¼
Þ
¼
¼
Þ
¼
ð
8
:
38
Þ
ð
ð
þ
K
q
S
1
K
M
¼
K
1
þ
K
2
K
1
where
. The constant,
K
M
, is called the Michaelis constant, as mentioned in the
beginning of this chapter.
The maximum complex
ES
from Eq. (8.38), with
q
s
¼
q
s
ð
0
Þ
, is approximately
q
S
¼
q
S
ð0Þ
q
E
ð
0
Þ
q
E
ð
0
Þ
q
ES
max
¼
ð
8
:
39
Þ
þ
K
q
S
þ
K
M
q
S
ð
1
1
0
Þ
Further, since
q
E
¼
q
E
ð
0
Þ
q
ES
, we have
Þ
q
E
ð
0
Þ
q
E
ð
0
Þ
¼
q
E
¼
q
E
ð
0
ð
8
:
40
Þ
þ
q
S
K
M
þ
K
M
q
S
1
1
To find a quasi-steady-state approximation for
q
S
, we use Eq. (8.34);
q
S
¼
K
1
q
S
q
E
þ
:
K
1
q
ES
¼
q
ES
þ
K
2
q
ES
With
q
ES
¼
0 and
q
ES
from Eq. (8.38), we have
q
ES
¼
K
2
q
E
ð
0
Þ
q
S
¼
K
ð
8
:
41
Þ
2
þ
K
M
q
S
1
Now, we have only two parameters that describe the change in substrate in Eq. (8.41),
which takes the place of the set of differential equations in Eq. (8.34). Moreover, we will
show that these two parameters can be estimated directly from data. Rearranging
Eq. (8.41) gives
þ
K
M
q
S
1
dq
S
¼
K
q
E
ð
0
Þ
dt
ð
8
:
42
Þ
2
and after integrating both sides of Eq. (8.42) yields
0
1
q
S
ð
0
Þ
@
A
¼
K
2
q
E
ð
ð
q
S
ð
0
Þ
q
S
Þ þ
K
M
ln
0
Þ
t
q
S
or
ð
8
:
43
Þ
0
0
1
1
1
K
2
q
E
ð
q
S
ð
0
Þ
@
@
A
A
t
¼
ð
q
S
ð
0
Þ
q
S
Þ þ
K
M
ln
0
Þ
q
S
