Biomedical Engineering Reference
In-Depth Information
We can eliminate [
E
] and [
ES
] from (7.22), (7.23), and (7.24) and we deduce
the rate of the reaction
k E
[
]
2
0
(7.25)
V
=
æ
k
+
k
ö
1
2
-
1
1
+
ç
÷
è
k
ø
[ ]
S
1
Introducing the notations
V
=
k E
[
]
(7.26)
max
2
0
and
k
+
k
2
-
1
K
=
(7.27)
m
k
1
We obtain the Michaelis-Menten law
V
max
(7.28)
V
=
K
S
m
1
+
[ ]
By remarking that the concentration of substrate [
S
] decreases with the concen-
tration of product [
P
] according to
[ ]
S
=
[ ]
S
-
[
P
]
(7.29)
0
and if we recall from (7.21) that
V
is the rate of production of
P
, integration of (7.28)
gives the relation between the concentration of product [
P
] and substrate [
S
]
[ ]
S
0
(7.30)
V
t
=
[
P K Ln
S
]
+
max
m
[ ]
-
[
P
]
0
The constant has been adjusted so to have [
P
] = 0 at
t
= 0. Relation (7.30) is
implicit. The kinetics of
P
derived from (7.30) is schematically represented in Figure
7.16.
It is easy to see that the Michaelis-Menten law can be cast under the form
æ
ö
1
1
K
=
1
+
(7.31)
ç
÷
V V
è
[ ]
S
ø
max
This form is called the Lineweaver-Burk expression of the Michaelis-Menten
relation. It is convenient to determine the kinetic constants
K
m
and
V
max
. If we
rewrite (7.31) under the form
1
1
K
1
[ ]
=
+
V V
V
S
max
max
we see that the plot of the reciprocal velocity 1
/V
against the reciprocal substrate
concentration 1
/
[
S
] is linear; the intercept is 1
/V
max
and the slope is
K
m
/
V
max
(Figure 7.17).
