Environmental Engineering Reference
In-Depth Information
The enzyme-substrate complex E-S is assumed to be at pseudo-steady-state. Conser-
vation of enzyme concentration is evident from the above mechanism, and we have
[
E
]
0
=[
E
]+[
E
−
S
]
. For the substrate, we have
[
S
]
0
=[
S
]+[
P
]
, with the caveat
that
[
E
]≈[
E
−
S
][
S
]
. The rate of formation of the product is given by
d
]
d
t
=−
[
P
d
[
]
d
t
=
S
r
=
k
2
[
E
−
S
]
.
(5.196)
The rate of formation of [E-S] is given by
d
[
E
−
S
]
=
[
]
0
−[
−
]
[
]−
(k
−
1
+
[
−
]
k
1
(
E
E
S
)
S
k
2
)
E
S
.
(5.197)
d
t
Applying the pseudo steady-state approximation,
k
1
[
]
0
[
]
E
S
[
−
]
ss
=
E
S
K
2
.
(5.198)
k
1
[
S
]+
k
−
1
+
The expression for the rate can now be written as
V
max
[
S
]
r
=
(5.199)
K
m
+[
S
]
with
V
max
=
k
2
)/k
1
. The equation given above is called the
Michaelis-Menten equation
. Note the similarity of the equation to that derived for
k
2
[E]
0
and
K
m
=
(k
+
−
1
V
max
V
max
/2
Slope =
V
max
[S]/
K
m
[S]
1/2
=
K
m
[S]/mol·m
-3
FIGURE 5.19
Variation in an enzyme-catalyzed reaction rate with substrate concentration as
per the Michaelis-Menten equation.
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