Environmental Engineering Reference
In-Depth Information
heterogeneous catalysis obeying the Langmuir isotherm. Figure 5.19 represents the
equation. The value of
r
used is generally the initial reaction rate. The limiting condi-
tions of the equation are of special interest. When
[
S
]
K
m
, that is, at high substrate
concentration,
r
→
V
max
,themaximumrateachieved.Atlowsubstrateconcentration,
[
(V
max
/K
m
)
[S],thatis,therateisdirectlyproportionaltothesubstrate
concentration. It is seen from the rate equation that when
r
S
]
K
m
,
r
→
]
1
/
2
,
which is the substrate concentration at which the enzyme reaction rate is one half
of the maximum rate. The parameters
V
max
and
K
m
therefore characterize the kinet-
ics of a reaction catalyzed by enzymes. The following example is an illustration of
how these constants can be obtained from experimental data on enzyme-catalyzed
substrate conversions.
→
V
max
/
2,
K
m
→[
S
E
XAMPLE
5.20 E
STIMATIONOFTHE
M
ICHAELIS
-M
ENTEN
K
INETIC
P
ARAMETERS
FROM
E
XPERIMENTAL
D
ATA
The enzymatic degradation of a pollutant to CO
2
and H
2
O was measured over a range
of pollutant concentration [S] (mol/L). The initial rates are given below:
[S] (mol/L)
r
(mM/min)
0.002
3.3
0.005
6.6
0.010
10.1
0.017
12.4
0.050
16.6
Obtain the Michaelis-Menten constants.
The Michaelis-Menten equation can be recast in three different forms to obtain linear
fits of the experimental data whereby the values of
K
m
and
V
max
can be determined
from the slopes and intercepts of the plots. These are:
a. Lineweaver-Burke plot: A plot of 1
/r
versus 1
/
[
S
]
:
1
r
=
K
m
V
max
1
[
S
]
+
1
V
max
.
(
5.200
)
The slope is
K
m
/V
max
and the intercept is 1
/V
max
.
b. Langmuir plot: A plot of
[
S
]
/r
versus [S]:
[
S
r
=
1
V
max
[
S
]+
K
m
V
max
.
(
5.201
)
The slope is 1/
V
max
and the intercept is
K
m
/V
max
.
c. Eadie-Hofstee plot: A plot of
r
versus
r/
[
S
]
:
r
=−
K
m
r
[
S
]
+
V
max
.
(
5.202
)
The slope is
−
K
m
and the intercept is
V
max
.
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