section 6.2
Catalysis
91
dependence on [ET] alone when [S] is large.
Furthermore, the rate constants are
kcat
for the
first-order reaction and
kcat/Km
for the second-order
reaction in which
Km
is the value of [S] when
v = l/2Vmax. Since /ccat is the rate constant for the
conversion of ES to product and enzyme, it is a
measure of how active the enzyme is in carrying out
the reaction
after the enzyme has bound to the
substrate.
The constant
kcat/Km,
however, measures
the total activity of the enzyme, which includes the
ability of the enzyme to bind to a particular substrate.
11.
Km
(the
Michaelis constant)
equals [S] when the
substrate concentrations results in the velocity that is
half-maximal (Vmax/2). To demonstrate that
Km
= [S] when v = Vmax/2, we substitute for v in
Equation (6.4):
Vmax
_
Vmax[S]
~2~ ~ Km
+ [S]
(Km + [S])(Vmax) = 2Vmax[S]
Km
+ [S] = 2[S]
whence, as indicated above,
F IG U R E 6 -5
Two linear plots for the same data obtained for an enzyme-catalyzed
reaction that obeys Michaelis-Menten kinetics.
then
Km =
[S] when
V =
From Equation (6.4), it can be seen that v increases
with increase in Vmax at constant [S] and
Km
and that
v decreases with increase in
Km
at constant [S] and
Vmax*
Linear Plots for Michaelis-Menten Expression
Because straight-line plots are easier to evaluate than
curves, it is convenient to reformulate Equation (6.4) to
yield straight-line plots. Two such reformulations can
be performed. The
Lineweaver-Burk plot
is a double-
reciprocal plot, obtained by taking reciprocals of both
sides of Equation (6.4) and rearranging:
1 =
Km
+ [S]
v
[S]Vmax
1
=
Km
1
1
v
Vmax [S]
Vmax
(6.5)
(
6
.
6
)
According
to
Equation
(
6
.
6
),
if the
system
obeys
Michaelis-Menten kinetics, a plot of 1/v versus 1/[S] is
a straight line having a slope of
Km/
Vmax and an intercept
of 1 /Vmax on the 1/v axis. Furthermore, the intercept on
the 1/[S] axis (occurring when 1/v = 0) is —
1
/
Km.
This
is seen by setting 1 /v equal to zero. If
0
= —
—
+
—
,
V„ax [51
Vmax’
J _
=
(1/ Vmax)
[S]
(ATm/Vmax)
1
_
1
[S
] ~ K ^ '
Figure 6-5a shows a Lineweaver-Burk plot. The disad-
vantage of this plot is that it depends on less precisely
determined points obtained at low values of [S], whereas
the more accurate points obtained at high values of [S]
cluster and so are less valuable in establishing the linear
plot.
The
Eadie—Hofstee plot
represents an improvement
over the Lineweaver-Burk plot in that the experimen-
tal points are usually more equally spaced. We obtain
the equation for the Eadie-Hofstee plot by rearranging
Equation (6.4):
'*m + [S]\ _
VI
[S]
J
- ^max
,
-
Т/
У
[S]
)
^
—
*inax
v =
~
K m
1
is]1
+ Vmax
( 6 J )
From Equation (6.7), a plot of v versus v / [S] has a slope
of —
Km,
an intercept of Vmax on the v axis, and an intercept
of
Vmax/Km
on the V/ [S] axis (Figure 6-5b).
