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Enzyme Kinetics
In 1913, Michaelis and Menten published the idea that enzymes and
substrates formed reasonably stable complexes with their
substrates that then subsequently undergo reaction. This is
represented by the following reaction scheme that is in every
biochemistry textbook.
E + S
E
.
S
E + P
K
M
k
cat
This mechanism states that enzyme and substrate are at true
equilibrium with the E
.
S complex, and that the E
.
S complex
undergoes reaction to E and P. How do we get from this mechanism
to the MichaelisMenten equation, shown below?
MichaelisMenten Equation:
d[P]
dt
=
v
i
=
k
cat
.
[E]
tot
.
[S]
K
M
+
[S]
=
V
max
.
[S]
K
M
+
[S]
(1)
First, we can simply write down by inspection a differential rate
equation for the appearance of products:
d[P]
dt
=
[ES] k
cat
(
2
)
In and of itself, this equation does not do us a lot of good
because we can not experimentally measure [E
.
S].
What can we measure? We can easily measure [E]
tot
by one of
several means for measuring protein concentration and [S], but
not [E] or [E
.
S]. Thus, we have to get the equation in terms of
[E]
tot
and [S].
The following derivation is the simplest possible for the
MichaelisMenten mechanism and does not involve the application
of the steady state assumption.
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We take note of two relationships:
K
M
=
[E][S]
[E
.
S]
and
[E] = [E]
tot
 [E
.
S]
Stick the second into the first and you get:
K
M
=
([ E]
tot
−
[E
.
S])[S]
[E
.
S]
Rearranging this to solve for [E
.
S]:
K
M
[E
.
S] = ([E]
tot
[E
.
S]) [S] = [E]
tot
[S]  [E
.
S] [S]
then
K
M
[E
.
S] + [E
.
S] [S] = [E]
tot
[S]
Factoring out [E
.
S],
[E
.
S] (K
M
+ [S]) = [E]
tot
[S]
Dividing both sides by (K
M
+ [S]) we get
[E
.
S]
=
[E]
tot
[S]
K
M
+
[S]
We've got [E
.
S] in terms of measurables. Now stick this
expression for [E
.
S] into (2) and what do you get?
d[P]
dt
=
v
i
=
k
cat
.
[E]
tot
.
[S]
K
M
+
[S]
You have now derived the MichaelisMenten equation.
What are the assumptions of this mechanism and the derivation?
The main one is that E and S are at equilibrium with E
.
S. The
second one is that the catalytic step is irreversible. The latter
3
is pretty good if there is no product present, since product
coming off the enzyme will be approximately infinitely diluted.
The first is not always so good, and we examine this below
(BriggsHaldane mechanism). (The unmentioned assumption is,
again, that [S]
free
= [S]
tot
).
The MichaelisMenten equation has the form of a rectangular
hyperbola. It has a saturation term, [S]/(K
M
+ [S]), that gives
the fraction of the total enzyme in the reactive ES form. This is
multiplied by [E]
tot
to give [ES], which is then multiplied by
k
cat
, the rate constant for the reaction of the ES complex.
The [S]/(K
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This note was uploaded on 05/04/2008 for the course BIS 102 taught by Professor Hilt during the Fall '08 term at UC Davis.
 Fall '08
 Hilt
 Biochemistry, Enzymes

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