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by the difference of these two rates:
d
[
P
]
v
=
=
v
f
−
v
b
(2.5)
dt
=
k
1
[
A
][
B
]−
k
2
[
P
]
.
(2.6)
In this equation
k
1
is a second order rate constant and
k
2
is a first order rate constant.
2.3.1
Enzymatic Reactions and the Michaelis-Menten Equation
Enzymes are specific proteins that catalyze reactions. An enzyme can increase the
rate of a reaction up to 10
12
-fold [
7
], compared to the spontaneous reaction with-
out the enzyme. The enzyme first binds to its substrate (reactant), forms a complex
(an enzyme-substrate complex), and performs a chemical operation on it. Then it
releases from the complex, resulting in conversion of the substrate into a product.
Enzymes stay unchanged after the reaction. Some enzymes bind to a single substrate
while others can bind to multiple substrates and combine them to produce a final
product.
Consider now the enzyme catalyzed reaction in Eq. (
2.7
), which involves three
individual reactions.
S
k
1
k
2
ES
k
3
E
+
−→
P
+
E
.
(2.7)
k
1
−→
Thefirstreactionis
E
ES
, where an enzyme
E
binds to a substrate
S
and forms
an enzyme-substrate complex
ES
with an associated rate constant
k
1
. Since this is a
reversible reaction,
ES
can break down into
E
and
S
+
S
ES
k
2
. The associated
rate constant for this backward reaction is
k
2
. The third reaction is
ES
k
3
(
−→
E
+
S
)
E
in
which the enzyme
E
releases from
ES
, producing a product
P
with a rate constant
k
3
.
The differential equation describing the dynamics of the concentration of the
enzyme-substrate complex
ES
is the difference between the gain and loss terms.
ES
is produced with the first reaction and consumed with the second and third reactions.
Hence, we have
−→
P
+
d
[
ES
]
=
k
1
[
E
][
S
]−
(
k
2
+
k
3
)
[
ES
]
.
(2.8)
dt
The dynamics of the product
P
are modeled by Eq. (
2.9
). This equation has a single
term, since
P
is produced by the third reaction and is not consumed by any of the
reactions.
d
[
P
]
=
k
3
[
ES
]
.
(2.9)
dt
If the total concentration of the enzyme stays constant over the duration of this reac-
tion, we can write,
E
0
=[
E
]+[
ES
]
,
(2.10)
where
E
0
represents the initial enzyme concentration.
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