Derivation of the Michaelis-Menten Equation
The Michaelis-Menten equation is an important equation in biochemistry and as
such it is imperative that you understand the derivation of this equation. By
understanding the derivation, you will have insight into the assumptions that went into
this model, and therefore you will have a better appreciation for the proper use of this
equation as well as the limitations of this model. In the following sections you will see
two different derivations of the Michaelis-Menten equation. When one is learning a
subject for the first time, it often helps to have the same or similar information presented
from alternative perspectives. One way might be clearer to you whereas the other way
might be clearer to someone else. That is ok! You should familiarize yourself with both
approaches, and then settle on the one that you prefer. The first derivation was adapted
from “An Introduction to Enzyme Kinetics” by Addison Ault (J. Chem. Ed. 1974, 51,
381 – 386).
First Derivation. We start with the kinetic mechanism shown in equation (eq) 1:
k1 k3
E + S ES E + P
k2 (1)
In eq 1, E is enzyme, S is substrate, ES is the enzyme-substrate complex, and P is
product. This equation includes the assumption that during the early stages of the reaction
so little product is formed that the reverse reaction (product combining with enzyme and
re-forming substrate) can be ignored (hence the unidirectional arrow under k3). Another
assumption is that the concentration of substrate is much greater than that of total enzyme
([S] >> [Et]), so it can essentially be treated as a constant.
1
, From General Chemistry we can equate the rate of this process (k3[ES]) to the
change in product concentration as a function of time (d[P]/dt), or, equivalently, we can
designate the rate with an italicized v (v) as follows in eq 2:
d [P]
v k 3 [ES] (2)
dt
Because the concentration of the enzyme•substrate complex ([ES]) cannot be measured
experimentally, we need an alternative expression for this term. Because the enzyme that
we add to the reaction will either be unbound (E) or bound (ES) we can express the
fraction of bound enzyme as follows:
[ES] [ES]
(3)
[E t ] [ES] [E ]
In eq 3 Et is the concentration of total enzyme, and the other variables are as defined
above. If we multiply both sides of eq 3 by Et we arrive at eq 4:
[E t ][ES]
[ES] (4)
[ES] [E]
If we multiply the numerator and denominator of the right-hand side of eq 4 by 1/[ES],
we are, in effect, multiplying by one and we do not change the value of this expression.
When we do this we obtain eq 5:
[E t ]
[ES] (5)
[E]
1
[ES]
We have almost achieved our goal of isolating [ES]. Next, we need to come up with an
alternative expression for the ratio [E]/[ES]. We do this by recalling that a major
assumption in enzyme kinetics is the steady-state assumption. Basically, it says the rate
of change of [ES] as a function of time is zero: d[ES]/dt = 0. Another way to express the
2
The Michaelis-Menten equation is an important equation in biochemistry and as
such it is imperative that you understand the derivation of this equation. By
understanding the derivation, you will have insight into the assumptions that went into
this model, and therefore you will have a better appreciation for the proper use of this
equation as well as the limitations of this model. In the following sections you will see
two different derivations of the Michaelis-Menten equation. When one is learning a
subject for the first time, it often helps to have the same or similar information presented
from alternative perspectives. One way might be clearer to you whereas the other way
might be clearer to someone else. That is ok! You should familiarize yourself with both
approaches, and then settle on the one that you prefer. The first derivation was adapted
from “An Introduction to Enzyme Kinetics” by Addison Ault (J. Chem. Ed. 1974, 51,
381 – 386).
First Derivation. We start with the kinetic mechanism shown in equation (eq) 1:
k1 k3
E + S ES E + P
k2 (1)
In eq 1, E is enzyme, S is substrate, ES is the enzyme-substrate complex, and P is
product. This equation includes the assumption that during the early stages of the reaction
so little product is formed that the reverse reaction (product combining with enzyme and
re-forming substrate) can be ignored (hence the unidirectional arrow under k3). Another
assumption is that the concentration of substrate is much greater than that of total enzyme
([S] >> [Et]), so it can essentially be treated as a constant.
1
, From General Chemistry we can equate the rate of this process (k3[ES]) to the
change in product concentration as a function of time (d[P]/dt), or, equivalently, we can
designate the rate with an italicized v (v) as follows in eq 2:
d [P]
v k 3 [ES] (2)
dt
Because the concentration of the enzyme•substrate complex ([ES]) cannot be measured
experimentally, we need an alternative expression for this term. Because the enzyme that
we add to the reaction will either be unbound (E) or bound (ES) we can express the
fraction of bound enzyme as follows:
[ES] [ES]
(3)
[E t ] [ES] [E ]
In eq 3 Et is the concentration of total enzyme, and the other variables are as defined
above. If we multiply both sides of eq 3 by Et we arrive at eq 4:
[E t ][ES]
[ES] (4)
[ES] [E]
If we multiply the numerator and denominator of the right-hand side of eq 4 by 1/[ES],
we are, in effect, multiplying by one and we do not change the value of this expression.
When we do this we obtain eq 5:
[E t ]
[ES] (5)
[E]
1
[ES]
We have almost achieved our goal of isolating [ES]. Next, we need to come up with an
alternative expression for the ratio [E]/[ES]. We do this by recalling that a major
assumption in enzyme kinetics is the steady-state assumption. Basically, it says the rate
of change of [ES] as a function of time is zero: d[ES]/dt = 0. Another way to express the
2
