Michaelis–Menten kinetics
Saturation curve for an enzyme showing the relation between the concentration of substrate and rate.
Single-substrate mechanism for an enzyme reaction.
k1,
k-1 and
k2 are the rate constants for the individual steps.
As enzyme-catalysed reactions are saturable, their rate of catalysis does not show a linear response to increasing substrate. If the initial rate of the reaction is measured over a range of substrate concentrations (denoted as [S]), the reaction rate (
v) increases as [S] increases, as shown on the right. However, as [S] gets higher, the enzyme becomes saturated with substrate and the rate reaches
Vmax, the enzyme's maximum rate.
The
Michaelis–Menten kinetic model of a single-substrate reaction is shown on the right. There is an initial
bimolecular reaction between the enzyme E and substrate S to form the enzyme–substrate complex ES. Although the enzymatic mechanism for the
unimolecular reaction 
can be quite complex, there is typically one rate-determining enzymatic step that allows this reaction to be modelled as a single catalytic step with an apparent unimolecular rate constant
kcat. If the reaction path proceeds over one or several intermediates,
kcat will be a function of several elementary rate constants, whereas in the simplest case of a single elementary reaction (e.g. no intermediates) it will be identical to the elementary unimolecular rate constant
k2. The apparent unimolecular rate constant
kcat is also called
turnover number and denotes the maximum number of enzymatic reactions catalysed per second.
The
Michaelis–Menten equation[9] describes how the (initial) reaction rate
v0 depends on the position of the substrate-binding
equilibrium and the rate constant
k2.
(Michaelis–Menten equation)
with the constants
This Michaelis–Menten equation is the basis for most single-substrate enzyme kinetics. Two crucial assumptions underlie this equation (apart from the general assumption about the mechanism only involving no intermediate or product inhibition, and there is no
allostericity or
cooperativity). The first assumption is the so called quasi-steady-state assumption (or pseudo-steady-state hypothesis), namely that the concentration of the substrate-bound enzyme (and hence also the unbound enzyme) changes much more slowly than those of the product and substrate and thus the change over time of the complex can be set to zero
![d{[}ES{]}/{dt} \; \overset{!} = \;0](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_uyRtBgPqUWwWw9xUSnho8AKmjiRlGiRqsbWSC38wgcM9L0Uw3bhiifW0ZFr8IbKAwGAAvAJdFKc4B5S3deDEiwamJJEb-lZrLtxQCn6Z1-FGz2Sdf2qZ7vFk835u2z0C62wOLwSRlYEd9R1D7K=s0-d)
. The second assumption is that the total enzyme concentration does not change over time, thus
![{[}E{]}_\text{tot} = {[}E{]} + {[}ES{]} \; \overset{!} = \; \text{const}](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_vpYQmCn7Q3Z3S2uin4CIp3U9aDFnYtdfOtOLQUNs5w9MDLcTEG14xAKwk2DVQSLavgetQSMeeiLidkGDsGfJMPFyLj8_89nyWhqSv9os58i2DTFVwEPfN93WYQTmenUnfMVpIsdYH8reAIXP5W9Q=s0-d)
. A complete derivation can be found
here.
The Michaelis constant
KM is experimentally defined as the concentration at which the rate of the enzyme reaction is half
Vmax, which can be verified by substituting [S] =
Km into the Michaelis–Menten equation and can also be seen graphically. If the rate-determining enzymatic step is slow compared to substrate dissociation (

), the Michaelis constant
KM is roughly the
dissociation constant KD of the ES complex.
If
[S] is small compared to
KM then the term
![[S] / (K_M + [S]) \approx [S] / K_M](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_uGC6AxnMCN9TvlKizna0jD6i-6DeFXllheVmrEdKzCV19fP51xAfvS5sQuRXdzcT2DbVh8NG4WsGQdGJt4fPHrQGwVhVYbvzGmQzcdjOPOg_DhTNMHzpYxByrDha3aUJClLikFpquydGw-xqWfEg=s0-d)
and also very little ES complex is formed, thus
![[E]_0 \approx [E]](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_t4oLNd7b40NPqDhkApnTKz8jjVhcUFdS1uTsIgxvRvs2gXndYyG-TW5YWBEY2q2Vij2Lk159KMMKUPyxpYeRBsswQRu4LhWTmktLEho_-8uSlZhXGSE1wSiRt758atg3OO0eXav1PJIK8XQazEww=s0-d)
. Therefore, the rate of product formation is
Thus the product formation rate depends on the enzyme concentration as well as on the substrate concentration, the equation resembles a bimolecular reaction with a corresponding pseudo-second order rate constant
k2 / KM. This constant is a measure of catalytic efficiency. The most efficient enzymes reach a
k2 / KM in the range of 10
8 - 10
10 M−1 s−1. These enzymes are so efficient they effectively catalyse a reaction each time they encounter a substrate molecule and have thus reached an upper theoretical limit for efficiency (diffusion limit); these enzymes have often been termed
perfect enzymes.
[10]
