Enzyme-catalyzed reactions are common in biochemical processes. We discuss the model that Michaelis and Menten have proposed in 1913 to describe the reaction kinetics. It is the most widely used model in systems biology and it is called after the inventors of the model: Michaelis-Menten kinetics.

In this model, a substrate S and a free enzyme E first reversibly combine to a substrate-enzyme complex SE. The complex SE may also convert into a product P and the free enzyme. The reaction scheme is represented as follows: \[\text{S}+\text{E }{\mathop{\rightleftharpoons}\limits_{k_{2}}^{k_1}} \text{ SE }{\mathop{\longrightarrow}\limits_{}^{k_3}} \text{ E} + \text{P}\] It is assumed in all reaction steps are elementary.

There are two ways to proceed:

1. an instantaneous equilibrium of substrate and complex;
2. a quasi steady state approximation.

The methods lead to a similar result. We discuss them both. Here, we introduce the following abbreviations for the concentration: \[s=[\text{S}], e =[\text{E}], c = [\text{SE}], p=[\text{P}]\] The differential equations that define the kinetics are \[\begin{aligned} \frac{\dd e}{\dd t}&= (k_{2}+k_3)c - k_1 s\,e \\ \\ \frac{\dd s}{\dd t}&= k_{2}c - k_1s\,e\\ \\ \frac{\dd c}{\dd t}&= k_1s\,e-(k_2+k_{3})c \\ \\ \frac{\dd p}{\dd t}&= k_{3}c\end{aligned}\] The total amount of enzyme, free or bound in the complex, is constant, say \(e_0\). So we have \(c+e = e_0\).

1. Instantaneous equilibrium

Leonor Michaelis and Maud Menten assumed that the equilibrium between substrate complex is established very quickly. This means that they assumed that the rate constants \(k_1\) and \(k_{2}\) are very large compared to the rate constant \(k_3\). In equilibrium, the second differential equation leads under this assumption to \[k_1s\,e=k_{2}c\] Please note that this is in fact an approximation! Substitution of \(c+e=e_0\) yields (check yourself!): \[c=\frac{e_0s}{K_1+s}\] with constant \[K_1=\frac{k_{2}}{k_1}\tiny.\] The reaction rate \(r\) is given by \[r=\frac{\dd p}{\dd t}=k_3c=\frac{{k_3}{e_0}s}{K_1+s}=\frac{V_{\max}\cdot s}{K_1+s}\] where \(V_{\max}=k_3e_0\) is the maximum reaction rate, which is achieved at high substrate concentrations (saturation).

2. The quasi steady state approximation

George Briggs and John Haldane assumed that the formation rate and conversion rate of the complex must be equal. In other words, \[\frac{\dd c}{\dd t}=0\] From the third reaction equation follows \[k_1s\,e=(k_2+k_{3})c\] Substitution of \(c+e=e_0\) yields (check yourself): \[c=\frac{{e_0}s}{K_m+s}\] with the so-called Michaelis-Menten constant \[K_m=\frac{k_2+k_{3}}{k_1}\tiny.\] The reaction rate \(r\) is then given by \[r=\frac{V_{\max}\cdot s}{K_m+s}\] where \(V_{\max}=k_3e_0\). As you can see, a similar formula as before, but now with another constant in the denominator. The constant \(K_1\) which had been used by Michaelis and Menten, is a limiting case of the Michaelis-Menten constant \(K_m\).