Reversible Michaelis-Menten Equation

Overview

The irreversible Michaelis-Menten equation is very important to the study of enzyme kinetics. However, it is not applicable to many complicated experimental systems, especially in vivo situations. In many situations, the reaction is reversible and/or reaction product is present at a non-negligable concentration. In these cases, the reverse reaction must explicitely be considered.

Assumptions

The MM equation is based on a set of assumptions that must be appreciated. Significant violation of these assumptions can invalidate the use of the model.

• Mass action kinetics is valid
• Enzyme complexes are at pseudo steady state

Derivation

Assume a reaction mechanism based on the following figure.

Let [S] represent the concentration of free substrate. Let [E] represent the concentration of free enzyme. Let [ES] represent the concentration of enzyme/substrate complex. Let [P] represent the concentration of free product. Let [E]_T represent the total amount of enzyme in the system.

Write a dynamic mass balance equation for the enzyme/substrate complex based on mass action kinetics.

Assume the concentration of enzyme/substrate complex is at pseudo steady state.

Using the fact that [E]_T = [E] + [ES], substitute for [E].

Rearrange the equation and solve for [ES].

Divide the numerator and denomenator by k₁.

Define v = k₃[ES] and substitute in [ES].

Define K_M = (k₂ + k₃)/k₁ and V = k₃[E]_T and substitute these.

This is the Michaelis-Menten equation. It is important to recognize that even though we have derived the MM equation assuming a certain reaction mechanism, the mathematical form of the MM equation can used to describe many other mechanisms.