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   Michaelis Menten Equation

Michaelis-Menten

A == B --- C

A reacts with E, and turns into B, with rate constant k1, which turns into C and regenerates E, with rate constant k2. Now, B can turn back into A and E, with rate constant k-1. The Michaelis-Menten mechanism for enzymic catalysis follows this path.

Analytical solutions to the equations for these reactions are not available, but numerical solutions may readily be found. The mechanism may be analysed using the Steady State Approximation to give the following expression for the rate.

Michaelis-Menten equation

where KM is called the Michaelis Constant.
Michaelis Constant

This expression is the Briggs-Haldane analysis of the reaction. The Michaelis-Menten analysis incorporates the additional assumption that k2 is much less than k-1, so the Pre-Equilibrium Approximation can be used instead of the Steady State Approximation. This leads to a very similar result, but KM has the simpler form k-1/k1. For a full account, see "Enzyme Structure and Mechanism", A. R. Fersht, 1988, pp98-101.

The rate equation can be integrated in two ways:

Approximation I: Re-introducing the assumption that [E] is small, set [A] = [A]0 - [C]. The Briggs-Haldane equation can now be integrated:
equation
This works well except at the beginning of the reaction, when [C] and [B] are comparable in size. For this region, it is better to use an alternative approximation.

Approximation II: Returning to the unsimplified rate equations, substitute [E] = [E]0 - [B] and [A] = [A]0. This will be a good approximation when the amount of product C is small. The equations can now be integrated.
equation

The Applet below demonstartes these two approximations. Approximation I works well, except at the beginning of the plot, when the amount of product is much smaller than the amount of starting material. In this initial region, Approximation II is much more effective, and this is the region which is usually of interest in studies of enzyme kinetics.

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