*Derivation of Michaelis-Menten equation*

**Contents**

- Outcomes
- Introduction
- The equilibrium approximation
- The steady state approximation
- Further reading
- Return to module guide

At the end of this section, you should be able to

- State and justify the assumptions on which the derivation of the Michaelis-Menten equation is based
- Derive the Michaelis-Menten equation using these assumptions
- Explain the significance of the parameters
*V*_{max}and*K*_{m}.

**Introduction**

The derivation of the Michaelis-Menten equation will be based on reactions in which there is a single substrate and conditions where the initial substrate concentration is very much greater than that of the enzyme. A corollary of this assumption is that the concentration of free substrate is approximately equal to the total substrate concentration. This is useful when carrying out experimental measurements.

Under these conditions, there are two approaches to the derivation of the Michaelis-Menten equation. Both are based on the principle that the rate of the reaction E + S ó ES is very much faster than the rate of the reaction ES à E + P. The two approaches are

- The equilibrium approximation
- The steady state approximation

**The equilibrium approximation**

In the equilibrium approximation, it is assumed that the relative reaction rates of the two steps are such that the reaction E + S ó ES remains at, or very close to equilibrium. The equilibrium constant for this reaction is given by

5.10 |

Now the fraction, *F* of the enzyme in the form ES is given by

5.11 |

By combining equations 5.10 and 5.11 we get

5.12 |

The rate of product formation will be a maximum when conditions are such that all the Enzyme is in the form
ES. Under these conditions, the rate will be *V _{max}*. Hence, under conditions when there is some
free enzyme present, the rate, V

V _{0} = V·_{max}F |
5.13 |

Hence, combining 5.12 and 5.13 gives

5.14 |

Which is essentially the Michaelis-Menten equation.

This derivation provides one definition of the significance of the parameter *K _{m}*. It is the
equilibrium constant for the reaction E + S ó ES.

**The steady state approximation**

The steady state approximation assumes that, after a short initial period, that the concentration of the ES complex is constant. In practice, the ES concentration declines slowly with time, as shown in Fig. 5.11, but the rate of decline is slow enough for the approximation to be practically valid.

**Fig. 5.21**

Now for the reaction

The rate of production of ES must equal its rate of breakdown. ** ie**.

5.21 |

Rearranging equation 5.21 gives the concentration of ES

5.22 |

As with the equilibrium approximation, the fraction of the enzyme in the form ES is given by

5.23 |

Combining 5.22 and 5.23 gives

5.24 |

As with the equilibrium approximation, the rate, *V*_{0} is given by;

V _{0} = V·_{max}F |
5.25 |

Hence, combining 5.24 and 5.25, we get

5.26 |

In this case, the Michaelis-Menten parameter *K _{m}* is defined as (

In some sources (eg. Fersht, 1985), *V*_{max} is defined in terms of the relationship

V _{max} = k_{cat}·[E]_{0} |
5.27 |

Where *k*_{cat} is a first order rate constant and [*E*]_{0} is the initial concentration
of the enzyme (before any binding has ocurred).

**Further Reading**

Fersht A, (1985), *Enzyme Structure and Mechanism, 2 ^{nd} Ed^{n}*, W H Freeman & Co.

Price, N C & Dwek, R A, (1979), *Principles and Problems in Physical Chemistry for Biochemists, 2 ^{nd}
Ed^{n},* Oxford Science Publications.

Produced by Geoff Walker

Last modified 21 September 2001