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.

where K_{M} is called the Michaelis Constant.

This expression is
the *Briggs-Haldane* analysis of the reaction.
The Michaelis-Menten analysis incorporates the additional assumption
that k_{2} is much less than k_{-1}, so the
*Pre-Equilibrium Approximation* (see
demo3) can be used
instead of the *Steady State Approximation*. This leads to a very similar result,
but K_{M} has the simpler form k_{-1}/k_{1}.
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:

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.

Type:

`demo4 k_{1} k_{-1} k_{2} R`

to plot the changes in concentrations for systems of this type,
calculated numerically. *R* is the initial ratio of *A*
to *E*, and so must be between zero and one.
The results of both *Approximation I* and *Approximation II*
are plotted on the graphs. *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.

This example uses a 4:1 ratio of substrate A to the catalyst E (*R = 0.2*).
This is much more catalyst that is common in an enzymic reaction. However,
the Michaelis-Menten analysis (cyan line) gives a very good approximation
to the rate of formation of the product C. The systematic error arises because
the initial formation of B is not instantaneous, and so the numeric solution
(dark blue line) for the formation of C lags behind the *Approximation I* line.
*Approximation II* (brown line) follows the numeric solution very closely at the
start of the reaction.

A stoichiometric amount of catalyst E is used in this example. The Both approximations give a reasonable description of the result, in the regions for which they are valid.

With a 9:1 ratio of substrate A to the catalyst E, the *Approximation I*
line falls on top of the more exact solution.

These graphs were created using Gnuplot

© 1997 J M Goodman