Biochemical reactions are typically catalysed by enzymes. A common feature of these reactions is that the initial concentration of the enzyme is much smaller than that of the substrate. Their ratio defines a small dimensionless constant . A model for the simplest type of reaction of this type was introduced by Leonor Michaelis and Maud Menten in 1913. Following my habit of identifying links between the subjects of different posts on this blog whenever I can, I mention that Michaelis was an assistant to Paul Ehrlich in Berlin at one time.
My primary source for the following discussion is the well-known book of Murray on mathematical biology. Another useful source is a paper by Segel and Slemrod (SIAM Review, 3, 466). When the basic chemical assumptions are translated into mathematics a system of four ordinary differential equations is obtained. One of these decouples and a conservation law can be used to eliminate another. The result is a system of two ODE. The unknowns are the concentrations of the substrate and of the complex made by the combination of substrate and enzyme. If these equations are written in dimensionless variables (dimensionless concentration of substrate) and (dimensionless concentration of complex) they take the schematic form . Recall that is small. The sledgehammer method is then to say: let us set equal to zero. If this is done the second equation becomes algebraic. If it is solved for and the result substituted into the first equation then an ODE for alone results. It is called the uptake equation. The question is now what solutions of the uptake equation have to do with solutions of the original system.
There are natural initial conditions for the original system coming from its interpretation. Since it is a system of two first order equations there are two conditions. For the uptake equation, on the other hand, it is only possible to prescribe one condition, which is the initial condition for . The original system together with the preferred initial data determine a unique solution for and for any fixed . Nevertheless considering the limit may be useful for extracting interesting information from the solution. In fact there is a short time interval after the initial time where varies very rapidly. This time, which is of order , is so short that it is not possible to measure the evolution of the concentration experimentally in this interval. The quantity of most practical interest is in a sense the limit of the time derivative of as but since the measurements are done at times greater than this limit can be evaluated using the uptake equation. At the times of measurement the solution of the uptake equation gives a good approximation to the genuine solution although the two are very different near . It is not equal to the corresponding derivative computed from the solution of the original system with the natural initial conditions. The mathematical techniques which play a role in analysing more precisely what is going on here, and why certain formal procedures give the right answer, are singular perturbation theory and matched asymptotic expansions.