## The Michaelis Menten equation

In 1926 J. B. Sumner(1) reported the crystallization of an enzyme, urease, and thus convinced most chemists and biologists that enzymes are distinct, albeit complex, chemical species that are able to be purified to homogeneity. This concept, coupled with that of Michaelis and Menten in 1913,® that enzymes form specific complexes with their reactants, paved the way for a detailed understanding of the chemical mechanisms of individual types of enzymes. One of the simplest experiments that can be carried out on a solution of a particular enzyme is to study the rate at which it converts its substrate(s) to product(s). This process can be studied by using a physical recording device such as a spectrophotometer; the chemical reaction either directly or indirectly develops a chromophore, thus enabling a record of the time dependence of the concentrations of at least one of the reactants. Alternatively, an NMR spectrometer can be used, most often without the requirement for additional reactants to generate a detectable chromophore, since almost invariably the NMR spectrum of the product(s) will be different from that of the substrate(s). Experimentally, the effect of substrate concentrations on the rate of the reaction is measured, and because the products of the reaction might inhibit or activate the enzyme as they accumulate during the reaction, or the enzyme may be unstable in the conditions of the assay medium, it is common practice to measure the initial velocity, v0. This initial velocity is measured as the slope of the progress curve of the reaction after extrapolating it back to t = 0. When the substrate concentration, [A]0, is much greater than the enzyme concentration, the overall rate of the reaction is not only proportional to the concentration of the enzyme, but the plot of v0 versus [A]0 has the form of a rectangular hyperbola. The equation describing the rectangular hyperbola is called the Michaelis-Menten equation:

This equation has the property that when [A]0 is very large, v0= Vmax (hence it is called the maximum velocity), and when v0 = Vmax / 2, the experimental value of [A]0 is equal to Km, and this is called the Michaelis constant.

Q: Using Mathematica construct a plot of the Michaelis-Menten equation over a domain of concentrations of A, for an enzyme having a Vmax of 1 mmol s-1 and a Km of 1 mM.

A: The following Mathematica Cell has the requisite series of functions to perform this task. Recall that items such as the semantics of delayed evaluation of equations, and of the Plot function, are given in Chapter 1 and also in the Mathematica Help Menu.

PlotRange -> {0, 1 x 10-6 } , AxesLabel -> { " [A] ", "v0 " }] ; Figure 2.1. The rectangular hyperbolic form of the Michaelis-Menten equation. The ordinate is in units of mmol s-1 and the abscissa is in units of mmol L-1.