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The Journal of Online Mathematics and Its Applications
Volume 7, October 2007, Article ID 1611

Some Mathematical and Statistical Aspects of Enzyme Kinetics
by Michel Helfgott and Edith Seier
Most calculus or differential equations courses utilize examples taken from physics, often discussing them in great detail. Chemistry, however, is seldom utilized to illustrate mathematicalconcepts. This tendency should be reversed because chemistry, especially chemical kinetics, provides the opportunity to apply mathematics readily. We will analyze some basic ideas behind enzyme kinetics, which allow us to deal with separable and linear differential equations as well as realize the need to use power series to Approximate e x and ln(1 x) close to the origin, and to apply the recentlydefined Lambert W function.
The models studied in this context require the estimation of parameters based on experimental data, which in turn allows us to discuss simple and multiple linear regression, transformations and non-linear regression and their implementation using statistical software.

1. Introduction
Enzymes are mainly proteins that catalyze biochemical reactions, which otherwisewould proceed very slowly. They are essential to life. Their kinetics began to be understood at the beginning of the 20th century. It was observed that a typical enzyme converts a substrate into a product according to the chemical formula
Assuming that we are dealing with a single-step reaction we will have
This is due to the law of mass action, which ascertains that the rate at which asingle-step chemical reaction proceeds is proportional to the product of the concentration of reactants. Thus increasing, the initial concentration of substrate, and keeping the amount of enzyme concentration constant, we could increase without limits the initial rate at which the product is formed. This conclusion is not in agreement with observations: reaches a value beyond which the addition of moresubstrate does not
Increase the rate of initial formation of the product. To circumvent this and other difficulties scientists postulated the existence of an intermediate compound, which achieved rapidly equilibrium with the reactants and decomposed gradually producing a molecule of the product and regenerating a molecule of enzyme. That is to say,

Let us assume that the reversible processhas rate constantsand for the forward and backward reaction respectively, while the irreversible process is governed by the rate constant . Due to the above-mentioned equilibrium we have, so But the enzyme exists either as free enzyme or forming part of the intermediate compound, thus where is the total concentration of enzyme.
Therefore which in turn leads to . Finally we get
WhereSince the rate of formation of the product is given by and according to the law of mass action , we reach the expression
In particular (1).

Where and
A close look at (1) allows us to conclude that if we increase So , keeping ET constant, eventually it will be much greater than K . So vo will tend to the limiting rate k2ET , which we denote Vmax following common usage among biochemists.Thus (2)

This relationship is known as the Michaelis-Menten equation, honoring Leonor Michaelis and Maud Menten, who in 1913 published a groundbreaking paper on enzyme kinetics. They were two early pioneers in a relatively new field.
If we consider vo as a function of So (keeping ET constant), the following graph, shared by all functions of the form , can be drawn:

Figure 1. Initial rate as afunction of initial concentration of substrate

The reader may note that Michaelis-Menten equation predicts the appearance of the phenomenon of saturation because, no matter how much substrate we add, the initial rate cannot surpass the limiting rate Vmax . By the early 1920s solid experimental evidence supporting Michaelis-Menten equation had accumulated. But the existence of an equilibrium...
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