Derivation of Michaelis-Menten Equation

Enzyme kinetics is the study of the rates of enzyme-catalyzed reactions. In order to study the activity of any enzyme, we must carry of an experiment in which we vary the concentration of substrate in the presence of some particular enzyme. Each time we increase the concentration of substrate, we must measure the reaction velocity of the enzyme. The reaction velocity describes the rate at which the enzyme operates on the substrate. Once we obtain enough date points, we can plot this on the xy-plane. The curve that we obtain tells us that initially, when the concentration of the substrate is still relatively low, the rate of the enzyme is directly proportional to the substrate concentration. However, as we continue to increase the substrate concentration, the slope of the curve begins to level off as the curve approaches the maximal velocity asymptotically. The maximum velocity describes the highest rate at which the enzyme can operate on the substrate. This corresponds to the point at which all the active sites on the enzyme mixture are occupied with the substrate. Since this curve is so important in understanding and studying the rates of enzymes, we will derive the mathematical equation that describes it. This mathematical equation is called the Michaelis-Menten equation. In order to derive the equation, we need to look at the reaction when the reaction is still at its beginning stage; that is, when the time is about zero. This will allow us to make the simplification that the reverse of the product formation reaction proceeds at a negligible rate. This has to do with the fact that at the beginning very little product is actually formed. A little further in the derivation, we are also going to assume the steady-state condition. This condition assumes that the concentration of the intermediate, namely the enzyme-substrate complex, does not change. With this assumption, we see that the rate of formation of the enzyme-substrate complex is equation to the rate of dissociation of the complex.

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