Enzyme kinetics is a bunch of really super exciting math that describes how enzymes behave. If you've ever sat in a math class wondering what the point of any of it was, then your teacher should have told you about enzyme kinetics!

Enzyme kinetics are called Michaelis-Menten kinetics. Named after the scientists Leonor Michaelis and Maud Menten. Maud Menten's story is a tale similar to that of other great female scientists of the 20th century. One of the first women to earn an MD in her native Canada, Menten moved to Germany where she worked with Leonor Michaelis, as at the time women were barred from being researchers in Canada. She subsequently returned to work as a professor in Canada, by which time women had rather more career options in scientific research.

It was while working with Michaelis that she derived the key fact that the rate of enzyme activity was dependent on the concentration of substrate. An enzyme will metabolize a certain amount of substrate in a certain period of time, so in a solution containing a fixed amount of enzyme, the addition of more substrate increases the rate of catalysis, as there is more substrate available for the fixed amount of enzyme to metabolize. However, once the substrate concentration reaches a certain concentration, it saturates the active sites of the enzyme (Figure 1). Once this happens, no extra substrate may be metabolized by the enzyme, since there are no active sites available. The rate at this point is the maximum rate of the enzyme activity, or V_max. This can be likened to a factory, a factory can produce a certain amount of goods with a certain amount of raw materials, and can produce more goods with more raw materials, even calling in extra labour if needed, however at a certain point, the factory cannot process any more raw material, as there aren't enough machines or staff available.

Figure 1: As the enzyme and substrate have to interact on a 1:1 basis for the reaction to take place, the rate of enzyme catalyzed reaction is fundamentally dependent on the concentration of substrate, and its relation to the concentration of enzyme. At low concentrations (left) the enzyme is not saturated, and there is more enzyme available to catalyze further reactions, so the rate of reaction is limited by a lack of substrate. When the substrate concentration saturates the available enzyme (middle) the rate of reaction is at its maximum, as all available enzyme is catalyzing the reaction. At higher substrate concentrations (right) there is no more enzyme available to catalyze extra substrate, so the addition of extra substrate does not increase the rate of reaction, it instead proceeds at the maximum rate.

Owing to the concentration of substrate being crucial for the rate of enzyme activity, and the substrate concentration falling as the reaction progresses (as substrate is converted to product), it is most important to consider the initial rate of the reaction, or V₀. As the reaction progresses, the concentration of substrate will fall and thus the rate of reaction will also fall, so it is only the V₀ that reflects the rate of reaction at the concentration of substrate being tested at the beginning of the experiment. All derivations of enzyme activity thus measure the initial rate of reaction.

An enzyme catalyzed reaction can be generalized as described by (Figure 2). Where [S] is the substrate, [E] is the enzyme, [ES] is the enzyme-substrate complex and [P] is the product. The substrate binds to the enzyme to form a complex, which in turn is metabolized into the product. The formation of [ES] and [E] + [S] are reversible, while the catalysis of the [ES] to the [P] is not. This fact means that there are constants for the rates of formation of both the [ES] and [E] + [S], known as the k_f and k_r respectively, while only one rate for the formation of [P], the k_kat.

Figure 2: The enzyme catalyzed reaction has three steps, the enzyme and substrate (E + S) step, where the enzyme and substrate are separate entities. These bind together to form the enzyme-substrate complex (ES). These two forms are interchangeable, and the rate of conversation between them are the forward rate (k_f) and reverse rate (k_r). The ES then progresses to the enzyme and product (E + P) step, where the enzyme has catalyzed the substrate and releases the product, thus recycling the enzyme for more catalysis. This step is not reversible, and is measured as the rate of catalysis (k_cat) of the enzyme. The k_f, k_r and k_cat together are used to calculate the K_M, an important parameter for measuring the enzyme catalysis.

These parameters that describe the flow of compounds through the enzyme catalzyed reaction can be used to calculate the Michaelis constant (K_M) using (Equation 1). The Michaelis constant is a summary metric of how the enzyme metabolizes substrate to product, this along with V_max are the most important descriptors of the enzyme activity, and together they are part of the Michaelis-Menten equation (Equation 2) that describes how the rate of enzyme activity relates to the maximum rate of enzyme activity, the Michaelis constant, and the concentration of substrate in the reaction.

Eq. 1: \(K_M = {k_r + k_{cat} \over k_f}\)

Eq. 2: \(v = k_{max}{[S] \over K_M + [S]}\)

It is this equation that is the basis behind the study of enzymes and their properties. Any given equation catalyzed reaction will have a K_M and a V_max that can be derived, and it is these parameters that are changed by the addition of enzyme inhibitors.