Michaelis-Menten Kinetics

represents the minimal equation needed to describe a simple one-substrate, one-product reaction catalyzed by an enzyme. This assumes the reverse reaction between E and P is negligible and is a simple, first-order reaction whose rate is determined solely by the concentration of ES and the value of k 2. The reaction rate (same as velocity or rate of formation of products) can be written as

> V = k 2[ES]

[ES] is usually not a measurable concentration. Easily measurable items are the substrate (or product) and the total concentration of enzyme, which is the sum of the free enzyme and complexed enzyme. That is

> [E]t = [E] + [ES],

where [E]t is total enzyme, [E] is free enzyme, and [ES] is complexed enzyme.

In general, it is incorrect to assume that E and S are in equilibrium with ES, because some ES is continually being drained off to make P. (When k 2 is much less than k-1, the assumption is reasonable, however.)

In 1925, Briggs and Haldane proposed a model that avoided the equilibrium assumption. It assumes that the more ES that is present, the faster ES will dissociated either to products or back to reactants. Therefore, when the reaction is started by mixing enzymes and substrates, the ES concentration builds up at first, but quickly reaches a steady state, in which it remains almost constant. The steady state will persist until almost all of the substrate has been consumed (Figure 11.14). If one measures rates only after the steady state has been established and before [ES] has changed much, reaction velocity can be calculated by assuming steady state conditions.

In the steady state, rates of formation and breakdown of ES are equal,

Rearranging 11.17 gives equation 11.18,

Combining the three rate constants of equation 11.18 into one, K M, yields

> K M = (k-1 + k 2)/k 1

Equation 11.18 becomes

> K M[ES] = [E][S]

Because [E] = [E]t - [ES],

> K M[ES] = [E]t - [S] - [ES][S]

Solving for [ES],

> [ES] = [E]t[S]/(K M+[S])

Substituting this into the earlier velocity equation,

> V = k 2[E]t[S]/(K M+[S])

This last equation is the Michaelis-Menten equation, and K M is called the Michaelis constant. K M has units of concentration and, because it is a ratio of the rate constants of a reaction, K M is characteristic of the reaction. A given enzyme acting upon a given substrate has a distinct K M.

Figure 11.15 shows a plot of velocity (V) versus substrate concentration ([S]). Note that at high substrate concentrations ([S] >> K M), the velocity approaches a maximum (called Vmax). At this point, the enzyme molecules are saturated with substrate. Note also that the substrate concentration where V = V max/2 corresponds to the K M.

At V max, [S] >> K M, so [S] + K M can be approximated simply by [S]. Thus, the velocity equation simplifies to

> V max = k 2[E]t

Substituting this back into the velocity equation yields

> V = V max[S]/(K M + [S])

For a multistep reaction more complicated than the one assumed above, the Michaelis-Menten equation must be modified. Consider the reaction

For this reaction, the k 2 in the Michaelis-Menten equation must be replaced by a more general constant, called k cat. Here,

> V = k cat[E]t[S]/( K M + [S])

k cat incorporates the rate constants for all the reactions between ES and E + P. For the two-step reaction above, k cat = k 2. For more complex reactions, k cat depends on which steps in the process are rate-limiting.

See also: K M, K CAT, and K CAT/K M, Analysis of Kinetic Data