The Michaelis–Menten mechanism of enzyme catalysis
Experimental studies of enzyme kinetics are typically conducted by monitoring the initial rate of product formation in a solution in which the enzyme is present at very low concentration. Indeed, enzymes are such efficient catalysts that significant accelerations may be observed even when their concentration is more than three orders of magnitude smaller than that of the substrate. The principal features of many enzyme-catalysed reactions are as follows:
1 For a given initial concentration of substrate, [S]0, the initial rate of product for mation is proportional to the total concentration of enzyme, [E]0.
2 For a given [E]0 and low values of [S]0, the rate of product formation is propor tional to [S]0.
3 For a given [E]0 and high values of [S]0, the rate of product formation becomes independent of [S]0, reaching a maximum value known as the maximum velocity, vmax.
The Michaelis–Menten mechanism accounts for these features. According to this mechanism, an enzyme–substrate complex is formed in the first step and either the substrate is released unchanged or after modification to form products:
E+S⇌ES ka, k′a
ES →P kb
We show in the following Justification that this mechanism leads to the Michaelis Menten equation for the rate of product formation
v =
where KM = (k′ a + kb)/ka is the Michaelis constant, characteristic of a given enzyme acting on a given substrate.
Justification 23.2 The Michaelis–Menten equation
The rate of product formation according to the Michaelis–Menten mechanism is v =kb [ES]
We can obtain the concentration of the enzyme–substrate complex by invoking the steady-state approximation and writing
It follows that
[ES] =
[E][S]
where [E] and [S] are the concentrations of free enzyme and substrate, respectively. Now we define the Michaelis constant as
KM=
and note that KM has the same units as molar concentration. To express the rate law in terms of the concentrations of enzyme and substrate added, we note that [E]0 = [E] +[ES]. Moreover, because the substrate is typically in large excess relative to the enzyme, the free substrate concentration is approximately equal to the initial substrate concentration and we can write [S] ≈ [S]0. It then follows that:
[ES] =
We obtain eqn 23.17 when we substitute this expression for [ES] into that for the rate of product formation (v = kb[ES]).
Equation 23.17 shows that, in accord with experimental observations (Fig. 23.10):
1 When [S]0<< KM, the rate is proportional to [S]0
v =
2 When [S]0>> KM, the rate reaches its maximum value and is independent of [S]0:
v =vmax=kb[E]0
Substitution of the definitions of KM and vmax into eqn 23.17 gives:
v =
We can rearrange this expression into a form that is amenable to data analysis by linear regression:
A Lineweaver–Burk plot is a plot of 1/v against 1/[S]0, and according to eqn 23.22 it should yield a straight line with slope of KM/vmax, a y-intercept at 1/vmax, and an x intercept at −1/KM (Fig. 23.11). The value of kb is then calculated from the y-intercept and eqn 23.20b. However, the plot cannot give the individual rate constants ka and k′ a that appear in the expression for KM. The stopped-flow technique described in Section 22.1b can give the additional data needed, because we can find the rate of formation of the enzyme–substrate complex by monitoring the concentration after mixing the enzyme and substrate. This procedure gives a value for ka, and k′ a is then found by combining these results with the values of kb and KM.
Fig. 23.10 The variation of the rate of an enzyme-catalysed reaction with substrate concentration. The approach to a maximum rate, vmax, for large [S] is explained by the Michaelis–Menten mechanism.
Fig. 23.11 A Lineweaver–Burk plot for the analysis of an enzyme-catalysed reaction that proceeds by a Michaelis–Menten mechanism and the significance of the intercepts and the slope.
