Question

What kind of kinetics is observed in an enzymatic reaction, under conditions where the substrate concentration is low compared to the Km?

Answer 1

As enzyme-catalysed reactions are saturable, their rate of catalysis does not show a linear response to increasing substrate. If the initial rate of the reaction is measured over a range of substrate concentrations (denoted as [S]), the reaction rate (v) increases as [S] increases, as shown on the right. However, as [S] gets higher, the enzyme becomes saturated with substrate and the rate reaches V_max, the enzyme's maximum rate.

The Michaelis–Menten kinetic model of a single-substrate reaction is shown on the right. There is an initial bimolecular reaction between the enzyme E and substrate S to form the enzyme–substrate complex ES but the rate of enzymatic reaction increases with the increase of the substrate concentration up to a certain level but then an increase in substrate concentration does not cause any increase in reaction rate as there no more E available for reacting with S and the rate of reaction becomes dependent on ES and the reaction becomes unimolecular reaction. Although the enzymatic mechanism for the unimolecular reaction can be quite complex, there is typically one rate-determining enzymatic step that allows this reaction to be modelled as a single catalytic step with an apparent unimolecular rate constant k_cat. If the reaction path proceeds over one or several intermediates, k_cat will be a function of several elementary rate constants, whereas in the simplest case of a single elementary reaction (e.g. no intermediates) it will be identical to the elementary unimolecular rate constant k₂. The apparent unimolecular rate constant k_cat is also called turnover number and denotes the maximum number of enzymatic reactions catalysed per second.

The Michaelis–Menten equation^[9] describes how the (initial) reaction rate v₀ depends on the position of the substrate-binding equilibrium and the rate constant k₂.

    (Michaelis–Menten equation)

with the constants

This Michaelis–Menten equation is the basis for most single-substrate enzyme kinetics. Two crucial assumptions underlie this equation (apart from the general assumption about the mechanism only involving no intermediate or product inhibition, and there is no allostericity or cooperativity). The first assumption is the so-called quasi-steady-state assumption (or pseudo-steady-state hypothesis), namely that the concentration of the substrate-bound enzyme (and hence also the unbound enzyme) changes much more slowly than those of the product and substrate and thus the change over time of the complex can be set to zero . The second assumption is that the total enzyme concentration does not change over time, thus . A complete derivation can be found here.

The Michaelis constant K_M is experimentally defined as the concentration at which the rate of the enzyme reaction is half V_max, which can be verified by substituting [S] = K_m into the Michaelis–Menten equation and can also be seen graphically. If the rate-determining enzymatic step is slow compared to substrate dissociation (), the Michaelis constant K_M is roughly the dissociation constant K_D of the ES complex.

If is small compared to then the term and also very little ES complex is formed, thus . Therefore, the rate of product formation is

Thus the product formation rate depends on the enzyme concentration as well as on the substrate concentration, the equation resembles a bimolecular reaction with a corresponding pseudo-second order rate constant . This constant is a measure of catalytic efficiency. The most efficient enzymes reach a in the range of 10⁸– 10¹⁰ M⁻¹ s⁻¹. These enzymes are so efficient they effectively catalyse a reaction each time they encounter a substrate molecule and have thus reached an upper theoretical limit for efficiency (diffusion limit); and are sometimes referred to as kinetically perfect enzymes