The binary complex ES is commonly referred to as the ES complex, the initial encounter complex, or the Michaelis complex. As described above, formation of the ES complex represents a thermodynamic equilibrium, and is hence quantifiable in terms of an equilibrium dissociation constant, Kd, or in the specific case of an enzyme-substrate complex, KS, which is defined as the ratio of reactant and product concentrations, and also by the ratio of the rate constants koff and kon (see Appendix 2):
(2.1)
The equilibrium dissociation constant KS has units of molarity and its value is inversely proportional to the affinity of the substrate for the enzyme (i.e., the lower the value of KS, the higher the affinity). The value of KS can be readily converted to a thermodynamic free energy value by the use of the familiar Gibbs free energy equation:
(2.2)
where R is the ideal gas constant and T is temperature in degrees Kelvin (note that for use in Equation 2.2 the value of KS is expressed as molar, not μM nor nM). Similar thermodynamic relationships hold for the reversible interactions of inhibitors with enzymes, as will be described in Chapter 3.
Thus, as described by Equation (2.1), the equilibrium dissociation constant depends on the rate of encounter between the enzyme and substrate and on the rate of dissociation of the binary ES complex.
According to collision theory, there are three factors that limit the rate of bimolecular encounters: the frequency of collisional events (Z); a steric factor (ρ), which takes into account that only a fraction of collisional events will occur with both molecules properly oriented relative to one another for reaction; and the activation energy of reaction (Eact or ΔG‡). The rate of productive encounters is thus given by the following equations:
(2.3)
and
(2.4)
where ra and rb are the molecular radii of the two colliding molecules, a and b, and η is the viscosity of the medium in which the reaction is taking place. The maximum rate of reaction would occur when Eact is zero and all encounters occur with both molecules properly oriented for reaction (i.e., ρ = 1). Thus the maximum rate is given by Z (Equation 2.4). This rate reflects the encounter frequency of the two molecules and is referred to as the diffusion controlled rate. For a low molecular weight ligand and a protein target, the diffusion limit on association (i.e., kon) has been calculated to be in the range of 1 × 109 M−1s−1 (Fersht, 1999). Similarly, the upper limit for the dissociation rate of a bimolecular complex (i.e., koff) has been determined to be between 109 and 1012 s−1. These extreme rates, however, are not seen in experimental measures of protein-ligand complex association and dissociation, largely because of the influence of steric parameters and activation energies (vide supra) as well as a significant impact of conformational dynamics of the macromolecular target on reaction (see Chapters 6 and 8).
Table 2.1 illustrates how the combination of these two rate constants can influence the overall value of Kd (in general) for any equilibrium binding process. One may think that association between the enzyme and substrate (or other ligands) is exclusively rate-limited by collisional encounters. However, as described above and further in Chapters 6 and 8, this is not always the case. Sometimes conformational adjustments of the enzyme’s active site must occur prior to productive ligand binding, and these conformational adjustments may occur on a time scale slower that diffusion. Likewise the rate of dissociation of the ES complex back to the free reactant state can vary significantly from one enzyme to another. This dissociation process is counterproductive to catalysis, as it competes with the forward process of bound substrate transformation to products.
