Enzyme-Inhibitor Binding Equilibria

As we have seen before, the enzymatic reaction begins with the reversible binding of substrate (S) to the free enzyme ( ) to form the ES complex, as quantified by the dissociation constant Ks. The ES complex thus formed goes on to generate the reaction product(s) through a series of chemical steps that are collectively defined by the first-order rate constant kCM. The first mode of inhibitor interaction that can be con- 

Evaluation of Enzyme Inhibitors in Drug Discovery, by Robert A. Copeland ISBN 0-471-68696-4 Copyright 2005 by John Wiley Sons, Inc. 

1 In this book we will use the symbol K, for the dissociation constant of the El complex, and aA) for the dissociation constant of the ESI complex (or subsequent species). The reader should note that different authors used different symbols for these dissociation constants. Hence in the enzymology literature one may find the dissociation constant for the El complex symbolized as K Ka, KEi, etc. Likewise the dissociation constant for the ESI complex may be symbolized as aK, K, Kis and KEsi- 

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Figure 2 Two thermodynamic models that link perturbations in enzyme-inhibitor binding equilibria to values for interaction energies. E, enzyme I, inhibitor E l, enzyme-inhibitor complex AAGinteract energetic contribution of the perturbed interaction to complex formation AGso1v,e energy of desolvation for the enzyme AGsoiv, energy of desolvation for the inhibitor AGresoiv, energy of resolvation of the E l complex.

Thus, every El complex reduces the amount of enzyme available for catalysis, regardless of where the inhibitor binds. As shown in Eq. (2.60), inhibition is a reversible process. The degree of reversibility depends on the ratio k k i or in other words, on the inhibitor binding equilibrium constant, K . 

Fig. 1 Equilibrium between reactants (E + I) and noncovalent complex (E- I ) in a typical enzyme-inhibitor binding process...

The basic kinetic properties of this allosteric enzyme are clearly explained by combining Monod s theory and these structural results. The tetrameric enzyme exists in equilibrium between a catalytically active R state and an inactive T state. There is a difference in the tertiary structure of the subunits in these two states, which is closely linked to a difference in the quaternary structure of the molecule. The substrate F6P binds preferentially to the R state, thereby shifting the equilibrium to that state. Since the mechanism is concerted, binding of one F6P to the first subunit provides an additional three subunits in the R state, hence the cooperativity of F6P binding and catalysis. ATP binds to both states, so there is no shift in the equilibrium and hence there is no cooperativity of ATP binding. The inhibitor PEP preferentially binds to the effector binding site of molecules in the T state and as a result the equilibrium is shifted to the inactive state. By contrast the activator ADP preferentially binds to the effector site of molecules in the R state and as a result shifts the equilibrium to the R state with its four available, catalytically competent, active sites per molecule. 

As we have just seen, the initial encounter complex between an enzyme and its substrate is characterized by a reversible equilibrium between the binary complex and the free forms of enzyme and substrate. Hence the binary complex is stabilized through a variety of noncovalent interactions between the substrate and enzyme molecules. Likewise the majority of pharmacologically relevant enzyme inhibitors, which we will encounter in subsequent chapters, bind to their enzyme targets through a combination of noncovalent interactions. Some of the more important of these noncovalent forces for interactions between proteins (e.g., enzymes) and ligands (e.g., substrates, cofactors, and reversible inhibitors) include electrostatic interactions, hydrogen bonds, hydrophobic forces, and van der Waals forces (Copeland, 2000). 

As we described in Chapter 3, the binding of reversible inhibitors to enzymes is an equilibrium process that can be defined in terms of the common thermodynamic parameters of dissociation constant and free energy of binding. As with any binding reaction, the dissociation constant can only be measured accurately after equilibrium has been established fully measurements made prior to the full establishment of equilibrium will not reflect the true affinity of the complex. In Appendix 1 we review the basic principles and equations of biochemical kinetics. For reversible binding equilibrium the amount of complex formed over time is given by the equation... 

If the inhibition is found to be rapidly reversible, we must next determine if the approach to equilibrium for the enzyme-inhibitor complex is also rapid. As described in Chapter 4, some inhibitors bind slowly to their target enzymes, on a time scale that is long in comparision to the time scale of the reaction velocity measurement. The effect of such slow binding inhibition is to convert the linear progress curve seen in the absence of inhibitor to a curvilinear function (Figure 5.10). When nonlinear progress curves are observed in the presence of inhibitor, the analysis of... 

The hallmark of slow binding inhibition is that the degree of inhibition at a fixed concentration of compound will vary over time, as equilibrium is slowly established between the free and enzyme-bound forms of the compound. Often the establishment of enzyme-inhibitor equilibrium is manifested over the time course of the enzyme activity assay, and this leads to a curvature of the reaction progress curve over a time scale where the uninhibited reaction progress curve is linear. We saw... 

Binding of a reversible inhibitor to an enzyme is rapidly reversible and thus bound and unbound enzymes are in equilibrium. Binding of the inhibitor can be to the active site, or to a cofactor, or to some other site on the protein leading to allosteric inhibition of enzyme activity. The degree of inhibition caused by a reversible inhibitor is not time-dependent the final level of inhibition is reached almost instantaneously, on addition of inhibitor to an enzyme or enzyme-substrate mixture. 

Inhibition is termed reversible if, when the concentration of the inhibitor is reduced, the extent of inhibition is reduced (i.e. the dissociation constant for the binding of the inhibitor to the enzyme is not extremely high). Most inhibitors bind either to the enzyme or the enzyme substrate complex. In both cases, the binding is an equilibrium process, as follows ... 

Consider the standard Uni Uni mechanism (E + A EX E + P). A noncompetitive inhibitor, I, can bind reversibly to either the free enzyme (E) to form an El complex (having a dissociation constant K s), or to the central complex (EX) to form the EXl ternary complex (having a dissociation constant Xu). Both the slope and vertical intercept of the standard double-reciprocal plot (1/v vx. 1/[A]) are affected by the presence of the inhibitor. If the secondary replots of the slopes and the intercepts (thus, slopes or vertical intercepts vx [I]) are linear (See Nonlinear Inhibition), then the values of those dissociation constants can be obtained from these replots. If Kis = Xu, then a plot of 1/v vx 1/[A] at different constant concentrations of the inhibitor will have a common intersection point on the horizontal axis (if not. See Mixed-Type Inhibition). Note that the above analysis assumes that the inhibitor binds in a rapid equilibrium fashion. If steady-state binding conditions are present, then nonlinearity may occur, depending on the magnitude of the [I] and [A] terms in the rate expression. See also Mixed Type Inhibition... 

The inhibitors are generally monoanions or neutral molecules capable of deprotonation to yield anionic species. These neutral inhibitors (mostly sulfonamides) (189) bind to the active site Zn-H O. Monovalent anions like I-, CN-, SCN-, N3, NCO-, SH- etc. (190,191), inhibit the catalyzed reaction of CA enzyme by binding directly to the metal ion either by displacing H2O to yield a tetra-coordinate metal ion or by adding to the coordination sphere to yield a penta-coordinate metal ion with H2O as the fifth ligand (see Table V). In some cases an equilibrium between these two coordination geometries is also reported. 

FIGURE 6-15 Three types of reversible inhibition, (a) Competitive inhibitors bind to the enzyme s active site, (b) Uncompetitive inhibitors bind at a separate site, blit bind only to the ES complex. K, is the equilibrium constant for inhibitor binding to E K is the equilibrium constant for inhibitor binding to ES. (c) Mixed inhibitors bind at a separate site, but may bind to either E or ES. 

The process of reversible inhibition is described by an equilibrium interaction between enzyme and inhibitor. Most inhibition processes can be classified as competitive or noncompetitive, depending on how the inhibitor impairs enzyme action. A competitive inhibitor is usually similar in structure to the substrate and is capable of reversible binding to the enzyme active site. In contrast to the substrate molecule, the inhibitor molecule cannot undergo chemical transformation to a product however, it does interfere with substrate binding. A noncompetitive inhibitor does not bind in the active site of an enzyme but binds at some other region of the enzyme molecule. Upon binding of the noncompetitive inhibitor, the enzyme is reversibly converted to a nonfunctional conformational state, and the substrate, which is fully capable of binding to the active site, is not converted to product. 

Reversible inhibition occurs rapidly in a system which is near its equilibrium point and its extent is dependent on the concentration of enzyme, inhibitor and substrate. It remains constant over the period when the initial reaction velocity studies are performed. In contrast, irreversible inhibition may increase with time. In simple single-substrate enzyme-catalysed reactions there are three main types of inhibition patterns involving reactions following the Michaelis-Menten equation competitive, uncompetitive and non-competitive inhibition. Competitive inhibition occurs when the inhibitor directly competes with the substrate in forming the enzyme complex. Uncompetitive inhibition involves the interaction of the inhibitor with only the enzyme-substrate complex, while non-competitive inhibition occurs when the inhibitor binds to either the enzyme or the enzyme-substrate complex without affecting the binding of the substrate. The kinetic modifications of the Michaelis-Menten equation associated with the various types of inhibition are shown below. The derivation of these equations is shown in Appendix S.S. 

Reversible inhibitors bind an enzyme through weak, intermolecular forces and establish an equilibrium of being bound or unbound to the enzyme. A competitive inhibitor binds at the active site and prevents the substrate from binding. A noncompetitive inhibitor binds an allosteric site on the enzyme and prevents conversion of the substrate to product. Uncompetitive inhibitors bind the enzyme-substrate complex and make it inactive. All three types of inhibitors show characteristic, distinctive features in a Lineweaver-Burk plot. 

The effect of an uncompetitive inhibitor on the Km of a substrate deserves some commentary. A lower Km implies that an enzyme has been inhibited by increasing the enzyme s affinity for its substrate. This seemingly counterintuitive statement can be made clearer by examining the equilibria in Scheme 4.14. By binding the enzyme-substrate complex, the inhibitor forces some free enzyme to bind substrate in order to maintain equilibrium concentrations of all species in solution. The inhibitor does indeed increase the affinity of the enzyme for its substrate (i.e., decrease Km). 

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