Enzyme Inhibition—Equations

The mathematical expressions relating reaction rate and inhibitor concentration are often rather complicated, but there are four simple equations that are extensions of the Michaelis-Menten formula. These merit special consideration because the kinetics of many enzymes can be satisfactorily described by them. In the equations in Table 9.1, [I] denotes the inhibitor concentration and K and K are inhibition constants, the units of which are those of a dissociation equilibrium constant (mmolL-1). Mechanisms that are consistent with these equations are described in Sect. 9.10. 

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Figure 5.5 Concentration-response plots for enzyme inhibition with Hill coefficients (h) of 1 (A) and 3 (B). Data simulated using Equation (5.4).

To account for differences in the Hill coefficient, enzyme inhibition data are best ht to Equation (5.4) or (5.5). In measuring the concentration-response function for small molecule inhibitors of most target enzymes, one will hnd that the majority of compounds display Hill coefficient close to unity. However, it is not uncommon to hnd examples of individual compounds for which the Hill coefficient is signihcandy greater than or less than unity. When this occurs, the cause of the deviation from expected behavior is often reflective of non-ideal behavior of the compound, rather than a true reflection of some fundamental mechanism of enzyme-inhibitor interactions. Some common causes for such behavior are presented below. 

If a drug is a substrate of an enzyme, it will also be a competitive inhibitor of that enzyme, but it may be a competitive inhibitor without being a substrate. This is because the rate of product formation is determined by k3 of the Michaelis-Menten equation while the rate of ES substrate dissociation and degree of enzyme inhibition is determined by the ratio of A/a i as discussed above. If A 3 is very small it will not be experimentally measurable however, the enzyme will still be bound and occupied as determined by ki/k. ... 

An adjacent tnfluoromethyl group sharply increases the electrophilic character of the carbonyl carbon Compounds that readily form hydrates and hemiacetals show a time-dependent reversible inhibition of the enzyme acetylcholinesterase (equation 2), in which the tight complex makes inhibition only partially reversible [75] In comparison with a nonfluonnated analogue, several aliphatic ketones flanked by CF3 and CF2 groups, are exceptionally potent reversible inhibitors of acetylcholinesterase, as documented by comparison of inhibition constants Kv shown in equation 3 [76]... 

If there is no reaction between the free enzyme and the inhibitor, and the inhibitor binds only to the (A.E) complex, Equation (5.154) becomes the uncompetitive inhibition equation ... 

The characteristics of the double reciprocal plots given by Equation (5.149), Equation (5.154), and Equation (5.155) determine what kind of enzyme inhibition may occur competitive, noncompetitive, or uncompetitive. In a given concentration of enzyme and inhibitor, the substrate concentration is changed and the double reciprocal plot of 1/V against 1/[A] is drawn. Figure 5.24a illustrates the double... 

Table 9.1. Rate Equations for the Four Types of Enzyme Inhibition...

The equations given in Table 9.1 simply describe the inhibition behaviors of enzymes and thus can be called phenomenological expressions. However, it is important to describe basic mechanisms, in the way that Michaelis and Menten did for a single enzyme. The mechanisms account for the form of the inhibition equations. 

For each of the four types of enzyme inhibition given in Table 9.1, derive the Lineweaver-Burk equations and draw archetypal graphs. 

The positive p values of these correlations for reactions of substituted phenyl phosphates may indicate a common feature in these reactions where a nucleophilic attack on phosphorus is a critical step. The magnitudes of the p values of Equations 13 and 14 are very close to each other but significantly larger than that of Equation 12. Thus, the electronic effect of the substituents on the insect mortality can be related mostly to an effect on the enzyme inhibition and plays only a minor role in the... 

By contrast, the enzyme-inhibition data of Table II correlated quite well with the partition and hydrolysis data (Equation 3). 

A different form of inhibition arises when the inhibitor binds to a second site on the enzyme, separate from the active site, and in doing so it modifies the enzyme, inhibiting its activity. This mode of inhibition is termed non-competitive inhibition (Figure 8-8), and unlike in competitive inhibition, there is often no structural similarity between the substrate and inhibitor. The simplest case of non-competitive inhibition, which is illustrated in Figure 8-8, is that the inhibitor binds with equal affinity to the free and substrate-bound forms of the enzyme, and that the inhibitor completely abolishes catalytic activity (fc at = 0). With these assumptions, the kinetic equation takes the form ... 

Mammalian alpha-amylases are activated by monovalent anions, especially chloride. Equations, similar to those for enzyme inhibition (see p. 290), may be written for the formation of enzyme—activator complexes, and kinetic parameters may be derived. Dissociation constants of salivary alpha-amylase—anion complexes have been determined. ... 

MichaeUs-Menten kinetics predict that as the concentration of the substrate increases, the rate increases hyperbolically. However, some enzymes exist in which a maximum velocity is obtained at low substrate concentration, but further increases in the substrate concentration lead to a decrease in velocity. This effect is known as substrate inhibition and can eventually lead to complete enzyme inhibition or partial enzyme inhibition. It is thought that substrate inhibition occurs if two substrate molecules bind to the enzyme simultaneously in an incorrect orientation and produce an inactive E S S complex, analogous to that discussed for uncompetitive inhibition. The rate of the enzyme reaction that undergoes substrate inhibition is given by Equation 17, where K represents the... 

The dose-response equations deployed to analyse most enzyme inhibition data are dependent upon there being a steady state (the degree of inhibition not changing during the assay), approximately equal concentrations of free and total inhibitor, and a one to one stoichiometry for binding and inhibition.29 Active compounds follow a range of kinetic mechanisms, which may be classified in various ways, for example according to dependence upon ATP or reversibility. Below, I describe some of the most common mechanisms. 

Thus, only if fmcYpvrere 0.5 would the AUG ratio double by GYP inhibition. However, as described earlier, the extent of clearance inhibition associated with enzyme inhibition actually depends upon the Ki value for the perpetrator. If a victim drug is metabolized by a single GYP enzyme, and if metabolism accounts for >50% of the victim drug s total clearance, then the AUG ratio arising from inhibition of that enzyme by a perpetrator is given by the Rowland-Matin equation ... 

If we now want to work out the initial-rate equation for this inhibition scheme, the equation (la or lb) relating [E] to [EA] still stands unaltered, as does Eqn. 3 relating v to [EA], What has to expand, however, is Eqn. 2, the enzyme conservation equation, since we now have a third enzyme species, El, to consider. We thus need an equation relating [Elj to the concentrations of the other enzyme forms. Since the route from E to El is a dead end , we may assume that equilibrium is established between E, I and El. This will be governed by a dissociation constant K-, so that we may write... 

The mechanisms of enzyme inhibition fall into three main types, and they yield particular forms of modified Michaelis-Menten equations. These can be derived for single-substrate/single-product enzymic reactions using the steady-state analysis of Sec. 5.10, as follows. 

It is apparent from Scheme 4.3 that the inhibitor cannot only compete with substrate for binding to the enzyme but bind to an enzyme molecule that subsequently binds a substrate molecule also or to an enzyme-substrate complex to affect catalytic turnover. These multiple binding mechanisms help explain the effects of mixed inhibition on both Vmax and K. The effects of mixed inhibition on the velocity of the reaction can be described by the following mixed inhibition equation (Eq. 4.15) ... 

The general rule for writing the rate equation according to the equilibrium treatment of enzyme inhibition kinetics for the forward direction is as follows ... 

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