Enzymes (Second Edition)

270 REVERSIBLE INHIBITORS

8.2 MODES OF REVERSIBLE INHIBITION

8.2.1 Competitive Inhibition

Competitive inhibition refers to the case of the inhibitor binding exclusively to the free enzyme and not at all to the ES binary complex. Thus, referring to the scheme in Figure 8.1, complete competitive inhibition is characterized by values of and 0. In competitive inhibition the two ligands (inhibitor and substrate) compete for the same enzyme form and generally bind in a mutually exclusive fashion; that is, the free enzyme binds either a molecule of inhibitor or a molecule of substrate, but not both simultaneously. Most often competitive inhibitors function by binding at the enzyme active site, hence competing directly with the substrate for a common site on the free enzyme, as depicted in the cartoon of Figure 8.2A. In these cases the inhibitor usually shares some structural commonality with the substrate or transition state of the reaction, thus allowing the inhibitor to make similar favorable interactions with groups in the enzyme active site. This is not, however, the only way that a competitive inhibitor can block substrate binding to the free enzyme. It is also possible (although perhaps less likely) for the inhibitor to bind at a distinct site that is distal to the substrate binding site, and to induce some type of conformation change in the enzyme that modifies the active site so that substrate can no longer bind. The observation of competitive inhibition therefore cannot be viewed as prima facie evidence for commonality of binding sites for the inhibitor and substrate. The best that one can say from kinetic measurements alone is that the two ligands compete for the same form of the enzyme — the free enzyme.

When the concentration of inhibitor is such that less than 100% of the enzyme molecules are bound to inhibitor, one will observe residual activity due to the population of free enzyme. The molecules of free enzyme in this population will turn over at the same rate as in the absence of inhibitor, displaying the same maximal velocity. The competition between the inhibitor and substrate for free enzyme, however, will have the effect of increasing the concentration of substrate required to reach half-maximal velocity. Hence the presence of a competitive inhibitor in the enzyme sample has the kinetic effect of raising the apparent K of the enzyme for its substrate without affecting the value of V ; this kinetic behavior is diagnositic of competitive inhibition. Because of the competition between inhibitor and substrate, a hallmark of competitive inhibition is that it can be overcome at high substrate concentrations; that is, the apparent K of the inhibitor increases with increasing substrate concentration.

8.2.2 Noncompetitive Inhibition

‘‘Noncompetitive inhibition’’ refers to the case in which an inhibitor displays binding affinity for both the free enzyme and the enzyme—substrate binary

Figure 8.2 Cartoon representations of the three major forms of inhibitor interactions with enzymes: (A) competitive inhibition, (B) noncompetitive inhibition, and (C) uncompetitive inhibition.

complex. Hence, complete noncompetitive inhibition is characterized by a finite value of and 0. This form of inhibition is the most general case that one can envision from the scheme in Figure 8.1; in fact, competitive and uncompetitive (see below) inhibition can be viewed as special, restricted cases of noncompetitive inhibition in which the value of is infinity or zero, respectively. Noncompetitive inhibitors do not compete with substrate for binding to the free enzyme; hence they bind to the enzyme at a site distinct from the active site. Because of this, noncompetitive inhibition cannot be overcome

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by increasing substrate concentration. Thus, the apparent effect of a noncompetitive inhibitor is to decrease the value of V without affecting the apparent K for the substrate. Figure 8.2B illustrates the interactions between a noncompetitive inhibitor and its enzyme target.

The enzymological literature is somewhat ambiguous in its designations of noncompetitive inhibition. Some authors reserve the term ‘‘noncompetitive inhibition’’ exclusively for the situation in which the inhibitor displays equal affinity for both the free enzyme and the ES complex (i.e., 1). When the inhibitor displays finite but unequal affinity for the two enzyme forms, these authors use the term ‘‘mixed inhibitors’’ (i.e., is finite but not equal to 1). Indeed, the first edition of this book used this more restrictive terminology. In teaching this material to students, however, I have found that ‘‘mixed inhibition’’ is confusing and often leads to misunderstandings about the nature of the enzyme—inhibitor interactions. Hence, we shall use noncompetitive inhibition in the broader context from here out and avoid the term ‘‘mixed inhibition.’’ The reader should, however, make note of these differences in terminology to avoid confusion when reading the literature.

8.2.3 Uncompetitive Inhibitors

Uncompetitive inhibitors bind exclusively to the ES complex, rather than to the free enzyme form. The apparent effect of an uncompetitive inhibitor is to decrease V and to actually decrease K (i.e., increase the affinity of the enzyme for its substrate). Therefore, complete uncompetitive inhibitors are characterized by 1 and 0 (Figure 8.2C).

Note that a truly uncompetitive inhibitor would have no affinity for the free enzyme; hence the value of K would be infinite. The inhibitor would, however, have a measurable affinity for the ES complex, so that K would be finite. Obviously this situation is not well described by the equilibria in Figure 8.1. For this reason many authors choose to distinguish between the dissociation constants for [E] and [ES] by giving them separate symbols, such as K and K , K and K , and K and K (the subscripts in this latter nomenclature refer to the effects on the slope and intercept values of double reciprocal plots, respectively). Only rarely, however, does the inhibitor have no affinity whatsoever for the free enzyme. Rather, for uncompetitive inhibitors it is usually the

case that K K . Thus we can still apply the scheme in Figure 8.1 with the

condition that 1.

8.2.4 Partial Inhibitors

Until now we have assumed that inhibitor binding to an enzyme molecule completely blocks subsequent product formation by that molecule. Referring to the scheme in Figure 8.1, this is equivalent to saying that 0 in these cases. In some situations, however, the enzyme can still turn over with the inhibitor bound, albeit at a far reduced rate compared to the uninhibited enzyme. Such situations, which manifest partial inhibition, are characterized by

0 1. The distinguishing feature of a partial inhibitor is that the activity of the enzyme cannot be driven to zero even at very high concentrations of the inhibitor. When this is observed, experimental artifacts must be ruled out before concluding that the inhibitor is acting as a partial inhibitor. Often, for example, the failure of an inhibitor to completely block enzyme activity at high concentrations is due to limited solubility of the compound. Suppose that the solubility limit of the inhibitor is 10 M, and at this concentration only 80% inhibition of the enzymatic velocity is observed. Addition of compound at concentrations higher that 10 M would continue to manifest 80% inhibition, as the inhibitor concentration in solution (i.e., that which is soluble) never exceeds the solubility limit of 10 M. Hence such experimental data must be examined carefully to determine the true reason for an observed partial inhibition. True partial inhibition is relatively rare, however, and we shall not discuss it further. A more complete description of partial inhibitors has been presented elsewhere (Segel, 1975).

8.3 GRAPHIC DETERMINATION OF INHIBITOR TYPE

8.3.1 Competitive Inhibitors

A number of graphic methods have been described for determining the mode of inhibition of a particular molecule. Of these, the double reciprocal, or Lineweaver—Burk, plot is the most straightforward means of diagnosing inhibitor modality. Recall from Chapter 5 that a double reciprocal plot graphs the value of reciprocal velocity as a function of reciprocal substrate concentration to yield, in most cases, a straight line. As we shall see, overlaying the double-reciprocal lines for an enzyme reaction carried out at several fixed inhibitor concentrations will yield a pattern of lines that is characteristic of a particular inhibitor type. The double-reciprocal plot was introduced in the days prior to the widespread use of computer-based curve-fitting methods, as a means of easily estimating the kinetic values K and V from the linear fits of the data in these plots. As we have described in Chapter 5, however, systematic weighting errors are associated with the data manipulations that must be performed in constructing such plots.

To avoid weighting errors and still use these reciprocal plots qualitatively to diagnose inhibitor modality, we make the following recommendation. To diagnose inhibitor type, measure the initial velocity as a function of substrate concentration at several fixed concentrations of the inhibitor of interest. To select fixed inhibitor concentrations for this type of experiment, first measure the effect of a broad range of inhibitor concentrations with [S] fixed at its K value (i.e., measure the Langmuir isotherm for inhibition (see Section 8.4) at [S] K ). From these results, choose inhibitor concentrations that yield between 30 and 75% inhibition under these conditions. This procedure will ensure that significant inhibitor effects are realized while maintaining sufficient signal from the assay readout to obtain accurate data.

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With the fixed inhibitor concentrations chosen, plot the data in terms of velocity as a function of substrate concentration for each inhibitor concentration, and fit these data to the Henri—Michaelis—Menten equation (Equation

into the reciprocal equation (Equation 5.34) to obtain a linear function, and

plot this linear function for each inhibitor concentration on the same doublereciprocal plot. In this way the double-reciprocal plots can be used to determine inhibitor modality from the pattern of lines that result from varying inhibitor concentrations, but without introducing systematic errors that could compromise the interpretations.

Let’s walk through an example to illustrate the method, and to determine the expected pattern for a competitive inhibitor. Let us say that we measure the initial velocity of our enzymatic reaction as a function of substrate concentration at 0, 10, and 25 M concentrations of an inhibitor, and obtain the results shown in Table 8.1.

If we were to plot these data, and fit them to Equation 5.24, we would obtain a graph such as that illustrated in Figure 8.3A. From the fits of the data we would obtain the following apparent values of the kinetic constants:

Table 8.1 Hypothetical velocity as a function of substrate concentration at three fixed concentrations of a competitive inhibitor

Figure 8.3 Untransformed (A) and double-reciprocal (B) plots for the effects of a competitive inhibitor on the velocity of an enzyme catalyzed reaction. The lines drawn in (B) are obtained by applying Equation 5.24 to the data in (A) and using the apparent values of the kinetic constants in conjunction with Equation 5.34. See text for further details.

If we plug these values of V and K into Equation 5.34 and plot the resulting linear functions, we obtain a graph like Figure 8.3B.

The pattern of straight lines with intersecting y intercepts seen in Figure 8.3B is the characteristic signature of a competitive inhibitor. The lines intersect at their y intercepts because a competitive inhibitor does not affect the apparent value of V , which, as we saw in Chapter 5, is defined by the y intercept in a double-reciprocal plot. The slopes of the lines, which are given by K /V , vary among the lines because of the effect imposed on K by the inhibitor. The degree of perturbation of K will vary with the inhibitor concentration and will depend also on the value of K for the particular

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inhibitor. The influence of these factors on the initial velocity is given by:

or, taking the reciprocal of this equation, we obtain:

Now, comparing Equation 8.11 to Equation 5.34, we see that the slopes of the double-reciprocal lines at inhibitor concentrations of 0 and i differ by the factor (1 [I]/K ). Thus, the ratio of these slope values is:

Thus, in principle, one could measure the velocity as a function of substrate concentration in the absence of inhibitor and at a single, fixed values of [I], and use Equation 8.13 to determine the K of the inhibitor from the doublereciprocal plots. This method can be potentially misleading, however, because it relies on a single inhibitor concentration for the determination of K .

A more common approach to determining the K value of a competitive inhibitor is to replot the kinetic data obtained in plots such as Figure 8.3A as the apparent K value as a function of inhibitor concentration. The x intercept of such a ‘‘secondary plot’’ is equal to the negative value of the K , as illustrated in Figure 8.4, using the data from Table 8.1.

In a third method for determining the K value of a competitive inhibitor suggested by Dixon (1953), one measures the initial velocity of the reaction as a function of inhibitor concentration at two or more fixed concentrations of substrate. The data are then plotted as 1/v as a function of [I] for each substrate concentration, and the value of K is determined from the x-axis value at which the lines intersect, as illustrated in Figure 8.5. The Dixon plot (1/v as a function of [I]) is useful in determining the K values for other inhibitor types as well, as we shall see later in this chapter.

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8.3.2 Noncompetitive Inhibitors

We have seen that a noncompetitive inhibitor has affinity for both the free enzyme and the ES complex; hence the dissociation constants from each of these enzyme forms must be considered in the kinetic analysis of these inhibitors. The most general velocity equation for an enzymatic reaction in the presence of an inhibitor is:

and this is the appropriate equation for evaluating noncompetitive inhibitors. Comparing Equations 8.14 and 8.10 reveals that the two are equivalent whenis infinite. Under these conditions the term [S](1 [I]/ K ) reduces to [S], and Equation 8.14 hence reduces to Equation 8.10. Thus, as stated above, competitive inhibition can be viewed as a special case of the more general case of noncompetitive inhibition.

In the unusual situation that K is exactly equal to K (i.e., is exactly 1), we can replace the term K by K and thus reduce Equation 8.14 to the following simpler form:

Equation 8.15 is sometimes quoted in the literature as the appropriate equation for evaluating noncompetitive inhibition. As stated earlier, however, this reflects the more restricted use of the term ‘‘noncompetitive.’’

The reciprocal form of Equation 8.14 (after some canceling of terms) has the form:

As described by Equation 8.16, both the slope and the y intercept of the double-reciprocal plot will be affected by the presence of a noncompetitive inhibitor. The pattern of lines seen when the plots for varying inhibitor concentrations are overlaid will depend on the value of . When exceeds 1, the lines will intersect at a value of 1/[S] less than zero and a value of 1/v of greater than zero (Figure 8.6A). If, on the other hand, 1, the lines will intersect below the x and y axes, at negative values of 1/[S] and 1/v (Figure 8.6B). If 1, the lines converge at 1/[S] less than zero on the x axis (i.e., at 1/[v] 0)

Figure 8.6 Patterns of lines in the double-reciprocal plots for noncompetitive inhibitors for (A)

1 and (B) 1.

To obtain the values of K and K , two secondary plots must be construc-

ted. The first of these is a Dixon plot of 1/V (i.e., at saturating substrate

concentration) as a function of [I], from which the value of K can be determined as the x intercept (Figure 8.7A). In the second plot, the slope of the double-reciprocal lines (from the Lineweaver—Burk plot) are plotted as a function of [I]. For this plot, the x intercept will be equal to K (Figure 8.7B). Combining the information from these two secondary plots allows determination of both inhibitor dissociation constants from a single set of experimental data.