Enzymes (Second Edition)

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Figure 8.7 Secondary plots for the determination of the inhibitor constants for a noncompetitive inhibitor. (A) 1/V is plotted as a function of [I], and the value of K is determined from the x intercept of the line. (B) The value of K is determined from the x intercept of a plot of the slope of the lines from the double-reciprocal (Lineweaver—Burk) plot as a function of [I].

8.3.3 Uncompetitive Inhibitors

Both V and K are affected by the presence of an uncompetitive inhibitor. The form of the velocity equation therefore contains the dissociation constantK in both the numerator and denominator:

Figure 8.8 Pattern of lines in the double-reciprocal plot of an uncompetitive inhibitor.

The reader will observe that Equation 8.18 is another special case of the more general equation given by Equation 8.14.

With a little algebra, it can be shown that the reciprocal form of Equation

We see from equation 8.19 that the slope of the double-reciprocal plot is independent of inhibitor concentration and that the y intercept increases steadily with increasing inhibitor. Thus, the overlaid double-reciprocal plot for an uncompetitive inhibitor at varying concentrations appears as a series of parallel lines that intersect the y axis at different values, as illustrated in Figure 8.8.

For an uncompetitive inhibitor, the x intercept of a Dixon plot will be equal to K (1 K /[S]). At first glance this relationship may not look particularly convenient. If, however, one is working at saturating conditions, where [S] K , the value of K /[S] becomes very small and can be assumed to be zero. Under these conditions, the x intercept of the Dixon plot will be equal to

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K . Thus, under conditions of saturating substrate, one can determine the value of K directly from the x intercept of a Dixon plot, as described earlier for the case of noncompetitive inhibition.

8.3.4 Global Fitting of Untransformed Data

The best method for determining inhibitor modality and the values of the inhibitor constant(s) is to fit directly and globally all the plots of velocity versus [S] at several fixed inhibitor concentrations to the untransformed equations for competitive (Equation 8.10), noncompetitive (Equation 8.14), and uncompetitive inhibition (Equation 8.18). From analysis of the statistical parameters for goodness of fit (typically ), one can determine which model of inhibitor modality best describes the experimental data as a complete set and simultaneously determine the value of the inhibitor constant(s). This type of global fitting analysis has only recently become widely available. The commercial programs GraphFit and SigmaPlot, for example, allow this type of global fitting [i.e., fitting multiple curves that conform to the functional form y f (x, z), where x is substrate concentration and z is inhibitor concentration]. Cleland (1979) also published the source code for FORTRAN programs that allow this type of global data fitting. The reader is strongly encouraged to make use of these programs if possible.

8.4 DOSE--RESPONSE CURVES OF ENZYME INHIBITION

In many biological assays one can measure a specific signal as a function of the concentration of some exogenous substance. A plot of the signal obtained as a function of the concentration of exogenous substance is referred to as a dose—response plot, and the function that describes the change in signal with changing concentration of substance is known as a dose—response curve (Figure 8.9). These plots have the form of a Langmuir isotherm, as introduced in Chapter 4. We have already seen that such plots can be conveniently used to follow protein—ligand binding equilibria. The same plots are used to follow saturable events in a number of other biological contexts, such as effects of substances on cell growth and proliferation. Dose—response plots also can be used to follow the effects of an inhibitor on the initial velocity of an enzymatic reaction at a fixed concentration of substrate. The concentration of inhibitor required to achieve a half-maximal degree of inhibition is referred to as the IC value (for inhibitor concentration giving 50% inhibition), and the equation describing the effect of inhibitor concentration on reaction velocity is related to the Langmuir isotherm equation as follows:

Figure 8.9 Dose—response plot of enzyme fractional activity as a function of inhibitor concentration. Note that the inhibitor concentration is plotted on a log scale. The value of the

IC for the inhibitor can be determined graphically as illustrated.

where v is the initial velocity in the presence of inhibitor at concentration [I] and v is the initial velocity in the absence of inhibitor.

The observant reader will note two differences between the form of Equation 8.20 and that of the standard Langmuir isotherm equation (Equation 4.23). First, we have replaced the dissociation constant K (or in the case of enzyme inhibition, K ) with the phenomenological term IC . This is because the concentration of inhibitor that displays half-maximal inhibition may be displaced from the true K by the influence of substrate concentration, as we shall describe shortly. The second difference between Equations 4.23 and 8.20 is that we have inverted the ratio of [I] and IC . This is because the standard Langmuir isotherm equation tracks the fraction of ligand-bound receptor molecules. The term v /v in Equation 8.20 is referred to as the fractional activity remaining at a given inhibitor concentration. This term reflects the fraction of free enzyme, rather than the fraction of inhibitor-bound enzyme. Considering mass conservation, the fraction of inhibitor-bound enzyme is related to the fractional activity as 1 (v /v ). Hence, we could recast Equation 8.20 in the more traditional form of the Langmuir isotherm as follows:

Dose—response plots are very widely used for comparing the relative inhibitor potencies of multiple compounds for the same enzyme, under well-controlled

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conditions. The method is popular because it permits analysts to determine the IC by making measurements over a broad range of inhibitor concentrations at a single, fixed substrate concentration. A range of inhibitor concentrations spaning several orders of magnitude can be conveniently studied by means of the twofold serial dilution scheme described in Chapter 5 (Section 5.6.1), with inhibitor being varied in place of substrate here. This strategy is very convenient when many compounds of unknown and varying inhibitory potency are to be screened.

In the pharmaceutical industry, for example, one may wish to screen several thousand compounds as potential inhibitors to find those that have some potency against a particular target enzyme. These compounds are likely to span a wide range of IC values. Thus, one would set up a standard screening protocol in which the initial velocity of an enzymatic reaction is measured over five or more logs of inhibitor concentrations. In this way the IC values of many of the compounds could be determined without any prior knowledge of the range of concentrations required to effect potent inhibition of the enzyme.

The IC value is a practical readout of the relative effects on enzyme activity of different substances under a specific set of solution conditions. In many instances, it is the net effect of the inhibitor on enzyme activity, rather than its true dissociation constant for the enzyme, that is the ultimate criterion by which the effectiveness of a compound is judged. In some situations, a K value cannot be rigorously determined because of a lack of knowledge or control over the assay conditions; many times, in these cases, the only measure of relative inhibitor potency is an IC value. For example, consider the task of determining the relative effectiveness of a series of inhibitors for a target enzyme in a cellular assay. Often, in these cases, the inhibitor is added to the cell medium and the effects of inhibition are measured indirectly by a readout of biological activity that is dependent on the activity of the target enzyme. In a cellular situation like this, one often does not know either the substrate concentration in the cell or the relative amounts of enzyme and substrate (recall that in vitro we set up our steady state conditions so that [S] [E], but this is not necessarily the case in the cell). Also, in these situations, one does not truly know the effective concentration of inhibitor within the cell that is causing the degree of inhibition being measured. This is because the cell membrane may block the transport of the bulk of added inhibitor into the cell. Moreover, cellular metabolism may diminish the effective concentration of inhibitor that reaches the target enzyme. Because of these uncontrollable factors in the cellular environment, often it is necessary to report the effectiveness of an inhibitor as an IC value.

Despite their convenience and popularity, IC value measurements can be misleading if used inappropriately. The IC value of a particular inhibitor can change with changing solution conditions, so it is very important to report the details of the assay conditions along with the IC value. For example, in the case of competitive inhibition, the IC value observed for an inhibitor will depend on the concentration of substrate present in the assay, relative to the K of that substrate. This is illustrated in Figure 8.10 for a competitive

Figure 8.10 Effect of substrate concentration on the IC value of a competitive inhibitor.

inhibitor under conditions of [S] K and [S] 10 K . Thus, in comparing a series of competitive inhibitors, it is important to ensure that the IC values are measured at the same substrate concentration. For the same reasons, it is not rigorously correct to compare the relative potencies of inhibitors of different modalities by use of IC values. The IC values of a noncompetitive and a competitive inhibitor will vary with substrate concentration, but in different ways. Hence, the relative effectiveness observed in vitro under a particular set of solution conditions may not be the same relative effectiveness observed in vivo, where the conditions are quite different. Whenever possible, therefore, the K values should be used to compare the inhibitory potency of different compounds.

It is possible to take advantage of the convenience of IC measurements and still report inhibitor potency in terms of true K values when the mode of inhibition for a series of compounds is known, as well as the values of [S] and K . The relationship between the K , [S], K , and IC values can be derived from the velocity equations already presented. The derivations have been described in detail by Cheng and Prusoff (1973) for competitive, noncompetitive, and uncompetitive inhibitors. The reader is referred to the original paper for the derivations. Here we shall simply present the final forms of the relationships

For competitive inhibitors:

Equations 8.22—8.24, known as the Cheng and Prusoff relationships, can be conveniently used to convert IC values to K values. To ensure that the correct relationship can be applied, however, it is critical to know the mode of inhibition of the compounds being tested. It might thus seem that there is no great advantage to the use of the Cheng and Prusoff relationships if the mode of inhibition for each compound must be determined by Lineweaver—Burk analysis anyway. In many cases, however, one will wish to measure the relative inhibitory potency of a series of structurally related compounds. If these compounds represent small structural perturbations from a common parent molecule, it is often safe to assume that all the derivative molecules share the same mode of inhibition as the parent. In such situations, one could determine the mode of inhibition for the parent molecule only and then apply the appropriate Cheng and Prusoff relationship to the rest of the molecular series.

There is, of course, the possibility of an inadvertent change in the mode of inhibition as a result of the structural perturbations. This is usually not a great danger if the perturbations are minor, and one can spot-check by performing Lineweaver—Burk analysis on a subgroup of compounds representing a wide range of perturbations within the series. This is a common strategy used in development of structure—activity relationships for the determination of the key structural components in the inhibitory mechanism shared by a series of related compounds, as described next, in Section 8.6. Many scientists, however, consider the K values derived by application of the Cheng and Prusoff relationships to be less accurate than those obtained by the more traditional methods described earlier. There is lower confidence in the former results partly because the effects of the inhibitor are examined at only a single, fixed substrate concentration. Nevertheless, because of their convenience, the Cheng and Prusoff relationships are commonly used for high throughput inhibitor screening.

At the beginning of this chapter we mentioned that some inhibitors do not block completely the ability of the enzyme to turnover when bound to the inhibitor. These partial inhibitors will not display the same dose—response curves as full inhibitors because, for these compounds, one can never drive the

reaction velocity to zero, even at very high inhibitor concentrations. Rather, the dose—response curve for a partial inhibitor will be best fit by a more generalized form of Equation 8.20, given by:

where y is the fractional activity of the enzyme in the presence of inhibitor at concentration [I], y is the maximum value of y that is observed at zero inhibitor concentration (for fractional activity, this is 1.0), and y is the minimum value of y that can be obtained at high inhibitor concentrations.

Unlike the case of full inhibitors, the dose—response curve for a partial

is 0.05, so that even at very high inhibitor concentrations, the enzyme still displays 5% of its uninhibited velocity. When behavior of this type is observed, one must be very careful to ensure that the lack of complete inhibition is not an experimental artifact. For example, in densitometry measurements one often observes some finite background density that is difficult to completely subtract out and can give the appearance of partial inhibition when, in fact, full inhibition is taking place.

A more diagnostic signature of partial inhibition can be obtained by arranging the data as a Dixon plot. While all the full inhibitors discussed thus far yielded linear fits in Dixon plots, partial inhibitors typically display hyperbolic fits of the data in these plots (Figure 8.11B). In these cases one can extract the values of , K , and for the inhibitor, depending on the mode of partial inhibition that is taking place. These analyses are, however, beyond the scope of the present text. The reader who encounters this relatively unusual form of enzyme inhibition is referred to the text by Segel (1975) for a more comprehensive discussion of the data analysis.

8.5 MUTUALLY EXCLUSIVE BINDING OF TWO INHIBITORS

If two structurally distinct inhibitors, I and J, are found to both act on the same enzyme, it is possible for them to bind simultaneously to form an EIJ complex (or an ESIJ complex if both inhibitors are capable of binding to the ES complex). Alternatively, the two inhibitors may bind in a mutually exclusive fashion (i.e., competitive with each other) so that only an EI or an EJ complex can form. There are several tests by which it can be determined if the two inhibitors compete for binding to the enzyme.

The most direct way to measure exclusivity of inhibitor binding is by use of a radiolabeled or fluorescently labeled version of one of the inhibitors. If such labels are used to follow direct binding of the inhibitor to the enzyme, the

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Figure 8.11 Dose—response (A) and Dixon (B) plots for a partial inhibitor. The value of v /v in (A) reaches a nonzero plateau at high inhibitor concentrations. The hyperbolic nature of the Dixon plot in (B) is characteristic of partial inhibition.

ability of the second inhibitor to interfere with this binding can be directly measured as described in Chapter 4.

A number of kinetic measures have also been described to test the exclusivity of inhibitor interactions with a target enzyme (see Martines-Irujo et al., 1998, for a recent review). All these methods involve measuring the initial velocity of the enzyme at different combinations of the two inhibitors. The effects of two inhibitors on the velocity of an enzymatic reaction can be generally described by the following reciprocal relationship:

where v is the initial velocity in the presence of both inhibitors, K and K are

the dissociation constants for inhibitors I and J, respectively, and is an interaction term that defines the effect of the binding of one inhibitor on the affinity of the second inhibitor. If the two inhibitors bind in a mutually exclusive fashion, . If the two bind completely independently, 1. If the two inhibitors bind nonexclusively but influence each other’s affinity for the enzyme, then will be finite, but less than or greater than 1. When is less than 1, the binding of one inhibitor increases the affinity of the enzyme for the second inhibitor, and the binding of the two is said to be synergistic (i.e., exhibiting positive cooperativity). When is greater than 1, the binding of one inhibitor decreases the affinity of the enzyme for the second inhibitor, and in this case the binding is said to be antagonistic (i.e., exhibiting negative cooperativity).

Loewe (1957) has described the isobologram method for determining exclusivity of binding. In this analysis different concentrations of I and J are combined to yield the same fractional activity (v /v ). The different concentrations of I in these combinations are plotted on the y axis, and the corresponding concentrations of J are plotted on the x axis. If the binding of the two inhibitors is mutually exclusive, the data points on such a plot fall on a straight line. If, however, the two inhibitors bind nonexclusively, the data points will form an outwardly concave curve on the isobologram, the curvature depending on the value of . A number of other graphic methods have been described for this type of analysis (see, e.g., Chou and Talalay, 1977); of all these methods, the most popular is that of Yonetani and Theorell (1964).

In the Yonetani—Theorell method, the data are arranged as Dixon plots,

where 1/v is plotted as a function of [I] at varying fixed concentrations of J.

Consideration of Equation 8.26 will reveal that when is infinity, the data points will form a series of parallel lines when plotted by the method of Yonetani and Theorell (Figure 8.12A). This is an indication that the two inhibitors bind in a mutually exclusive fashion, competing with one another for the same enzyme form. If is 1, the two inhibitors bind independently, and the lines in the Yonetani—Theorell plot intersect on the x axis (Figure 8.12B). If exceeds 1, the two inhibitors antagonize each other’s binding, and the lines on the plot intersect below the x axis. Alternatively, if the two inhibitors are synergistic with one another, is less than 1 and the lines intersect above the x axis. For any Yonetani—Theorell plot in which the lines intersect (i.e.,

), the x-axis value at the point of intersection provides an estimate ofK when [I] is plotted on the x axis, or K when [J] is the variable inhibitor concentration. If the values of K and K are known from independent measurements, the value of is then easily calculated.

A common motivation for performing the analysis described in this section is to determine whether two structurally distinct inhibitors share a common binding site on the enzyme molecule. If two inhibitors are found to bind in a mutually exclusive fashion, through either kinetic analysis or direct binding measurements, it is tempting to conclude that they bind to the same site on the