As with binding, antagonists produce distinctive effects on dose-response curves for agonists. In all cases but non-competitive and pseudo-irrevesible antagonism, the effects are the same as those observed on tracer saturation binding curves. Thus, a competitive antagonist produces parallel dextral displacement of dose-response curves with no diminution of maximal response (see Fig. 8.2a). In this case, the affinity of the antagonist can be estimated by Schild analysis, whereby the dose-ratio (DR) for agonism (quantified as the ratio of EC5o [molar concentration producing 50% maximal response] obtained in the presence and absence of antagonist) is compared to the concentration of the antagonist according to the

Schild equation (Arunlakshana and Schild 1959):

The concentration of the antagonist is denoted as [B] and Kb is the equilibrium dissociation constant of the antagonist-receptor complex (reciprocal of affinity). Thus a regression of Log(DR-1) values upon Log[B] values should be linear with a slope of unity and an intercept of Log Kb (Arunlakshana and Schild 1959).

While Schild analysis is historically the most often utilized method to measure competitive antagonist affinity, it is not the optimal method. Another approach, namely a non-linear fit of the data (with visualization of the data with a 'Clark plot'—Stone and Angus 1976; Stone 1980; Lew and Angus 1996), does not over-emphasize control pEC50 (as does Schild analysis) thereby giving a more balanced estimate of antagonist affinity. While superior to the Schild method it should be noted that Schild analysis is rapid, intuitive, and can be used to detect non-equilibrium steady-states in the system that can corrupt estimates of affinity. Also non-linear regression requires matrix algebra to estimate the error of the pKb. While error estimates are given with many commercially available software packages for curve fitting, they are difficult to obtain without these (from first principles). In contrast, manual calculation with Schild analysis furnishes an estimate of the error for the pKb from the linear regression using all of the data.

For the non-linear procedure, the pEC50s (— log of the EC50 values) of the agonist dose-response curves are fit to the equation:

where [B] is the concentration of the antagonist and pKB and c are fitting constants. Note that the control pEC50 is used with [B] = 0. The relationship between the pEC50 and increments of antagonist concentration can be shown in a Clark plot of pEC50 versus -Log ([B] + 10-pKB). Constructing such a plot is useful because, although it is not used in any calculation of the pKb, it allows visualization of the data to ensure that the plot is linear and has a slope of unity (Lew and Angus 1996).

Competitive antagonists can also possess efficacy and thereby produce low levels of tissue response. If the maximal level of response is lower than the system maximum, these compounds are referred to as partial agonists. Providing a significantly greater response can be obtained with a full agonist in the presence of the partial agonist, the affinity of the partial agonist can be measured with Schild analysis. As a first approximation it should be noted that the EC50 of the partial agonist (molar concentration producing 50 per cent of the maximal response to the partial agonist) is equal to the Kb of the partial agonist. The positive efficacy of the partial agonist will introduce a minor error into the estimation, the magnitude of which is proportional to the efficacy of the partial agonist. Thus, the relationship between the EC50 of a partial agonist and its affinity is given by:

ETC Model

CTC Model

In the case of non-competitive and/or pseudo-irreversible antagonism, the receptor occupancy by the agonist is precluded by the antagonist causing a direct depression of the maximal receptor occupancy curve (Fig. 8.1c). However, due to the amplication factors in stimulus-response mechanisms, maximal tissue responses for high efficacy agonists can be attained, in some systems, with sub-maximal receptor occupancy, that is, a powerful agonist may need only 10% of the existing receptor population to produce maximal tissue response. Thus, until 90% of receptors are inactivated by antagonist, the maximal response for the agonist will be attained and the observed antagonism will resemble simple competitive antagonism until that point. Therefore, the characteristic pattern of non-competitive antagonism on agonist response is a mixture of dextral displacement and eventual depression of maximal response (see Fig. 8.5a).

There is another mechanism whereby the pattern shown in Fig. 8.5a can evolve, namely in cases of hemi-equilibria. Under equilibrium conditions, the relative receptor occupancies of an agonist and competitive antagonist adjust according to their respective affinities and relative concentrations. However, if insufficient time is allowed for this to occur, then a truncation of the response will ensue. This is especially the case with slowly dissociating antagonists since the antagonist occupancy must re-adjust to the presence of the agonist during the measurement of response. If this does not occur then the maximal asymptote of response is effectively truncated and a series of depressions of maximal responses with increasing concentrations of a competitive antagonist are observed. Thus, the curves are shifted to the right according to the Schild equation but with depressed maxima. While this apparently resembles non-competitive antagonism, it actually is due to competititve antagonism with insufficient time to observe equilibrium response.

Fig. 8.5 Depression of functional maximal responses by non-competitive antagonism (a) and hemi-equilibrium kinetics (b). (a) Inhibition of agonist response by a non-competitive antagonist in a system where receptor stimulus is highly amplified (i.e. 10% receptor occupancy by the agonist generates the system maximum response). Dextral displacement of the curve with little diminution of maximum can be observed before depression occurs. Numbers refer to values of [B]/Kb. (b) Effect of a slowly dissociating simple competitive antagonist on agonist response measured within a time frame insufficient to allow equilibration of the receptors with agonist and antagonist. If kinetic conditions would allow equilibrium to be achieved, the curves would be the dotted lines. Under hemi-equilibirum conditions, depressions of the maximal response are observed. Numbers refer to [B]/Kb.

Non-competitive antagonism can be quantified by the method of Gaddum (Gaddum et al. 1955) in functional studies. Thus, equiactive concentrations of agonist in the absence (denoted [A]) and presence (denoted [A']) of a given concentration of noncompetitive antagonist (the concentration must be sufficient to depress the maximal response to the agonist) are compared according to a double reciprocal equation of the form (Gaddum et al. 1955):

where Ka and Kb refer to the equilibrium dissociation constants of the agonist and antagonist receptor complexes, respectively and a the allosteric factor for the non-competitive antagonist (change in affinity of the agonist for the receptor imparted by the antagonist). Note that there may not be a change in the affinity of the receptor for the agonist (a may be unity) but that the receptor maybe made inoperative when occupied by the non-competitive antagonist. Equation (15) describes a straight line the slope ofwhich can be used to estimate the Kb of the antagonist with the equation:

Best results with this method are obtained when the maximum response is depressed below 50 per cent control levels. There are other transformations which circumvent the obvious weighting disadvantages of double reciprocal plots that can be used with this method (see Kenakin 1997b). Equiactive responses are most readily obtained by comparing the control and depressed concentration-response curves at various levels of response.

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