## Allosteric Modulator Titration Curves

In a modern context of high-throughput screening, running multiple curves to screen a single compound is considered inefficient. Therefore, it has become commonplace for researchers to simplify assay design. Rather than running multiple curves to an agonist in the presence of varying concentrations of a putative modulator (as described above), it is now commonplace to select a single concentration of agonist and examine the effects of a range of putative modulator concentrations on the response. For example, a researcher seeking a negative allosteric modulator may design an assay to look for inhibition of an EC80 concentration of agonist. Conversely, an assay designed to look for a positive allosteric modulator may utilize an EC20 concentration of agonist to allow for potentiation of the response (Fig. 12.5). Such curves are generally referred to as modulator concentration-response or titration curves, and are analyzed according to simple logistic functions to determine IC50 or EC50 values (Fig. 12.5). These potency values are then used as the basis for driving the SAR, a key part of a drug discovery program.

Clearly, the reduction in the number of data points in going from an assay of multiple curves to a single curve diminishes the information content that an assay yields. However, simply using the potency value for a compound disregards other information that provides detail pertaining to a modulator's mechanism of action. A previous study has examined the relationship between the properties of positive allosteric modulators according to the simple ATCM and the resultant modulator titration curves [32] . It is clear that for positive modulators, the potency of the titration curve is dictated by both its affinity and the strength of the cooperativity between the agonist and the modulator. This is akin to agonist potency being a product of its affinity and efficacy. For a modulator of fixed affinity, increasing values of a result in higher potencies. Conversely, as the cooperativity value tends to a value of 1 (neutral coopera-tivity), the resultant pEC50 value will tend to the pKB value for the modulator [32]. By extension, for a negative allosteric modulator, the pIC50 of the modulator titration curve will tend to the pKB value for progressively stronger degrees of negative cooperativity (as a ^ 0).

It is possible to extend this relationship using a simplified version of the operational model of allosterism discussed in the Section 12.3 . This model, shown in Fig. 12.3b, is based on the full version in Fig. 12.3a. It makes the assumption that the orthosteric agonist being used is a full agonist and that it is not possible for its maximal response to be enhanced or reduced. Therefore, it is limited to the analysis of positive allosteric modulators, where any effects of efficacy will be manifest as a leftward shift in the agonist curve. It estimates a net cooperativity parameter, ap, which represents the product of the modulator's effects on affinity and efficacy. In addition to the modulator affinity (KB), it also incorporates the ability of the allosteric ligand to activate the receptor in its own right, governed by the parameter tb. If the allosteric modulator exhibits no intrinsic agonist activity (tb = 0), then the model is essentially

Figure 12.5 (a) Effect of varying concentrations of a positive allosteric modulator on an agonist concentration-response curve, as modeled using the simplified operational model of allosterism in Fig. 12.3b. Also shown is the resultant modulator titration curve that the positive modulator would produce if screened against an EC20 of agonist, fitted using a four-parameter logistic equation. For the simulation, the following parameters were used: pEC5o = 6.0, pKe = 6.0, aP = 30, Tb = 0, B = 30nM - 3|M. (b) Effect of varying concentrations of a similar positive allosteric modulator that displays intrinsic agonist activity (tb = 1) on an agonist concentration-response curve. Also shown is the resultant modulator titration curve that the modulator would produce if screened against an EC20 of agonist, fitted using a four-parameter logistic equation.

Figure 12.5 (a) Effect of varying concentrations of a positive allosteric modulator on an agonist concentration-response curve, as modeled using the simplified operational model of allosterism in Fig. 12.3b. Also shown is the resultant modulator titration curve that the positive modulator would produce if screened against an EC20 of agonist, fitted using a four-parameter logistic equation. For the simulation, the following parameters were used: pEC5o = 6.0, pKe = 6.0, aP = 30, Tb = 0, B = 30nM - 3|M. (b) Effect of varying concentrations of a similar positive allosteric modulator that displays intrinsic agonist activity (tb = 1) on an agonist concentration-response curve. Also shown is the resultant modulator titration curve that the modulator would produce if screened against an EC20 of agonist, fitted using a four-parameter logistic equation.

identical to the ATCM, except that the net cooperativity factor, aP, replaces the affinity cooperativity factor, a.

Figure 12.5a shows the effect of a positive allosteric modulator with a pKB = 6.0, a net cooperativity, aP = 30, and no intrinsic agonist activity (tb = 0) on an agonist concentration-response curve. Also shown is the resultant modulator titration curve, assuming an EC20 of the agonist is used. The fitted pEC50 (6.9) is greater than the affinity (6.0) as described above. Figure 12.5b shows the effect of a similar positive allosteric modulator but which also displays a significant degree of intrinsic agonist activity (tb = 1) on an agonist concentration-response curve. It is interesting to note that the presence or absence of intrinsic agonist activity in a positive modulator molecule makes no discernible difference to the EC50 of the resultant titration curve (Fig. 12.5). This suggests that a modulator titration curve format is not well suited for detecting intrinsic agonist activity in a molecule.

A previous study [32] highlighted a method for analyzing modulator titration curves in conjunction with the orthosteric agonist curve to yield estimates of modulator affinity and cooperativity. However, this method assumes that the modulator behaves according to the ATCM (which, as we have already seen, is often not the case). However, it is relatively simple to extend this analysis to the simplified operational model of allosterism. By recasting the model in Fig. 12.3b such that the modulator concentration ([B]) is the independent variable on the x-axis and that the orthosteric agonist concentration ([A]) is constrained to a fixed value (FixAg), the model can be used to analyze modulator titration curves in conjunction with the orthosteric agonist curve.

Figure 12.6a shows the recast model, and Fig. 12.6b demonstrates the analysis of a positive modulator titration curve in conjunction with the agonist

Response = Basal +

(Em - Basal)(([A](Kb + aß[FixAg])) + Tb • ECso)n (([A](Kb + aß[FixAg])) + Tb[FixAg] • ECso)n + (ECao- (Kb[FixAg])n)

Figure 12.6 (a) The simplified operational model of allosterism, recast such that the modulator concentration, B, is the independent variable and the orthosteric agonist concentration is recast as a constant, FixAg. (b) Analysis of a positive allosteric modulator titration curve in conjunction with the orthosteric agonist concentration-response curve according to the recast equation. Using the global shared analysis feature of GraphPad Prism 5 (GraphPad Software, La Jolla, CA), the values of EC50, Basal, Em, Kb, ap, and transducer function slope (n) are shared across both data sets, whereas the orthosteric agonist concentration, FixAg, is set at 4 |M. Note that in the analysis, the values of EC50, Basal, Em, and transducer function slope refer to the agonist concentration-response curve, rather than the modulator curve.

Figure 12.6 (a) The simplified operational model of allosterism, recast such that the modulator concentration, B, is the independent variable and the orthosteric agonist concentration is recast as a constant, FixAg. (b) Analysis of a positive allosteric modulator titration curve in conjunction with the orthosteric agonist concentration-response curve according to the recast equation. Using the global shared analysis feature of GraphPad Prism 5 (GraphPad Software, La Jolla, CA), the values of EC50, Basal, Em, Kb, ap, and transducer function slope (n) are shared across both data sets, whereas the orthosteric agonist concentration, FixAg, is set at 4 |M. Note that in the analysis, the values of EC50, Basal, Em, and transducer function slope refer to the agonist concentration-response curve, rather than the modulator curve.

concentration-response curve. The allosteric modulator potentiates an EC2 0 of agonist (4 |M). Analysis according to the operational model yields an estimate of affinity (pKB = 6.0) and net cooperativity (ap = 6.3). This is a relatively simple method for analyzing modulator titration curves, which yields significant information that can be used to characterize allosteric modulators in a relatively high - throughput manner.

However, due to the limited number of data points compared to a full "Schild-type" approach, there are limitations to this analysis. First, it is generally not possible to estimate the degree of intrinsic agonist activity of an allosteric modulator (tb) using this approach, as demonstrated with the examples in Fig. 12.5 . In the analysis example shown in Fig. 12.6b, the value of tb was determined in a separate study and shown to be zero, and, thus, constrained as such. Therefore, this value needs to be independently determined in order for this analysis to work correctly. Second, by the nature of the model, it is assumed that the maximal asymptote of the modulator titration curve cannot exceed the maximal asymptote of the orthosteric agonist curve (as the agonist is assumed to be full). If this does happen, then the model cannot be used. Finally, for highly positive or highly negative cooperativity values, the maximum/minimum asymptote of the modulator titration curve approaches either the maximum asymptote of the agonist curve or zero, respectively, which makes an accurate estimation of both the KB and ap values very difficult.

However, even without applying this methodology, a useful empirical marker of modulator function is the maximum asymptote (for a positive modulator) or the minimum asymptote (for a negative modulator) of the titration curve. If a given positive modulator yields a titration curve with a higher maximal response than for another modulator, then theory suggests that it will have stronger degree of positive cooperativity with the agonist than the second modulator (whether affinity or efficacy mediated). Thus, simply using the fitted EC50 (or IC50) value in conjunction with the maximal (or minimal) response should provide sufficient information with which to drive an SAR program.

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