Estimation of competitive antagonist affinity by the Schild method and from functional inhibition curves using a novel form of the Gaddum equation.

The equilibrium dissociation constant of competitive antagonists represents the affinity of the receptor-ligand interaction, and it is a key characteristic of many therapeutic drugs or toxic compounds. Two commonly used methods by which the affinity of the antagonist can be estimated are Schild analysis and the Cheng-Prusoff method. However, both methods yield different results when applied to systems with slopes not equal to one. The Gaddum equation, which is fundamental for both methods, should be extended to incorporate the slope parameter of the dose-response curves and this extension should diminish the differences between the Schild and Cheng-Prusoff methods. In this study, we derived a novel form of the Gaddum equation with a slope parameter (Hill coefficient) of agonist dose-response curve. We also derived the subsequent equations for Schild and Cheng-Prusoff analysis and we validated the proposed model by the measurement of several known estrogen receptor competitive antagonists. Standardized in vitro yeast reporter gene assay (BMAEREluc/ERα) has been used for the measurements and the range of used antagonist concentrations was 1.37-46.03 μM. By applying our mathematical model, both Schild and Cheng-Prusoff methods provide more similar values of antagonist affinity than the original mathematical approach. The correctness of the model has also been demonstrated by the measurement of a partial agonist with a known receptor affinity. The presented mathematical model significantly reduces the differences in values calculated by the Cheng-Prusoff and Schild methods and yields more accurate estimations of antagonist affinity.