Agonist-antagonist interaction models with slope factor: implications of an alternative derivation

Objectives: The method developed by Schild [1] to determine the equilibrium dissociation constant of a competitive antagonist (K_B) has been a cornerstone of quantitative pharmacological research.  On the basis of this method, Waud [5], Leff and Dougal [4] proposed a model to describe the interaction of an agonist and a competitive antagonist in terms of an sigmoidal E_max model and antagonist potency and Schild slope factor (b).  A key feature of this model and the Schild equation is that they are ‘null methods’ and provide antagonist potency estimates independent of the slope of the E_max model (n_H).  Holford and Sheiner [3] proposed a generalized model for the competitive interaction between two ligands, which reduced to the case of an agonist and a competitive antagonists results in an equation similar to that of Waud, with the exception of the positions of the two slope factors, n_H and b.  Interestingly, it can be shown that the Holford and Sheiner model predicts that the Schild analysis is no longer a null method and is sensitive to the value of n_H. More recently, Zuideveld et al. [6] and Cheng [2] independently derived a third parameterization of the interaction model, with a slightly different effect of b compared to the reduced Holford and Sheiner model and again predicts dependency of the Schild method on n_H. Despite the fact that their work appears to challenge one of the most fundamental methods in pharmacology, Holford and Sheiner, Zuideveld et al. and Cheng failed to describe this implication of their model modifications.  This poster therefore describes the implication of the model as described by Zuideveld et al. and Cheng, using simulations and several examples from literature.

Methods & Results: Sensitivity analysis of the new model has shown that slope factor (n_H) of the agonist has to be large (>3) before a change would be observed in a Schild analysis plot. Simulations talking into account typical inter-individual variability and residual error further show that it is difficult to prove that the antagonist slope factor, b, deviates significantly from 1, which may be the reason that the vast majority of experimental data appears to support the null method assumption behind the Schild analysis.

Conclusions: Based on the model and literature examples, it is predicted that a significant increase in K_B is to be expected where antagonist are evaluated with agonist with an associated n_H > 1.

References:

[1] Arunlakshana O & Schild HO, (1959) Br.J.Pharm., 14:48-58.

[2] Cheng HC, (2002) J.Pharm.Tox.Meth., 46:61-71.

[3] Holford N & Sheiner L, (1981) CRC Critical reviews in bioeng., 5:273-322.

[4] Leff P & Dougall LG, (1993) TiPS, 14:110-112.

[5] Waud DR, (1975), In Smooth Muscle, Daniel & Edwin ed., Fleming Press

[6] Zuideveld KP, et al., (2002),  J.Pharm.Exp.Ther., 300:330-338.