Assessing the (a)symmetry of concentration-effect curves: empirical versus mechanistic models

Modeling the shape of concentration-effect curves is of prime importance in pharmacology. Geometric descriptors characterizing these curves (the upper and lower asymptotes, the mid-point, the mid-point slope, and the point of inflection) are used for drug comparison or for assessing the change in agonist function after a system modification. The symmetry or asymmetry around the mid-point of a concentration-effect curve is a fundamental property that, regretfully, is often overlooked because, generally, models yielding exclusively symmetric curves are used. In the present review, empirical and mechanistic models are examined in their ability to fit experimental data. The geometric parameters of a survey of empirical models, the Hill equation, a logistic variant that we call the modified Hill equation, the Richards function, and the Gompertz model are determined. To analyze the relationship between asymmetry and mechanism, some examples from the ionic channel field, in an increasing degree of complexity, are used. It is shown that asymmetry arises from ionic channels with multiple binding sites that are partly occupied. The operational model of agonism is discussed both in its empirical general formulation and including the signal transduction mechanisms through G-protein-coupled receptors. It is shown that asymmetry results from systems where receptor distribution is allowed. Developed mathematical models are compared for describing experimental data on alpha-adrenoceptors. The existence or not of a relationship between the shape of the curves and receptor reserve is discussed.