Isoboles
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Concentration addition as a classic null model for toxicology and pharmacology is based on Loewe's mathematical formulation and the linearity of the isoboles. Novel mathematical models, however, propose curved isoboles in certain conditions. This article aims to test the hypothesis of the curvature of isoboles in experimental measurements. With the assumption of linear isoboles, a partial agonist acts as an antagonist above its maximal effect level. The isoboles automatically convert to a positive slope. For curved isoboles, a partial agonist acts as an antagonist at higher effect levels than its maximal effect alone. The discrepancies between effect levels were studied with an estrogen receptor binding assay (BMAEREluc/ERα) using a mixture of 17β-estradiol and fulvestrant as a partial agonist. A mixture of 17β-estradiol and fulvestrant acts as a partial agonist and causes the diminishing of the effect level of 17β-estradiol at a significantly higher level than the maximal effect of their partial-agonistic dose-response curve. Measured, elevated effect levels were well predicted by the mathematical model. Nonlinear isoboles may change our understanding and definition of synergism or antagonism and prompt further attention in receptor theory.
Receptor ligands in mixtures may produce effects that are greater than the effect predicted from their individual dose-response curves. The historical basis for predicting the mixture effect is based on Loewe's concept and its mathematical formulation. This concept considers compounds with constant relative potencies (parallel dose-response curves) and leads to linear additive isoboles. These lines serve as references for distinguishing additive from nonadditive interactions according to the positions of the experimental data on or outside of the lines. In this paper, we applied a highly relevant two-state model for a description of the receptor-ligand interaction in the construction of the isobologram. In our model we consider partial agonists that have dose-response curve slopes differing from one. With this theoretical basis, we demonstrated that a combination of compounds with different efficacies leads to curved isoboles. This model should overwrite Tallarida's flawed assumption about isobolographic analysis of partial agonists and enhance our understanding of how the partial agonists contribute to the overall mixture effect.
