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The study of a receptor-ligand system generally relies on the analysis of itsdose-response (or concentration-effect) curve, which quantifies the relationbetween ligand concentration and the biological effect (or cellular response)induced when binding its specific cell surface receptor. Mathematical models ofreceptor-ligand systems have been developed to compute a dose-response curveunder the assumption that the biological effect is proportional to the numberof ligand-bound receptors. Given a dose-response curve, two quantities (ormetrics) have been defined to characterise the properties of theligand-receptor system under consideration: amplitude and potency (orhalf-maximal effective concentration, and denoted by EC$_{50}$). Both theamplitude and the EC$_{50}$ are key quantities commonly used inpharmaco-dynamic modelling, yet a comprehensive mathematical investigation ofthe behaviour of these two metrics is still outstanding; for a large (andimportant) family of receptors, called cytokine receptors, we still do not knowhow amplitude and EC$_{50}$ depend on receptor copy numbers. Here we make useof algebraic approaches (Gr\"obner basis) to study these metrics for a largeclass of receptor-ligand models, with a focus on cytokine receptors. Inparticular, we introduce a method, making use of two motivating examples basedon the interleukin-7 (IL-7) receptor, to compute analytic expressions for theamplitude and the EC$_{50}$. We then extend the method to a wider class ofreceptor-ligand systems, sequential receptor-ligand systems with extrinsickinase, and provide some examples.

Algebraic study of receptor-ligand systems: a dose-response analysis