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