Modeling of bias for the analysis of receptor signaling in biochemical systems

Ligand bias is a recently introduced concept in the receptor signaling field that underlies innovative strategies for targeted drug design. Ligands, as a consequence of conformational selectivity, produce signaling bias in which some downstream biochemical pathways are favored over others, and this contributes to variability in physiological responsiveness. Though the concept of bias and its implications for receptor signaling have become more important, its working definition in biochemical signaling is sufficiently imprecise as to impede the use of bias as an analytical tool. In this work, we provide a precise mathematical definition for receptor signaling bias using a formalism expressly applied to logistic response functions, models of most physiological behaviors. We show that signaling-response bias of biological processes may be represented by hyperbolae, or more generally as families of bias coordinates that index hyperbolae. Furthermore, we show bias is a property of a parametric mapping of these indexes into vertical strings that reside within a cylinder of stacked Poincare disks and that bias factors representing signaling probabilities are the radial distance of the strings from the cylinder axis. The utility of the formalism is demonstrated with logistic hyperbolic plots, by transducer ratio modeling, and with novel examples of Poincare disk plots of Gi and β-arrestin biased dopamine 2 receptor signaling. Our results provide a platform for categorizing compounds using distance relationships in the Poincare disk, indicate that signaling bias is a relatively common phenomenon at low ligand concentrations, and suggest that potent partial agonists and signaling pathway modulators may be preferred leads for signal bias-based therapies.