Modeling of Bias for the Analysis of Receptor Signaling in Biochemical Systems

Gprotein-coupled receptors (GPCRs) signal through multiple pathways that are regulated by G proteins andβ-arrestins.1,2 Many of these signaling pathways respond selectively to ligands that are able to stabilize preferred subsets of receptor conformations.3,4 As a result of conformational heterogeneity, small differences in ligand structure can dramatically shift signaling toward one response pathway and away from another.5 The ability of an agonist–receptor pair to produce a quantitative response, measured as efficacy, has been historically modeled by a transducer ratio parameter reflecting the total receptor concentration and the transduction of the agonist–receptor complex into a pharmacological response.6 Potent ligands having low transducer ratios may not be efficacious, and conversely, efficacious responses precipitated by large transducer ratios do not necessarily require potent ligands. Because ligand potencies and their associated transducer ratios can vary widely, a signaling bias may result in which different ligands produce variable degrees of response in a single pathway or a single ligand displays large differences in efficacy between two independent signaling pathways.7 A comprehensive review of qualitative and quantitative strategies for assessing ligand bias is found in ref (8). The available approaches similarly address bias within the confines of experiment and attempt to define it observationally or numerically, by data trends or bias factors, as a property that arises from the signaling paradigm. In contrast, an axiomatic formalism for bias could be developed in a manner that is independent of experiment and subsequently applied to a particular signaling paradigm. We believe that this latter approach allows a broader treatment of signaling bias and provides a more fundamental development and conceptual understanding of bias-dependent factors. Applying this strategy to logistic (sigmoid) response functions representative of most biological processes,6 we present a comprehensive, simple formalism for qualitative and quantitative signaling bias comparisons. In this formulation, hyperbolae represent the comparative responses of test ligands, and signaling biases are described by mappings of bias coordinates representing the hyperbolae from the unit square to a stack of Poincare unit disks. Bias factors are simple consequences of the map and the novel distance metric of the disk, and the distance between bias coordinates in the disk provides a quantitative means of characterizing and sorting ligands. Our analysis of comparative signaling bias, which can be applied to many signal transduction systems, was developed with G protein-coupled receptors in mind, and we illustrate the approach and its utility using dopamine 2 receptor signaling.