Universal properties of estimating many ligand species concentrations by many cellular receptor types

All organisms, and specifically cells, are faced with the task to sense the concentrations of a large variety of ligands using a limited set of receptors. As the number of receptor types is significantly smaller than ligands, the inference is impossible through deterministic steady-state chemical kinetics. Recently (Singh and Nemenman, 2017, PLOS Comp Biol), we have shown that the sequence of bound-unbound states of a receptor carries more information than its mean occupancy. This can be used to infer concentrations of several ligands from the activity of a single receptor, as long as the unbinding rates of the ligands are sufficiently different. Here, extending these ideas to a network of multiple receptors that bind multiple ligands, we show that the temporal sequences of binding-unbinding events again carry more information than the mean occupancy. The analysis of the Fisher information matrix shows that, for a random distribution of unbinding rates, one can estimate concentration of 3-10 times as many ligands as there are receptors, in realistic time. The spectrum of the Fisher matrix shows interesting scaling regimes with the ratio of the number of ligands to receptors, and the scaling is universal for different probability distribution of the unbinding rates.