Single-cell directional sensing at ultra-low chemoattractant concentrations from extreme first-passage events

We investigate single-cell directional sensing from diffusing chemoattractant signals released by a localized source. We focus on the low-concentration regime in which receptor activity is discrete and cellular decisions are made on timescales far shorter than those required for steady-state concentration profiles or receptor saturation to emerge. We derive analytic expressions for the joint distribution of receptor binding times and binding locations, conditional on the position of the source. We show that early binding events carry disproportionately more information about source directionality than later arrivals. Motivated by this observation, we propose and analyze several source-localization estimates that exploit early receptor binding statistics. For a directional estimate formed from an equally weighted average of the earliest impacts, we demonstrate that this estimate is accurate even from a small number of impacts, and derive an expression for the number of binding events that minimizes the mean squared error. Our results demonstrate that cells possess sufficient information to rapidly and accurately infer the directionality of sparse signals from a localized source through a small number of discrete receptor engagements, well before a steady-state gradient has been established.