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

TL;DR

This work investigates single-cell directional sensing from diffusing chemoattractant signals released by a localized source and shows that early binding events carry disproportionately more information about source directionality than later arrivals.

Abstract

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 occupancy 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. Our results demonstrate that, even with a small number of binding events, cells possess sufficient information to rapidly and accurately infer the directionality of a diffusing chemoattractant source.

Figures (6)

• Figure 1: Schematic of source detection from the discrete arrival of signaling molecules to a circular cell. At $t=0$, a source located $R>1$ cell lengths away emits $N$ diffusing molecules of diffusivity $D>0$. These signaling molecules diffuse freely before being absorbed at a surface receptor (GCPR). The extreme directional sensing problem is the following: given the arrival times $\{T_{j,N}\}_{j=1}^k$ and locations $\{\theta_{j,N}\}_{j=1}^k$ of the first $k$ impacts based on $N$ initial walkers, what estimates can the cell form for the directionality of the source?
• Figure 2: Recovery of the extreme distribution parameter for test case $a=0.5$ for $n = 10^3$. Distribution in errors $(\hat{a}-a)$ using the full times (blue) and the time differences (green). Each histogram is formed by $M= 10^4$ repetitions and for each we overlay the distribution $\mathcal{N}(0,1/\sqrt{nI(a)}))$ where the Fisher Information $I(a)$ is given in (\ref{['eq_final_fishers']}).
• Figure 3: Accuracy in the prediction (\ref{['eq:estimate_R']}) of source distance using the full arrival times and the differences in arrival times in one-dimension. Panel (a): The distribution of errors for source detection when $R=5$ and $N = 10^6$ The estimate is vastly more accurate using the full times (blue) than the time differences (green). Distributions formed from $M=10^4$ independent estimates. In panels (b-c) we plot the average error against $k$ for initial walker numbers ranging from $N = 10^3-10^7$.
• Figure 4: The distribution of impact angles $\theta_{k,N}$ for $k=10$, $D=1$, $\theta_0 = 0$ and $N= 10^6$ for various $R$. The histograms are based on $M=10^4$ impacts sampled directly from the case of Dirichlet boundary condition ($\kappa = \infty$). The solid curve is the predicted limiting distribution (\ref{['eqn:AngleDist']}) with good agreement observed.
• Figure 5: Distribution of angular estimates in predictions using the average angles $\Theta_k^{\text{avg}}$ (green histograms) defined in (\ref{['eqn:AvgAngle']}) and the differences estimate $\Theta_k^{\text{diff}}$ (blue histograms) defined in (\ref{['eq:EqnDiff']}).