Local Clustering and Global Spreading of Receptors for Optimal Spatial Gradient Sensing

TL;DR

This work investigates the optimal placement of cell-surface receptors through a theoretical model that minimizes uncertainty in gradient estimation, and reproduces the emergence of clusters that closely resemble those observed in real cells.

Abstract

Spatial information from cell-surface receptors is crucial for processes that require signal processing and sensing of the environment. Here, we investigate the optimal placement of such receptors through a theoretical model that minimizes uncertainty in gradient estimation. Without requiring a priori knowledge of the physical limits of sensing or biochemical processes, we reproduce the emergence of clusters that closely resemble those observed in real cells. On perfect spherical surfaces, optimally placed receptors spread uniformly. When perturbations break their symmetry, receptors cluster in regions of high curvature, massively reducing estimation uncertainty. This agrees with mechanistic models that minimize elastic preference discrepancies between receptors and cell membranes. We further extend our model to motile receptors responding to cell-shape changes and external fluid flow, demonstrating the relevance of our model in realistic scenarios. Our findings provide a simple and utilitarian explanation for receptor clustering at high-curvature regions when high sensing accuracy is paramount.

Figures (4)

• Figure 1: Uncertainty of gradient estimation by the cell, normalized by the initial value of $\delta \widehat{g}(X_0)$, of a spherical cell perturbed by the harmonic $Y_6^4$ to form protrusions, in the case of uniformly spread receptors (black) or when placed at their optimal location for each $\alpha$ (orange).
• Figure 2: Optimal receptor regions match those of minimal elastic energy on an spherical cell perturbed by the $Y_7^4$ harmonic. (a) Mean curvature ($H$) in black and Gaussian curvature ($K$) in gray along the surface of the cell shown on the right, at fixed azimuthal angle $\phi=0\ \mathrm{rad}$ (top) and polar angle $\theta = 0.4\pi\ \mathrm{rad}$ (bottom). A single orange vertical line is placed at each angle where a receptor is found along the surface path. (b) Direction alignment between the gradients from elastic energy and uncertainty minimization Eq. \ref{['eq:alignment']}, for homogeneously spread receptors (orange) and receptors located on half the surface (blue). Error bars show the standard deviation of 100 realizations, where receptor position is uniformly sampled at the entire or half cell surface. The diagrams show the individual alignment value (cosine similarity) of each receptor indicated by the color.
• Figure 3: Effects of reacting motile receptors to membrane dynamics. (a) Frames from three independent simulations of the motion form protrusions caused by the harmonic $Y_4^3$ perturbation, where the receptors are immobile (top), have low speed (middle) or move fast (bottom) as the membrane oscillates. (b) Average gradient estimation error on an oscillating motion where protrusions are grown and retracted (3 full oscillations). The inset shows the oscillations in error for the three cases displayed in (a) as the membrane changes shape, with the black line showing the reduction for static optimization.
• Figure 4: Cluster formation in the presence of flow for a cell perturbed by the $Y_3^2$ harmonic. (a) Streamlines of the simulated Stokes flow around the cell. (b) Resulting cluster on an environment without flow (top), cluster formation under the effect of front-facing fluid flow (middle) and cluster morphology for a side protrusion affected by flow (bottom).