Gradient sensing limit of an elongated cell with orientational control

Eukaryotic cells perform chemotaxis by determining the direction of chemical gradients based on stochastic sensing of concentrations at the cell surface. To examine the efficiency of this process, previous studies have investigated the limit of estimation accuracy for gradients. However, most studies have treated a circular cell shape, and the few considering elongated shapes assume the elongated direction as fixed. This leaves the question of how adaptive regulation of cell shape affects the estimation limit. Dynamics of cell shape during gradient sensing is biologically ubiquitous and can influence the estimation by altering the way the concentration is measured, and cells may strategically regulate their shape to improve estimation accuracy. To address this gap, we investigate the estimation limits in dynamic situations where elongated cells change their orientation adaptively depending on the sensed signal. We approach this problem by analyzing the stationary solution of the Bayesian nonlinear filtering equation. By applying diffusion approximation to the ligand-receptor binding process and the Laplace method for the posterior expectation under a high signal-to-noise ratio regime, we obtain an analytical expression for the estimation limit. This expression indicates that estimation accuracy can be improved by aligning the elongated direction perpendicular to the estimated direction, which is also confirmed by numerical simulations. Our analysis provides a basis for clarifying the interplay between estimation and control in gradient sensing and sheds light on how cells optimize their shape to enhance chemotactic efficiency.