Mixed analytical-stochastic simulation method for the recovery of a Brownian gradient source from probability fluxes to small windows

TL;DR

This paper introduces a numerical method to recover the position of a Brownian particle source from steady-state fluxes to small boundary windows, avoiding full trajectory tracking and applicable in biological sensing contexts.

Contribution

It develops an analytical-stochastic hybrid simulation approach and provides a formula for source localization based on flux measurements, advancing understanding of gradient sensing.

Findings

Validated the method with stochastic simulations

Derived asymptotic expressions for fluxes and splitting probabilities

Demonstrated source reconstruction in complex boundary configurations

Abstract

Is it possible to recover the position of a source from the steady-state fluxes of Brownian particles to small absorbing windows located on the boundary of a domain? To address this question, we develop a numerical procedure to avoid tracking Brownian trajectories in the entire infinite space. Instead, we generate particles near the absorbing windows, computed from the analytical expression of the exit probability. When the Brownian particles are generated by a steady-state gradient at a single point, we compute asymptotically the fluxes to small absorbing holes distributed on the boundary of half-space and on a disk in two dimensions, which agree with stochastic simulations. We also derive an expression for the splitting probability between small windows using the matched asymptotic method. Finally, when there are more than two small absorbing windows, we show how to reconstruct the position of the source from the diffusion fluxes. The present approach provides a computational first principle for the mechanism of sensing a gradient of diffusing particles, a ubiquitous problem in cell biology.