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

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.

The First 100 nm Inside the Pre-synaptic Terminal Where Calcium Diffusion Triggers Vesicular Release

“…An alternative is to replace such a rate by Brownian simulations, but this approach is in general heavy computationally. Recent hybrid simulations have been developed, where classical diffusion is used far away from a sensor, while near the boundary of a channel a Brownian representation is used (see Dobramysl and Holcman, 2018 for a description of such framework).…”

Reconstructing a point source from diffusion fluxes to narrow windows in three dimensions

“…Finding the source when the measured fluxes through the windows P 1 , P 2 and P 3 are given can be categorized into the general class of inverse problems. Contrary to the two-dimensional case [10,11], with three windows, the sum of the fluxes Φ 1 + Φ 2 + Φ 3 is strictly less than one because particles can escape to infinity in three dimensions. For this reason, the flux values provide separate and independent pieces of information which allows us to recover the source position with at least three windows.…”

“…We use a diffusion model to compute the fluxes through small windows either located on the boundary of half-space or the surface of a sphere in three dimensions. In previous work, we addressed the two-dimensional problem and assessed the sensitivity of direction sensing as a function of distance [10]. We found that long-distance sensing requires the confinement of the cell in a corridor [10].…”

“…In previous work, we addressed the two-dimensional problem and assessed the sensitivity of direction sensing as a function of distance [10]. We found that long-distance sensing requires the confinement of the cell in a corridor [10]. Furthermore, the reconstruction of the source position is possible, using at least three receptor windows, with a strong spatial heterogeneity in the associated uncertainty [11].…”

The first one hundred nanometers inside the pre-synaptic terminal where calcium diffusion triggers vesicular release

“…An alternative is to replace such a rate by Brownian simulations, but this approach is in general heavy computationally. Recent hybrid simulations have been developed, where classical diffusion is used far away from a sensor, while near the boundary of a channel a Brownian representation is used (see [32] for a description of such framework).…”