  Brownian simulations can be used to generate statistics relevant for studying
molecular interactions or trafficking. However, the concurrent simulation of
many Brownian trajectories at can become computationally intractable. Replacing
detailed Brownian simulations by a rate model was the basis of Gillespie's
algorithm, but requires one to disregard spatial information. However, this
information is crucial in molecular and cellular biology. Alternatively one can
use a hybrid approach, generating Brownian paths only in a small region where
the spatial organization is relevant and avoiding it in the remainder of the
domain. Here we review the recent progress of hybrid methods and simulations in
the context of cell sensing and guidance via external chemical gradients.
Specifically, we highlight the reconstruction of the location of a point source
in 2D and 3D from diffusion fluxes arriving at narrow windows located on the
cell. We discuss cases in which these windows are located on the boundary of
the 2D or 3D half-space, on a disk in free space, inside a 2D corridor, or a 3D
ball. The hybrid method in question performs Brownian simulations only inside a
region of interest. It uses the Neumann-Green's function for the mentioned
geometries to generate exact mappings exit and entry points when the trajectory
leaves the region, thus avoiding the explicit computation of Brownian paths in
an infinite domain. Matched asymptotics is used to compute the probability
fluxes to small windows and we review how such an approach can be used to
reconstruct the location of a point source and estimating the uncertainty in
the source reconstruction due to an additive perturbation present in the
fluxes. We also review the influence of various window configurations on the
source position recovery. Finally, we discuss potential applications in
developmental cell biology and possible computational principles.

Statistical Mechanics;Analysis of PDEs;Cell Behavior