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Brownian simulations can be used to generate statistics relevant for studyingmolecular interactions or trafficking. However, the concurrent simulation ofmany Brownian trajectories at can become computationally intractable. Replacingdetailed Brownian simulations by a rate model was the basis of Gillespie'salgorithm, but requires one to disregard spatial information. However, thisinformation is crucial in molecular and cellular biology. Alternatively one canuse a hybrid approach, generating Brownian paths only in a small region wherethe spatial organization is relevant and avoiding it in the remainder of thedomain. Here we review the recent progress of hybrid methods and simulations inthe context of cell sensing and guidance via external chemical gradients.Specifically, we highlight the reconstruction of the location of a point sourcein 2D and 3D from diffusion fluxes arriving at narrow windows located on thecell. We discuss cases in which these windows are located on the boundary ofthe 2D or 3D half-space, on a disk in free space, inside a 2D corridor, or a 3Dball. The hybrid method in question performs Brownian simulations only inside aregion of interest. It uses the Neumann-Green's function for the mentionedgeometries to generate exact mappings exit and entry points when the trajectoryleaves the region, thus avoiding the explicit computation of Brownian paths inan infinite domain. Matched asymptotics is used to compute the probabilityfluxes to small windows and we review how such an approach can be used toreconstruct the location of a point source and estimating the uncertainty inthe source reconstruction due to an additive perturbation present in thefluxes. We also review the influence of various window configurations on thesource position recovery. Finally, we discuss potential applications indevelopmental cell biology and possible computational principles.

Computational methods and fast hybrid stochastic simulations for triangulation sensing and identifying principles of cell navigation in the brain