Triangulation sensing: how cells recover a source from diffusing particles in three dimensions

How can cells embedded into a gradient concentration triangulate the position of the source and migrate toward their final destination? The source triangulation requires to recover the three dimensional coordinates of the source from the fluxes of diffusing cues at narrow windows (receptors) located on the surface of a cell. We develop here a method to address this question and we show in the limit of fast binding rate to the receptors, that at least three receptors are necessary. We solve the steady-state diffusion equation using an asymptotic approach, which agrees with hybrid stochastic-analytical simulations. Interestingly, with an accuracy of few percent, the source cannot be located if it is located at a distance tens of time the size of the cell. Finally, the precision of the source recovery increases with the number of receptors.