Optimal distributions of receptors on arbitrarily shaped cell surfaces

Efficient absorption of signaling molecules and carbohydrates through receptors on cell surfaces is crucial for various biological processes. While ubiquitous patterns of receptor distributions, including polar localization in rod-shaped cells, have been widely observed experimentally, their underlying evolutionary advantage is unclear. In this work, we study how spatial distributions of receptors on cell surfaces affect the total flux entering the cell. We innovate a method by which one can calculate the fluxes through all receptors using linear equations, which applies to arbitrarily shaped cells. Our theories recover previous results for spherical cells and further show that the flux through each receptor is spatially dependent in non-spherical cells. In particular, the fluxes are the highest near the poles in rod-shaped cells and the highest near the invagination in defective spherical cells. Surprisingly, we prove that the optimal receptor distribution on an arbitrarily shaped cell maximizing the total flux is precisely the charge density distribution on an ideal conductor of the same shape, which agrees with numerical simulations. Our work unveils the evolutionary origin of receptor localizations.