Diffusional screening in treelike spaces: an exactly solvable diffusion-reaction model.

A renormalization approach is used to derive an analytic formula for the total current crossing the reactive surface of a Cayley tree of cylindrical tubes under a Helmholtz-type approximation to the full diffusion-reaction problem. We provide analytic conditions for the emergence of a plateau in the current-a region of maximum insensitivity of the current to variations in either the reaction rate (membrane permeability) or the diffusivity. The occurrence of such a plateau is associated with a partial screening regime wherein most of the active surface is screened to incoming diffusing particles. Large trees trade efficiency for fault tolerance, a valuable feature which may provide robustness to mammalian respiratory systems and tolerance to catalytic poisoning in chemical reactors.