A hard sphere model for single-file water transport across biological membranes

We use Gürsey's statistical mechanics of a one-dimensional fluid to find a formula for the $$ ratio in the transport of hard spheres across a membrane through a narrow channel that can accommodate molecular movement only in single file. $$ is the membrane permeability for osmotic flow and $$ the permeability for exchange across the membrane in the absence of osmotic flow. The deviation of the ratio from unity indicates the degree of cooperative transport relative to ordinary diffusion of independent molecules. In contrast to an early idea that $$ must be equal to the number of molecules in the channel, regardless of the physical nature of the interactions among the molecules, we find a functional dependence on the fractional occupancy of the length of the channel by the hard spheres. We also attempt a random walk calculation for $$ individually, which gives a result for $$ as well when combined with the ratio.