Stationary-state distributions in membrane channels. I. Conservative flux systems

The population distribution of cations at binding sites in membrane channels is determined for stationary states characterized by a net ion flux. For conservative systems, the net flux conserves the total intra-channel cation population for an ensemble of channels. The requisite matrix formalism is developed and illustrated for some homogeneous channels with N single-ion occupancy states with chemical and electrochemical transmembrane gradients. The lowest eigen-value approximation, which is used effectively for systems with no net cation flux, is applied to these stationary-state systems for comparison with the exact solutions. The stationary states are characterized by special thermodynamic functions for entropy, internal energy, and free energy defined by using information theory. The changes in these parameters for the transition from the equilibrium to the stationary state depend on the ratio of the next flux to the transition velocity for nearest-neighbor transitions within the channel. The free energy changes with the square of this ratio while the internal energy changes linearly.