Calculating tracer currents through narrow ion channels: beyond the independent particle model

Discrete state models of single-file ion permeation through a narrow ion channel pore are employed to analyze the ratio of forward to backward tracer current. Conditions under which the well-known Ussing formula for this ratio hold are explored in systems where ions do not move independently through the channel. Building detailed balance into the rate constants for the model in such a way that under equilibrium conditions (equal rate of forward versus backward permeation events) the Nernst equation is satisfied, it is found that in a model where only one ion can occupy the channel at a time, the Ussing formula is always obeyed for any number of binding sites, reservoir concentrations of the ions and electric potential difference across the membrane which the ion channel spans, independent of the internal details of the permeation pathway. However, numerical analysis demonstrates that when multiple ions can occupy the channel at once, the nonequilibrium forward/backward tracer flux ratio deviates from the prediction of the Ussing model. Assuming an appropriate effective potential experienced by ions in the channel, we provide explicit formulae for the rate constants in these models.