Integral weak diffusion and diffusion approximations applied to ion transport through biological ion channels

In this article a theory is presented to calculate integral properties of biological ion channels (like currentvoltage and conductance-concentration relations). The qualitative form of these relations predicted by the theory agrees well with data measured in experiments. For instance, the saturation of the channel conductance with increasing external ion concentration is predicted for a class of ion channels (as, for instance, found for the gramicidin A, acetylcholine receptors, NMDA, and sarcoplasmic reticulum channels). In contrast to commonly used approaches such as the Eyring rate theory, this method is directly related to physical parameters of the ion channel such as the channel length and diameter, dielectric constant, ionic mobility, and minimal ionic concentration inside the channel. The theory starts from Nernst-Planck and Poisson equations. Using the method of phase trajectory (as proposed by Schottky) and the regional approximation, rather general expressions can be derived for integral channel quantities in the drift limit (|Vl > kBT/eo) in the presence of multiple ionic species. The theory predicts two typical types of conductance-concentration relations found experimentally: a monotone saturating conductance and a maximum in the conductance. The realized type of relation depends on the minimal ionic concentration inside the channel. In the present form the theory is restricted to narrow ion channels where the length exceeds its diameter. The ions are assumed to behave like structureless point charges at not too high ionic concentration