Biological ion channels are essential for maintaining life, and appear as a seemingly paradoxical combination of both large conductivity and strong selection between ionic species. This process involves many complicated interactions, and their inclusion in a multi-species conduction model remains a fundamental theoretical challenge. In this thesis, we derive the theory of multi-species ionic conduction through narrow biological channels, taking into account ion-ion, ion-water and ion-channel interactions. The theories we derive lead to new results that describe multi-species conduction in and far from equilibrium in KcsA, including the resolution of the conductivity-selectivity paradox.
The thesis builds on existing research on the physiological properties and structures of biological ion channels in deriving a first-principles, multi-species statistical and kinetic theory. The development of the statistical theory includes the derivation of the free energy, distribution and partition functions, as well as the statistical properties within the grand canonical ensemble. The conductivity of the channels is also derived using linear response theory and the generalised Einstein relation. The development of the kinetic theory involves the analysis of the transition rates, and the calculation of current and selectivity ratios. The kinetic model is then validated by comparing the theoretical currents with the currents measured experimentally for the Shaker and KcsA potassium channels in five different external data sets.
The main results of this thesis are: a derivation of the grand canonical ensemble for narrow channels with multiple binding sites and mixed-species bulk solutions; a derivation of the linear response theory of multi-species conduction in such channels; development of non-equilibrium multi-species kinetic equations, that describe the conductivity; the validation of the kinetic theory through comparison with experimental data sets; and the joint application of these derived theories to the multi-species conduction of KcsA in and far from equilibrium, which demonstrates the resolution of the conductivity-selectivity paradox. These results should be applicable to other narrow voltage-gated ion channels, and can describe multi-species conduction of neutral particles through zeolites.