Finite Ion Size Effects on Ionic Flows via Poisson–Nernst–Planck Systems: Higher Order Contributions

Ions are crowded in ion channels, and finite ion sizes play essential roles in the study of ionic flows through membrane channels. Some significant properties of ion channels, such as selectivity, rely on ion sizes critically. Following the work done in (SIAM J Appl Dyn Syst 12:1613–1648, 2013), we focus on the higher order (in the diameter of the cation), mainly the second order, contributions from finite ion sizes to ionic flows in terms of both the total flow rate of charges and the individual fluxes. This is particularly essential because the first-order terms approach zero when the left boundary concentration is close to the right one for the same ion species. The interplays between the first-order terms and the second-order terms are characterized. Furthermore, several critical potentials are identified, which play critical roles in examining the dynamics of ionic flows. Some can be experimentally estimated. The analysis could provide deep insights into the future studies of ionic flows through membrane channels.