Ion channels are pore-forming trans-membrane proteins that allow ions to enter/leave cell. There are many important cell functions involving ion channels, e.g., establishing and regulating action potential in neurons and myocytes. The average time for an ion passing through ion channel is in the order of ms, which is infeasible for molecular dynamics (MD) simulation so far. Continuum model like Poisson-Boltzmann equation (PB) and Poisson-Nernst-Planck (PNP) equations are popular to describe ion channel in equilibrium and non-equilibrium situations.

The X-ray crystallographic structures of distinct potassium channels reveal a common architecture of the pore. Four subunits are symmetrically arranged around the channel axis, with each subunit having at least two transmembrane helixes separated by a re-entrant P-loop and selectivity filter (SF). K channels are the most extensively studied family of ion channels, both experimentally and computationally, and KcsA structure has been the most popular one among K channels since it is the first K channel to be crystalized. Many computational and experimental data of KcsA is available for comparison.

SF of K channels is the essential element to their permeation and selectivity mechanisms. Thousands of millions of K ions per second can diffuse in single file down their electrochemical gradient across the membrane at physiological conditions. Each subunit contributes to SF with a conserved signature peptide, namely TVGYG in most of the channels. The carbonyl oxygens of the backbone of SF point toward the lumen and orchestrate the movements of ions in and out of the channel. These carbonyl oxygens together with the side-chain hydroxyl oxygen of a threonine residue define four ion-binding sites in SF, designated S1-S4 starting at the extracellular side. In addition, K ion can bind in the central water-filled cavity of pore and two alternate positions at the extracellular side of pore. SF is generally too narrow to accommodate a K ion with its hydration shell, and thus K ions must be dehydrated to enter SF, when attracted by the strong negative charges of carbonyl oxygens in SF. K ion must replace its solvation shell by the carbonyl oxygens in the backbone of SF. Each of these protein sites binds K ions with a tight-fitting cage of 8 carbonyl oxygen atoms that resembles the solvation shell of a hydrated K ion.

3D PB and PNP simulations of KcsA channel have been a challenging task, since (1) geometry is complicated especially the extremely narrow SF generally requiring high resolution of meshes distributed there; (2) mathematical models are complicated since modifications of PB/PNP are generally required to accommodate additional physical effects; (3) parameters such as dielectric constant and diffusion coefficient are generally unknown inside SF; (4) the strong negative charges of carbonyl oxygens in SF makes the numerical integration in time extremely stiff. Here, a PDB 3F7Y KcsA structure with filter part replaced by that of PDB 1K4C is used as the structure for simulation. Unlike all other KcsA PDB structures, this synthetic structure guarantees that the channel is open. 3D modified PB and PNP solvers have been developed by the author under the framework of method of lines (MOL). Governing equations are semi-discretized in space by 2nd order finite volume method under Cartesian grids with the cell edge value to cope with interface conditions. This semi-discretized system forms a system of ordinary differential algebraic equations (ODAE) that can be further integrated by popular ODAE solvers. Mathematical models simulated here are (I) classical PB, (II) modified PB with steric effect described by Bikerman model, and (III) modified PB as (II) with solvation energy included in addition. From simulation results, we found potassium ion is unrealistically crowded in SF for model (I). For model (II), though potassium ion is no more unrealistically crowded in SF due to the inclusion of steric effect, there is no room for water in SF. Model (III) delivers the most reasonable physical results, compared with MD simulations, among all three models by obtaining reasonable potassium concentration under steric effect and allowing water residence in SF at the same time.

Simulation results based on model (III) further explored the following: (1) S2 site is the most stable binding site from line density distribution in SF, which agrees with MD simulations; (2) Pile-up of K at cavity sites and S0 sites are exclusively found by model (III). It is due to saturated K inside SF remaining insufficient to balance the carbonyl oxygen negative charges over there. This clears up why water cavity of KcsA is a binding site for K observed from X-ray crystallography, though electrostatic attraction is actually weak there from the structure constituting water cavity. Cavity site is actually a false binding site; (3) As to Na-K selectivity, Na loses the competition to K on occupation of SF due to larger solvation energy. However, Na wins the occupation of S0 and cavity sites due to smaller size. This explains its interference with the conduction of K in KcsA channel, that has been observed in experiments.