Numerical Methods for a Poisson-Nernst-Planck-Fermi Model

TL;DR

This paper introduces advanced numerical methods for the Poisson-Nernst-Planck-Fermi model, capturing complex ion-water interactions and correlations in biological channels, with results aligning well with experimental data.

Contribution

It extends classical numerical schemes to include steric and correlation effects, enabling more accurate simulations of ion transport in biological channels.

Findings

PNPF model accounts for water and dielectric properties.

Extended SG method achieves optimal convergence.

Simulation results agree with experimental data.

Abstract

Numerical methods are proposed for an advanced Poisson-Nernst-Planck-Fermi (PNPF) model for studying ion transport through biological ion channels. PNPF contains many more correlations than most models and simulations of channels, because it includes water and calculates dielectric properties consistently as outputs. This model accounts for the steric effect of ions and water molecules with different sizes and interstitial voids, the correlation effect of crowded ions with different valences, and the screening effect of polarized water molecules in an inhomogeneous aqueous electrolyte. The steric energy is shown to be comparable to the electrical energy under physiological conditions, demonstrating the crucial role of the excluded volume of particles and the voids in the natural function of channel proteins. Water is shown to play a critical role in both correlation and steric effects in the model. We extend the classical Scharfetter-Gummel (SG) method for semiconductor devices to include the steric potential for ion channels, which is a fundamental physical property not present in semiconductors. Together with a simplified matched interface and boundary (SMIB) method for treating molecular surfaces and singular charges of channel proteins, the extended SG method is shown to exhibit important features in flow simulations such as optimal convergence, efficient nonlinear iterations, and physical conservation. Two different methods --- called the SMIB and multiscale methods --- are proposed for two different types of channels, namely, the gramicidin A channel and an L-type calcium channel, depending on whether water is allowed to pass through the channel. The PNPF currents are in accord with the experimental I-V (V for applied voltages) data of the gramicidin A channel and I-C (C for bath concentrations) data of the calcium channel.