Numerical methods for a Poisson-Nernst-Planck-Fermi model of biological ion channels

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. The generalized SG stability condition shows why the standard discretization (without SG exponential fitting) of NP equations may fail and that divalent Ca(2+) may cause more unstable discrete Ca(2+) fluxes than that of monovalent Na(+). 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. Numerical methods are first validated with constructed models whose exact solutions are known. The experimental data of both channels are then used to verify and explain novel features of PNPF as compared with previous PNP models. 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 with 10(-8)-fold bath concentrations that pose severe challenges in theoretical simulations.

A GPU Poisson–Fermi solver for ion channel simulations

“…The Poisson-Fermi (PF) model [9][10][11][12][13][14][15][16][17] is a fourth-order nonlinear partial differential equation (PDE) that, in addition to the electric effect, can be used to describe the steric, correlation, and polarization effects of water molecules and ions in aqueous solutions. Ions and water are treated as hard spheres with different sizes, different valences, and interstitial voids, which yield Fermi-like distributions that are bounded above for any arbitrary (or even infinite) electric potential at any location of the system domain of interest.…”

“…These effects and properties cannot be described by the classical Poisson-Boltzmann theory that consequently has been slowly modified and improved [18][19][20][21][22][23][24][25][26][27][28] for more than 100 years since the work of Gouy and Chapman [29,30]. It is shown in [11,17] However, in addition to the computational complexity of PB solvers for biophysical simulations, the PF model incurs more difficulties in numerical stability and convergence and is thus computationally more expensive than the PB model as described and illustrated in [9,13]. To reduce long execution times of PF solver on CPU, we propose here two GPU algorithms, one for linear algebraic system solver and the other for nonlinear PDE solver.…”

The Role of Proton Transport in Gating Current in a Voltage Gated Ion Channel, as Shown by Quantum Calculations

“…Calculations have been done for the gating mechanism, or the response to a change in voltage, as well as inactivation, selectivity, and conduction. There are essentially electrostatic physical models of ion permeation through the pore, largely proposed by Eisenberg and coworkers (120)(121)(122)(123); these tend to use a simplified model of the channel, in which many of the specific interactions are omitted, or lumped into a smaller number of parameters; it is interesting that electrostatics can account for so much of the description of ion permeation. That said, it still will not be sufficient for the explicit description of the interactions involved in ion transport, nor is that the intended purpose of such an essentially macroscopic view.…”

Poisson-Fermi modeling of ion activities in aqueous single and mixed electrolyte solutions at variable temperature

“…In the past few years, we have intensively investigated these two effects in a range of areas from electric double layers [17,18], ion activities [19], to biological ion channels [18,[20][21][22][23][24] and consequently developed an advanced theory -the Poisson-Fermi (PF) theory -that treats ions and water molecules as nonuniform hard spheres of any size with interstitial voids and includes many of the correlation effects of ions and water. We refer to our previous papers and references therein for a historical account of the literature of this theory.…”