Lattice Boltzmann method to simulate three-dimensional ion channel flow using fourth order Poisson–Nernst–Planck–Bikerman model

Abstract

Over the past three decades, the lattice Boltzmann method (LBM) has been applied to a vast range of hydrodynamic and non-hydrodynamic (e.g., ion transport) systems. In conjunction with the immersed boundary method (IBM), the LBM has been successfully implemented to solve systems with complex geometries. In this study, the immersed boundary–lattice Boltzmann method (IB-LBM) is implemented to simulate nanoscale ion transport. Traditionally, ion transport is described through the Poisson–Nernst–Planck (PNP) equations where ionic interactions are included. In the current paper, the fourth order Poisson–Nernst–Planck–Bikerman (4PNPBik) model has been used. In addition to ionic interactions, the 4PNPBik model includes the effects of the finite size of particles (ions and water) and interactions between ions and its surrounding medium. Applicability of the 4PNPBik model is demonstrated through comparison of the experimental and predicted ion activity. Implementation of the 4PNPBik model has been validated by comparing the predicted current–voltage curve with the analytical result. The transient receptor potential (TRP) ion channel of the vanilloid group (TRPV4) is used to demonstrate the applicability of this approach. The TRPV4 is a nonselective cation channel that prefers divalent cationic species over monovalent cations. In this study, this selectivity is demonstrated by comparing the concentration profiles of calcium, sodium, and chloride ions. Further, the role of the finite size of particles and nonlocal electrostatics is discussed by comparing the results obtained from the PNP and 4PNPBik models under identical initial and boundary conditions.

A detailed study of ion transport through the SARS-CoV-2 E protein ion channel

“…To conduct this analysis, experimentally obtained ion activity curves for NaCl and LiCl salts 39 have been compared with their analytical counterparts obtained upon solving the fourth order Poisson–Bikerman equations. It is to be noted that this analysis has previously been carried out 10,11 and we are repeating it here for the completeness of the study.…”

“…Screened forces for a system containing fixed/free charges in a dielectric medium are most commonly described by the van der Waals potential kernel : 10,11,24,28 which also satisfies the Laplace–Poisson kernel:…”

“…Please refer to ref. 34 and 11 for more details. For a D3Q19 lattice, w α and w̄ α can be calculated as:…”

“…Based on the Taylor series expansion of and further manipulations, one can obtain b 0 = c B 3 satisfies eqn (A5a). Substituting b 0 in eqn (A5b) one can obtain b 1 as:Charge density can be approximated as:Derivation for the PBik system 11 is explained next. The governing equation in Ω i ∪ Ω sh can be described aswhile the governing equation in Ω s :where .…”

“…In this study, we describe ion transport through the fourthorder Poisson-Nernst-Planck-Bikerman model (4PNPBik). 10,11 Traditional Poisson-Nernst-Planck equations describe the dynamics of ion transport in a mean-field sense where ions are assumed to be volumeless and electrostatic interactions for ions (fixed and free) are considered. The 4PNPBik model additionally includes the effects of the finite size of ions, finite size of water molecules and polarization of water molecules, etc.…”