Qualitative properties of ionic flows via Poisson-Nernst-Planck systems with Bikerman's local hard-sphere potential: Ion size effects

Jia, Yusheng; Liu, Weishi; Zhang, Mingji

doi:10.3934/dcdsb.2016022

Cited by 25 publications

(35 citation statements)

References 81 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Indeed, two steady state solutions of PNP-steric equations were proved for three types of ion species (two types of cations and one type of anion) and four types of ion species (two types of cations and their counter-ions) with specific assumptions on permanent charges in channel proteins. In Jia et al (2016), a quasi-one-dimensional steadystate PNP model modified with size effect was studied as a singularly perturbed differential system, with fixed boundary ion concentrations and electric potentials. The existence of solutions to the boundary value problem for small ion sizes was investigated with the ion sizes as small parameters.…”

Section: Mathematical Analysismentioning

confidence: 99%

A Review of Mathematical Modeling, Simulation and Analysis of Membrane Channel Charge Transport ☆

Chen

Wei

2017

Reference Module in Life Sciences

View full text Add to dashboard Cite

The molecular mechanism of ion channel gating and substrate modulation is elusive for many voltage gated ion channels, such as eukaryotic sodium ones. The understanding of channel functions is a pressing issue in molecular biophysics and biology. Mathematical modeling, computation and analysis of membrane channel charge transport have become an emergent field and give rise to significant contributions to our understanding of ion channel gating and function. This review summarizes recent progresses and outlines remaining challenges in mathematical modeling, simulation and analysis of ion channel charge transport. One of our focuses is the Poisson-Nernst-Planck (PNP) model and its generalizations. Specifically, the basic framework of the PNP system and some of its extensions, including size effects, ion-water interactions, coupling with density functional theory and relation to fluid flow models. A reduced theory, the PoissonBoltzmann-Nernst-Planck (PBNP) model, and a differential geometry based ion transport model are also discussed. For proton channel, a multiscale and multiphysics Poisson-Boltzmann-Kohn-Sham (PBKS) model is presented. We show that all of these ion channel models can be cast into a unified variational multiscale framework with a macroscopic continuum domain of the solvent and a microscopic discrete domain of the solute. The main strategy is to construct a total energy functional of a charge transport system to encompass the polar and nonpolar free energies of solvation and chemical potential related energies. Using the Euler-Lagrange variation, the coupled PNP equations and other transport equations are derived, whose solutions lead to the minimization of the total free energy and explicit profiles of electrostatic potential and densities of charge species. Current computational algorithms and tools for numerical simulations and results from mathematical analysis of ion channel systems are also surveyed.

show abstract

Section: Mathematical Analysismentioning

confidence: 99%

A Review of Mathematical Modeling, Simulation and Analysis of Membrane Channel Charge Transport ☆

Chen

Wei

2017

Reference Module in Life Sciences

View full text Add to dashboard Cite

show abstract

“…One common approach is to add an excess chemical potential to the potential of mean force. Such a correction is able to address the excluded volume effect [7,13,[20][21][22][23][24][25]34,35,41,43,45,48,49,52], dielectric effects [9,32,40,44], and ion-ion correlations [39,53]. For instance, the excluded volume effect is included by considering the entropy of solvent molecules, giving rise to a model with nonlinearly modified mobilities [25,43,49].…”

Section: Introductionmentioning

confidence: 99%

“…A more sophisticated strategy is to incorporate the excluded volume effect by adding an excess chemical potential, which is described by the density functional theory (DFT) [22,24,41,45], or by the Lennard-Jones potential accounting for hard-sphere repulsions [13,20,35]. To avoid computationally intractable integro-differential equations, local approximations of nonlocal integrals are employed to obtain local models [20,21,23,34,35,48].…”

Section: Introductionmentioning

confidence: 99%

A modified Poisson–Nernst–Planck model with excluded volume effect: theory and numerical implementation

Siddiqua

Wang

Zhou

2018

Communications in Mathematical Sciences

View full text Add to dashboard Cite

The Poisson-Nernst-Planck (PNP) equations have been widely applied to describe ionic transport in ion channels, nanofluidic devices, and many electrochemical systems. Despite their wide applications, the PNP equations fail in predicting dynamics and equilibrium states of ionic concentrations in confined environments, due to the ignorance of the excluded volume effect. In this work, a simple but effective modified PNP (MPNP) model with the excluded volume effect is derived, based on a modification of diffusion coefficients of ions. At the steady state, a modified Poisson-Boltzmann (MPB) equation is obtained with the help of the Lambert-W special function. The existence and uniqueness of a weak solution to the MPB equation are established. Further analysis on the limit of weak and strong electrostatic potential leads to two modified Debye screening lengths, respectively. A numerical scheme that conserves total ionic concentration and satisfies energy dissipation is developed for the MPNP model. Numerical analysis is performed to prove that our scheme respects ionic mass conservation and satisfies a corresponding discrete free energy dissipation law. Positivity of numerical solutions is also discussed and numerically investigated. Numerical tests are conducted to demonstrate that the scheme is of second-order accurate in spatial discretization and has expected properties. Extensive numerical simulations reveal that the excluded volume effect has pronounced impacts on the dynamics of ionic concentration and flux. In addition, the effect of volume exclusion on the timescales of charge diffusion is systematically investigated by studying the evolution of free energies and diffuse charges.

show abstract

“…Most of mathematical analyses of the PNP equations are based on singular perturbation methods, in which the Debye length is assumed to be much smaller than the dimensions of ion channels, giving rise to a small singular parameter. Such singular perturbation problems are solved mainly by two categories of methods: matched asymptotic expansions [1,[4][5][6]30,54,55,58,61] and geometric singular perturbation theory [11,12,26,27,35,[38][39][40]. For instance, Wang et al study the PNP equations using matched asymptotic analysis, and prove the existence and uniqueness of the solution to the PNP equations with two and three ionic species [61].…”

Section: Introductionmentioning

confidence: 99%

Computational Study on Hysteresis of Ion Channels: Multiple Solutions to Steady-State Poisson-Nernst-Planck Equations

Ding¹,

Sun²,

Wang³

et al. 2018

CiCP

View full text Add to dashboard Cite

The steady-state Poisson-Nernst-Planck (ssPNP) equations are an effective model for the description of ionic transport in ion channels. It is observed that an ion channel exhibits voltage-dependent switching between open and closed states. Different conductance states of a channel imply that the ssPNP equations probably have multiple solutions with different level of currents. We propose numerical approaches to study multiple solutions to the ssPNP equations with multiple ionic species. To find complete current-voltage (I-V ) and current-concentration (I-C) curves, we reformulate the ssPNP equations into four different boundary value problems (BVPs). Numerical continuation approaches are developed to provide good initial guesses for iteratively solving algebraic equations resulting from discretization. Numerical continuations on V , I, and boundary concentrations result in S-shaped and double S-shaped (I-V and I-C) curves for the ssPNP equations with multiple species of ions. There are five solutions to the ssPNP equations with five ionic species, when an applied voltage is given in certain intervals. Remarkably, the current through ion channels responds hysteretically to varying applied voltages and boundary concentrations, showing a memory effect. In addition, we propose a useful computational approach to locate turning points of an I-V curve. With obtained locations, we are able to determine critical threshold values for hysteresis to occur and the interval for V in which the ssPNP equations have multiple solutions. Our numerical results indicate that the developed numerical approaches have a promising potential in studying hysteretic conductance states of ion channels.

show abstract

Qualitative properties of ionic flows via Poisson-Nernst-Planck systems with Bikerman's local hard-sphere potential: Ion size effects

Cited by 25 publications

References 81 publications

A Review of Mathematical Modeling, Simulation and Analysis of Membrane Channel Charge Transport ☆

A Review of Mathematical Modeling, Simulation and Analysis of Membrane Channel Charge Transport ☆

A modified Poisson–Nernst–Planck model with excluded volume effect: theory and numerical implementation

Computational Study on Hysteresis of Ion Channels: Multiple Solutions to Steady-State Poisson-Nernst-Planck Equations

Contact Info

Product

Resources

About