Nonlinear Poisson–Nernst–Planck equations for ion flux through confined geometries

The mathematical modelling and simulation of ion transport through biological and synthetic channels (nanopores) is a challenging problem, with direct application in biophysics, physiology and chemistry. At least two major effects have to be taken into account when creating such models: the electrostatic interaction of ions and the effects due to size exclusion in narrow regions. While mathematical models and methods for electrostatic interactions are well-developed and can be transferred from other flow problems with charged particles, e.g. semiconductor devices, less is known about the appropriate macroscopic modelling of size exclusion effects.Recently several papers proposed simple or sophisticated approaches for including size exclusion effects into entropies, in equilibrium as well as off equilibrium. The aim of this paper is to investigate a second potentially important modification due to size exclusion, which often seems to be ignored and is not implemented in currently used models, namely the modification of mobilities due to size exclusion effects. We discuss a simple model derived from a self-consisted random walk and investigate the stationary solutions as well as the computation of conductance. The need of incorporating nonlinear mobilities in high density situations is demonstrated in an investigation of conductance as a function of bath concentrations, which does not lead to obvious saturation effects in the case of linear mobility.

[1]  J. Simonin,et al.  Real Ionic Solutions in the Mean Spherical Approximation. 3. Osmotic and Activity Coefficients for Associating Electrolytes in the Primitive Model , 1998 .

[2]  Matthew J. Simpson,et al.  Multi-species simple exclusion processes , 2009 .

[3]  O. A. Ladyzhenskai︠a︡,et al.  Linear and quasilinear elliptic equations , 1968 .

[4]  P. Bassanini,et al.  Elliptic Partial Differential Equations of Second Order , 1997 .

[5]  Bo Li.  Continuum electrostatics for ionic solutions with non-uniform ionic sizes , 2009 .

[6]  C. Schmeiser,et al.  Semiconductor equations , 1990 .

[7]  Martin Burger,et al.  Inverse Problems Related to Ion Channel Selectivity , 2007, SIAM J. Appl. Math..

[8]  Klemens Fellner,et al.  The entropy dissipation method for spatially inhomogeneous reaction–diffusion-type systems , 2008, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[9]  L. Ambrosio,et al.  Gradient Flows: In Metric Spaces and in the Space of Probability Measures , 2005 .

[10]  Roland Roth,et al.  Density functional theory for hard-sphere mixtures: the White Bear version mark II , 2006, Journal of physics. Condensed matter : an Institute of Physics journal.

[11]  Dirk Horstmann,et al.  F ¨ Ur Mathematik in Den Naturwissenschaften Leipzig from 1970 until Present: the Keller-segel Model in Chemotaxis and Its Consequences from 1970 until Present: the Keller-segel Model in Chemotaxis and Its Consequences , 2022 .

[12]  Kerry A. Landman,et al.  Diffusing Populations: Ghosts or Folks? , 2009 .

[13]  Augusto R. Lopez,et al.  Electrodiffusion of Ions in Alpha-Quartz , 1985 .

[14]  M. Wolfram,et al.  Mathematical modeling and simulation of nanopore blocking by precipitation , 2010, Journal of physics. Condensed matter : an Institute of Physics journal.

[15]  Dirk Gillespie,et al.  (De)constructing the ryanodine receptor: modeling ion permeation and selectivity of the calcium release channel. , 2005, The journal of physical chemistry. B.

[16]  K. Deimling.  Nonlinear functional analysis , 1985 .

[17]  Angela Stevens,et al.  The Derivation of Chemotaxis Equations as Limit Dynamics of Moderately Interacting Stochastic Many-Particle Systems , 2000, SIAM J. Appl. Math..

[18]  Robert S. Eisenberg,et al.  Coupling Poisson–Nernst–Planck and density functional theory to calculate ion flux , 2002 .

[19]  P. Grisvard.  Elliptic Problems in Nonsmooth Domains , 1985 .

[20]  Kevin J Painter,et al.  Continuous Models for Cell Migration in Tissues and Applications to Cell Sorting via Differential Chemotaxis , 2009, Bulletin of mathematical biology.

[21]  J. Kepler,et al.  Mathematical Modelling and Simulation of Ion Channels , 2009 .

[22]  G. M.,et al.  Partial Differential Equations I , 2023, Applied Mathematical Sciences.

[23]  Joel Lebowitz,et al.  Phase Segregation Dynamics in Particle Systems with Long Range Interactions II: Interface Motion , 1997, SIAM J. Appl. Math..

[24]  Robert S. Eisenberg,et al.  Ion flow through narrow membrane channels: part II , 1992 .

[25]  Ansgar Jüngel,et al.  Analysis of a Multidimensional Parabolic Population Model with Strong Cross-Diffusion , 2004, SIAM J. Math. Anal..

[26]  Robert S. Eisenberg,et al.  Ionic diffusion through confined geometries: from Langevin equations to partial differential equations , 2004 .

[27]  G. Burton.  Sobolev Spaces , 2013 .

[28]  Giambattista Giacomin,et al.  Phase segregation dynamics in particle systems with long range interactions. I. Macroscopic limits , 1997, comp-gas/9705001.

[29]  J. Ruppersberg.  Ion Channels in Excitable Membranes , 1996 .

[30]  Bo Li,et al.  Minimization of Electrostatic Free Energy and the Poisson-Boltzmann Equation for Molecular Solvation with Implicit Solvent , 2009, SIAM J. Math. Anal..

[31]  Ansgar Jüngel,et al.  A Parabolic Cross-Diffusion System for Granular Materials , 2003, SIAM J. Math. Anal..

[32]  B. Li,et al.  Continuum electrostatics for ionic solutions with non-uniform ionic sizes , 2009 .

[33]  K. Painter,et al.  Volume-filling and quorum-sensing in models for chemosensitive movement , 2002 .

[34]  Dirk Horstmann,et al.  Generalizing the Keller–Segel Model: Lyapunov Functionals, Steady State Analysis, and Blow-Up Results for Multi-species Chemotaxis Models in the Presence of Attraction and Repulsion Between Competitive Interacting Species , 2011, J. Nonlinear Sci..

[35]  J Andrew McCammon,et al.  Electrostatic Free Energy and Its Variations in Implicit Solvent Models , 2022 .

[36]  P. Turq,et al.  REAL IONIC SOLUTIONS IN THE MEAN SPHERICAL APPROXIMATION. 1. SIMPLE SALTS IN THE PRIMITIVE MODEL , 1996 .