Abstract
We study a quasi-one-dimensional classical Poisson–Nernst–Planck model for ionic flow through a membrane channel with two positively charged ion species (cations) and one negatively charged, and with zero permanent charges. We treat the model problem as a boundary value problem of a singularly perturbed differential system. Under the framework of the geometric singular perturbation theory, together with specific structures of this concrete model, the existence of solutions to the boundary value problem is established and, for a special case that the two cations have the same valences, we are able to derive approximations of the individual fluxes and the I–V (current–voltage) relation explicitly, from which, our two main focuses in this work, boundary layer effects on ionic flows and competitions between two cations, are analyzed in great details. Critical potentials are identified and their roles in characterizing these effects are studied. Nonlinear interplays among physical parameters, such as boundary concentrations and potentials, diffusion coefficients and ion valences, are characterized, which could potentially provide efficient ways to control and affect some biological functions. Numerical simulations are performed, and numerical results are consistent with our analytical ones.
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References
Abaid, N., Eisenberg, R.S., Liu, W.: Asymptotic expansions of I-V relations via a Poisson-Nernst-Planck system. SIAM J. Appl. Dyn. Syst. 7, 1507–1526 (2008)
Aboud, S., Marreiro, D., Saraniti, M., Eisenberg, R.S.: A Poisson P3M force field scheme for particle-based simulations of ionic liquids. J. Comput. Electr. 3, 117–133 (2004)
Alberts, B., Bray, D., Lewis, J., Raff, M., Roberts, K., Watson, J.D.: Molecular Biology of the Cell, 3rd edn. Garland, New York (1994)
Barcilon, V.: Ion flow through narrow membrane channels: Part I. SIAM J. Appl. Math. 52, 1391–1404 (1992)
Bates, P.W., Jia, Y., Lin, G., Lu, H., Zhang, M.: Individual flux study via steady-state Poisson-Nernst-Planck systems: effects from boundary conditions. SIAM J. Appl. Dyn. Syst. 16, 410–430 (2017)
Bazant, M.Z., Chu, K.T., Bayly, B.J.: Current-voltage relations for electrochemical thin films. SIAM J. Appl. Math. 65, 1463–1484 (2005)
Barcilon, V., Chen, D.-P., Eisenberg, R.S.: Ion flow through narrow membrane channels: Part II. SIAM J. Appl. Math. 52, 1405–1425 (1992)
Barcilon, V., Chen, D.-P., Eisenberg, R.S., Jerome, J.W.: Qualitative properties of steady-state Poisson-Nernst-Planck systems: perturbation and simulation study. SIAM J. Appl. Math. 57, 631–648 (1997)
Blum, L., Høye, J.S.: Mean spherical model for asymmetric electrolytes. 2. Thermodynamic properties and the pair correlation function. J. Phys. Chem. 81, 1311–1316 (1977)
Biesheuvel, P.M.: Two-fluid model for the simultaneous flow of colloids and fluids in porous media. J. Colloid Interface Sci. 355, 389–395 (2011)
Barthel, J., Krienke, H., Kunz, W.: Physical Chemistry of Electrolyte Solutions: Modern Aspects. Springer, New York (1998)
Bazant, M.Z., Kilic, M.S., Storey, B.D., Ajdari, A.: Towards an understanding of induced-charge electrokinetics at large applied voltages in concentrated solutions. Adv. Colloid Interface Sci. 152, 48–88 (2009)
Bates, P.W., Liu, W., Lu, H., Zhang, M.: Ion size and valence effects on ionic flows via Poisson-Nernst-Planck systems. Commun. Math. Sci. 15, 881–901 (2017)
Blum, L.: Mean spherical model for asymmetric electrolytes. Mol. Phys. 30, 1529–1535 (1975)
Brillantiv, N., Poschel, T.: Kinetic theory of Granular Gases. Oxford University Press, New York (2004)
Berry, S.R., Rice, S.A., Ross, J.: Physical Chemistry, 2nd edn. Oxford University Press, New York (2000)
Bazant, M., Thornton, K., Ajdari, A.: Diffuse-charge dynamics in electrochemical systems. Phys. Rev. E 70, 1–24 (2004)
Chazalviel, J.-N.: Coulomb Screening by Mobile Charges. Birkhauser, New York (1999)
Chen, D.P., Eisenberg, R.S.: Charges, currents and potentials in ionic channels of one conformation. Biophys. J. 64, 1405–1421 (1993)
Durand-Vidal, S., Turq, P., Bernard, O., Treiner, C., Blum, L.: New perspectives in transport phenomena in electrolytes. Phys. A 231, 123–143 (1996)
Chen, D., Eisenberg, R., Jerome, J., Shu, C.: Hydrodynamic model of temperature change in open ionic channels. Biophys. J. 69, 2304–2322 (1995)
Eisenberg, B., Hyon, Y., Liu, C.: Energy variational analysis of ions in water and channels: Field theory for primitive models of complex ionic fluids. J. Chem. Phys. 133, 104104 (2010)
Eisenberg, B.: Proteins, channels, and crowded Ions. Biophys. Chem. 100, 507–517 (2003)
Eisenberg, B., Liu, W.: Poisson-Nernst-Planck systems for ion channels with permanent charges. SIAM J. Math. Anal. 38, 1932–1966 (2007)
Eisenberg, B., Liu, W., Xu, H.: Reversal charge and reversal potential: case studies via classical Poisson-Nernst-Planck models. Nonlinarity 28, 103–127 (2015)
Ern, A., Joubaud, R., Leliévre, T.: Mathematical study of non-ideal electrostatic correlations in equilibrium electrolytes. Nonlinearity 25, 1635–1652 (2012)
Fawcett, W.R.: Liquids, Solutions, and Interfaces: From Classical Macroscopic Descriptions to Modern Microscopic Details. Oxford University Press, New York (2004)
Fair, J.C., Osterle, J.F.: Reverse Electrodialysis in charged capillary membranes. J. Chem. Phys. 54, 3307–3316 (1971)
Gillespie, D., Eisenberg, R.S.: Physical descriptions of experimental selectivity measurements in ion channels. Eur. Biophys. J. 31, 454–466 (2002)
Gillespie, D.: A singular perturbation analysis of the Poisson-Nernst-Planck system: Applications to Ionic Channels, Ph.D Dissertation, Rush University at Chicago (1999)
Gillespie, D., Nonner, W., Eisenberg, R.S.: Coupling Poisson-Nernst-Planck and density functional theory to calculate ion flux. J. Phys. Condens. Matter 14, 12129–12145 (2002)
Gillespie, D., Nonner, W., Eisenberg, R.S.: Crowded charge in biological ion channels. Nanotechnology 3, 435–438 (2003)
Gross, R.J., Osterle, J.F.: Membrane transport characteristics of ultra fine capillary. J. Chem. Phys. 49, 228–234 (1968)
Gillespie, D., Xu, L., Wang, Y., Meissner, G.: (De)constructing the Ryanodine receptor: modeling ion permeation and selectivity of the Calcium release channel. J. Phys. Chem. B 109, 15598–15610 (2005)
Henderson, L.J.: The Fitness of the Environment: An Inquiry Into the Biological Significance of the Properties of Matter. Macmillan, New York (1927)
Hodgkin, A.L., Huxley, A.F.: Propagation of electrical signals along giant nerve fibers. Proc. R. Soc. Lond. 140, 177–183 (1952)
Hodgkin, A.L., Huxley, A.F.: Currents carried by sodium and potassium ions through the membrane of the giant axon of Loligo. J. Physiol. 116, 449–472 (1952)
Hodgkin, A.L., Huxley, A.F.: The components of membrane conductance in the giant axon of Loligo. J. Physiol. 116, 473–496 (1952)
Hodgkin, A.L., Huxley, A.F.: The dual effect of membrane potential on sodium conductance in the giant axon of Loligo. J. Physiol. 116, 497–506 (1952)
Hodgkin, A.L., Huxley, A.F.: A quantitative description of membrane current and its application to conduction and excitation in nerve. J. Physiol. 117, 500–544 (1952)
Hyon, Y., Eisenberg, B., Liu, C.: A mathematical model for the hard sphere repulsion in ionic solutions. Commun. Math. Sci. 9, 459–475 (2010)
Hyon, Y., Fonseca, J., Eisenberg, B., Liu, C.: Energy variational approach to study charge inversion (layering) near charged walls. Discrete Contin. Dyn. Syst. Ser. B 17, 2725–2743 (2012)
Horng, T.L., Lin, T.C., Liu, C., Eisenberg, B.: PNP equations with steric effects: a model of ion flow through channels. J. Phys. Chem. B 116, 11422–11441 (2012)
Im, W., Roux, B.: Ion permeation and selectivity of OmpF porin: a theoretical study based on molecular dynamics, Brownian dynamics, and continuum electrodiffusion theory. J. Mol. Biol. 322, 851–869 (2002)
Jacoboni, C., Lugli, P.: The Monte Carlo Method for Semiconductor Device Simulation. Springer, New York (1989)
Ji, S., Liu, W.: Poisson-Nernst-Planck systems for ion flow with density functional theory for hard-sphere potential: I-V relations and critical potentials. Part I: Analysis. J. Dyn. Differ. Equ. 24, 955–983 (2012)
Ji, S., Liu, W., Zhang, M.: Effects of (small) permanent charges and channel geometry on ionic flows via classical Poisson-Nernst-Planck models. SIAM J. on Appl. Math. 75, 114–135 (2015)
Jia, Y., Liu, W., Zhang, M.: Qualitative properties of ionic flows via Poisson-Nernst-Planck systems with Bikerman’s local hard-sphere potential: ion size effects. Discrete Contin. Dyn. Syst. Series B 21, 1775–1802 (2016)
Jones, C.: Geometric Singular Perturbation Theory. Dynamical Systems (Montecatini Terme, 1994). Lecture Notes in Mathematics, vol. 1609, pp. 44–118. Springer, Berlin (1995)
Jones, C., Kaper, T., Kopell, N.: Tracking invariant manifolds up to exponentially small errors. SIAM J. Math. Anal. 27, 558–577 (1996)
Jones, C., Kopell, N.: Tracking invariant manifolds with differential forms in singularly perturbed systems. J. Differ. Equ. 108, 64–88 (1994)
Lin, T.C., Eisenberg, B.: Multiple solutions of steady-state Poisson- Nernst-Planck equations with steric effects. Nonlinearity 28, 2053–2080 (2015)
Lin, T.C., Eisenberg, B.: A new approach to the Lennard-Jones potential and a new model: PNP-steric equations. Commun. Math. Sci. 12, 149–173 (2014)
Lee, C.-C., Lee, H., Hyon, Y., Lin, T.-C., Liu, C.: New Poisson-Boltzmann type equations: one-dimensional solutions. Nonlinearity 24, 431–58 (2011)
Liu, W.: Geometric singular perturbation approach to steady-state Poisson-Nernst-Planck systems. SIAM J. Appl. Math. 65, 754–766 (2005)
Liu, W.: One-dimensional steady-state Poisson-Nernst-Planck systems for ion channels with multiple ion species. J. Differ. Equ. 246, 428–451 (2009)
Lin, G., Liu, W., Yi, Y., Zhang, M.: Poisson-Nernst-Planck systems for ion flow with density functional theory for local hard-sphere potential. SIAM J. Appl. Dyn. Syst. 12, 1613–1648 (2013)
Liu, W., Wang, B.: Poisson-Nernst-Planck systems for narrow tubular-like membrane channels. J. Dyn. Differ. Equ. 22, 413–437 (2010)
Liu, W., Tu, X., Zhang, M.: Poisson-Nernst-Planck systems for ion flow with density functional theory for hard-sphere potential: I-V relations and critical potentials. Part II: Numerics. J. Dyn. Differ. Equ. 24, 985–1004 (2012)
Liu, W., Xu, H.: A complete analysis of a classical Poisson-Nernst-Planck model for ionic flow. J. Differ. Equ. 258, 1192–1228 (2015)
Lu, H., Li, J., Shackelford, J., Vorenberg, J., Zhang, M.: Ion size effects on individual fluxes via Poisson-Nernst-Planck systems with Bikerman’s local hard-sphere potential: Analysis without electroneutrality boundary conditions. Discrete Contin. Dyn. Syst. B 23, 1623–1643 (2018)
Lundstrom, M.: Fundamentals of Carrier Transport, 2nd edn. Addison-Wesley, New York (2000)
Mason, E., McDaniel, E.: Transport Properties of Ions in Gases. Wiley, New York (1988)
Nadler, B., Schuss, Z., Singer, A., Eisenberg, B.: Diffusion through protein channels: from molecular description to continuum equations. Nanotechnology 3, 439–442 (2003)
Nonner, W., Eisenberg, R.S.: Ion permeation and glutamate residues linked by Poisson-Nernst-Planck theory in L-type Calcium channels. Biophys. J. 75, 1287–1305 (1998)
Noskov, S.Y., Im, W., Roux, B.: Ion Permeation through the \(z_1\)-Hemolysin Channel: theoretical studies based on Brownian Dynamics and Poisson-Nernst-Planck electrodiffusion theory. Biophys. J. 87, 2299–2309 (2004)
Park, J.-K., Jerome, J.W.: Qualitative properties of steady-state Poisson-Nernst-Planck systems: mathematical study. SIAM J. Appl. Math. 57, 609–630 (1997)
Park, J.-K., Jerome, J.W.: Qualitative properties of steady-state Poisson-Nernst-Planck systems: mathematical study. SIAM J. Appl. Math. 57, 609–630 (1997)
Rosenfeld, Y.: Free energy model for the inhomogeneous fluid mixtures: Yukawa-charged hard spheres, general interactions, and plasmas. J. Chem. Phys. 98, 8126–8148 (1993)
Roth, R.: Fundamental measure theory for hard-sphere mixtures: a review. J. Phys. Condens. Matter 22, 063102 (2010)
Rouston, D.J.: Bipolar Semiconductor Devices. McGraw-Hill Publishing Company, New York (1990)
Roux, B., Allen, T.W., Berneche, S., Im, W.: Theoretical and computational models of biological ion channels. Quat. Rev. Biophys. 37, 15–103 (2004)
Sakmann, B., Neher, E. (eds.): Single-Channel Recording. Plenum Press, New York (1995)
Selberherr, S.: Analysis and Simulation of Semiconductor Devices. Springer, New York (1984)
Singer, A., Gillespie, D., Norbury, J., Eisenberg, R.S.: Singular perturbation analysis of the steady-state Poisson-Nernst-Planck system: applications to ion channels. Eur. J. Appl. Math. 19, 541–560 (2008)
Schuss, Z., Nadler, B., Eisenberg, R.S.: Derivation of Poisson and Nernst-Planck equations in a bath and channel from a molecular model. Phys. Rev. E 64, 1–14 (2001)
Singer, A., Norbury, J.: A Poisson-Nernst-Planck model for biological ion channels-an asymptotic analysis in a three-dimensional narrow funnel. SIAM J. Appl. Math. 70, 949–968 (2009)
Streetman, B.G.: Solid State Electronic Devices, 4th edn. Prentice-Hall, Englewood Cliffs, NJ (1972)
Sasidhar, V., Ruckenstein, E.: Electrolyte osmosis through capillaries. J. Colloid Interface Sci. 82, 439–457 (1981)
Tin, S.-K., Kopell, N., Jones, C.: Invariant manifolds and singularly perturbed boundary value problems. SIAM J. Numer. Anal. 31, 1558–1576 (1994)
Tanford, C., Reynolds, J.: Nature’s Robots: A History of Proteins. Oxford University Press, New Work (2001)
Warner Jr., R.M.: Microelectronics: its unusual origin and personality. IEEE Trans. Electron. Dev. 48, 2457–2467 (2001)
Wang, X.-S., He, D., Wylie, J., Huang, H.: Singular perturbation solutions of steady-state Poisson-Nernst-Planck systems. Phys. Rev. E 89, 022722 (2014)
Wu, H., Lin, T.C., Liu, C.: Diffusion limit of kinetic equations for multiple species charged particles. Arch. Rational Mech. Anal. 215, 419–441 (2015)
Wei, G.W., Zheng, Q., Chen, Z., Xia, K.: Variational multiscale models for charge transport. SIAM Rev. 54, 699–754 (2012)
Zhang, J., Acheampong, D., Zhang, M.: Geometric singular approach to Poisson-Nernst-Planck models with excess chemical potentials: Ion size effects on individual fluxes. Comput. Math. Biophys. 5, 58–77 (2017)
Zhang, M.: Asymptotic expansions and numerical simulations of I-V relations via a steady-state Poisson-Nernst-Planck system. Rocky Mt. J. Math. 45, 1681–1708 (2015)
Zhang, M.: Boundary layer effects on ionic flows via classical Poisson-Nernst-Planck systems. Comput. Math. Biophys. 6, 14–27 (2018)
Acknowledgements
Z. Wen was supported in part by China Scholarship Council and the National Natural Science Foundation of China (No. 11701191), Fundamental Research Funds for the Central Universities(No. ZQN-802), Program for Innovative Research Team in Science and Technology in Fujian Province University, and Quanzhou High-Level Talents Support Plan (No. 2017ZT012), L. Zhang was supported in part by the National Natural Science Foundation of China (Nos. 11672270 and 12011530062), and M. Zhang was partially supported by MPS Simons Foundation (No. 628308).
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Wen, Z., Zhang, L. & Zhang, M. Dynamics of Classical Poisson–Nernst–Planck Systems with Multiple Cations and Boundary Layers. J Dyn Diff Equat 33, 211–234 (2021). https://doi.org/10.1007/s10884-020-09861-4
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DOI: https://doi.org/10.1007/s10884-020-09861-4