Skip to main content
SpringerLink
Log in

Geometric Singular Perturbation Approach to Poisson-Nernst-Planck Systems with Local Hard-Sphere Potential: Studies on Zero-Current Ionic Flows with Boundary Layers

  • Published:
Qualitative Theory of Dynamical Systems Aims and scope Submit manuscript

Abstract

We examine the dynamics of zero-current ionic flows via Poisson-Nernst-Planck systems with one cation, one anion and boundary layers. To account finite ion size effects in the analysis, we include Bikerman’s local hard-sphere model. Geometric singular perturbation theory is employed in our discussion, together with the specific structures of the model problem, we obtain the existence and local uniqueness result of the problem for zero-current state. More importantly, we are able to derive explicit expressions of the approximation to individual fluxes from the solutions. This allows us to further examine the effect on the zero-current ionic flows with boundary layers from finite ion sizes and diffusion coefficients by further employing regular perturbation analysis. The detailed analysis, particularly, the characterization of the nonlinear interplay between system parameters provides deep insights and better understandings of the internal dynamics of ionic flows through membrane channels.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

Data Availability

All data generated or analyzed during this study are included in this article.

Code Availability

Not applicable.

References

  1. Eisenberg, B.: Ions in Fluctuating Channels: Transistors Alive. Fluct. Noise Lett. 11, 76–96 (2012)

    Google Scholar 

  2. Eisenberg, B.: Crowded charges in ion channels. In: Rice, S.A. (ed.) Advances in chemical physics, pp. 77–223. John Wiley & Sons, Hoboken, NJ (2011)

    Google Scholar 

  3. Gillespie, G.: A singular perturbation analysis of the Poisson-Nernst-Planck system: applications to ionic channels. Ph.D Thesis, Rush University at Chicago, Chicago, IL (1999)

  4. Dworakowska, B., Dołowy, K.: Ion channels-related diseases. Acta Biochim Pol. 47, 685–703 (2000)

    Google Scholar 

  5. Unwin, N.: The structure of ion channels in membranes of excitable cells. Neuron 3, 665–676 (1989)

    Google Scholar 

  6. 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)

    MathSciNet  MATH  Google Scholar 

  7. Chen, D.-P., Eisenberg, R.S.: Charges, currents and potentials in ionic channels of one conformation. Biophys. J. 64, 1405–1421 (1993)

    Google Scholar 

  8. Burger, M.: Inverse problems in ion channel modelling. Inverse Problems 27, 083001 (2011)

    MathSciNet  MATH  Google Scholar 

  9. Burger, M., Eisenberg, R.S., Engl, H.: Inverse problems related to ion channel selectivity. SIAM J. Appl. Math. 67, 960–989 (2007)

    MathSciNet  MATH  Google Scholar 

  10. Bates, P.W., Chen, J., Zhang, M.: Dynamics of ionic flows via Poisson-Nernst-Planck systems with local hard-sphere potentials: competition between cations. Math. Biosci. Eng. 17, 3736–3766 (2020)

    MathSciNet  MATH  Google Scholar 

  11. Bates, P.W., Wen, Z., Zhang, M.: Small permanent charge effects on individual fluxes via Poisson-Nernst-Planck models with multiple cations. J. Nonlinear Sci. 31, 55 (2021)

    MathSciNet  MATH  Google Scholar 

  12. Chen, J., Wang, Y., Zhang, L., Zhang, M.: Mathematical analysis of Poisson- Nernst-Planck models with permanent charge and boundary layers: studies on individual fluxes. Nonlinearity 34, 3879–3906 (2021)

    MathSciNet  MATH  Google Scholar 

  13. Eisenberg, B., Liu, W.: Poisson-Nernst-Planck systems for ion channels with permanent charges. SIAM J. Math. Anal. 38, 1932–1966 (2007)

    MathSciNet  MATH  Google Scholar 

  14. Eisenberg, B., Liu, W., Xu, H.: Reversal charge and reversal potential: case studies via classical Poisson-Nernst-Planck models. Nonlinearity 28, 103–128 (2015)

    MathSciNet  MATH  Google Scholar 

  15. Ji, S., Liu, W.: Flux ratios and channel structures. J. Dyn. Differ. Equ. 31, 1141–1183 (2019)

    MathSciNet  MATH  Google Scholar 

  16. 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)

    MathSciNet  MATH  Google Scholar 

  17. 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)

    MathSciNet  MATH  Google Scholar 

  18. Liu, W.: Geometric singular perturbation approach to steady-state Poisson-Nernst-Planck systems. SIAM J. Appl. Math. 65, 754–766 (2005)

    MathSciNet  MATH  Google Scholar 

  19. Liu, W.: One-dimensional steady-state Poisson-Nernst-Planck systems for ion channels with multiple ion species. J. Differ. Equ. 246, 428–451 (2009)

    MathSciNet  MATH  Google Scholar 

  20. Liu, W., Xu, H.: A complete analysis of a classical Poisson-Nernst-Planck model for ionic flow. J. Differ. Equ. 258, 1192–1228 (2015)

    MathSciNet  MATH  Google Scholar 

  21. Mofidi, H., Liu, W.: Reversal potential and reversal permanent charge with unequal diffusion coefficients via classical Poisson-Nernst-Planck models. SIAM J. Appl. Math. 80, 1908–1935 (2020)

    MathSciNet  MATH  Google Scholar 

  22. 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)

    MathSciNet  MATH  Google Scholar 

  23. Wen, Z., Bates, P.W., Zhang, M.: Effects on I-V relations from small permanent charge and channel geometry via classical Poisson-Nernst-Planck equations with multiple cations. Nonlinearity 34, 4464–4502 (2021)

    MathSciNet  MATH  Google Scholar 

  24. Wen, Z., Zhang, L., Zhang, M.: Dynamics of classical Poisson-Nernst-Planck systems with multiple cations and boundary layers. J. Dyn. Diff. Equ. 33, 211–234 (2021)

    MathSciNet  MATH  Google Scholar 

  25. Zhang, L., Eisenberg, B., Liu, W.: An effect of large permanent charge: decreasing flux with increasing transmembrane potential. Eur. Phys. J. Special Topics 227, 2575–2601 (2019)

    Google Scholar 

  26. Zhang, M.: Competition between cations via Poisson-Nernst-Planck systems with nonzero but small permanent charges. Membranes 11, 236 (2021)

    Google Scholar 

  27. Eisenberg, B.: Proteins, channels, and crowded ions. Biophys. Chem. 100, 507–517 (2003)

    Google Scholar 

  28. Eisenberg, R.S.: From structure to function in open ionic channels. J. Memb. Biol. 171, 1–24 (1999)

    Google Scholar 

  29. Gillespie, D., Eisenberg, R.S.: Physical descriptions of experimental selectivity measurements in ion channels. Eur. Biophys. J. 31, 454–466 (2002)

    Google Scholar 

  30. Henderson, L.J.: The fitness of the environment: an inquiry into the biological significance of the properties of matter. Macmillan, New York (1927)

    Google Scholar 

  31. Noskov, S.Y., Berneche, S., Roux, B.: Control of ion selectivity in potassium channels by electrostatic and dynamic properties of carbonyl ligands. Nature 431, 830–834 (2004)

    Google Scholar 

  32. Barcilon, V.: Ion flow through narrow membrane channels: part I. SIAM J. Appl. Math. 52, 1391–1404 (1992)

    MathSciNet  MATH  Google Scholar 

  33. Hyon, Y., Eisenberg, B., Liu, C.: A mathematical model for the hard sphere repulsion in ionic solutions. Commun. Math. Sci. 9, 459–475 (2010)

    MathSciNet  MATH  Google Scholar 

  34. Hyon, Y., Fonseca, J., Eisenberg, B., Liu, C.: A new Poisson-Nernst-Planck equation (PNP-FS-IF) for charge inversion near walls. Biophys. J. 100, 578a (2011)

    Google Scholar 

  35. 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)

    Google Scholar 

  36. 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)

    Google Scholar 

  37. 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)

    MathSciNet  MATH  Google Scholar 

  38. Barcilon, V., Chen, D.-P., Eisenberg, R.S.: Ion flow through narrow membrane channels: part II. SIAM J. Appl. Math. 52, 1405–1425 (1992)

    MathSciNet  MATH  Google Scholar 

  39. 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)

    MathSciNet  MATH  Google Scholar 

  40. Cardenas, A.E., Coalson, R.D., Kurnikova, M.G.: Three-dimensional Poisson-Nernst-Planck theory studies: influence of membrane electrostatics on Gramicidin a channel conductance. Biophys. J. 79, 80–93 (2000)

    Google Scholar 

  41. Graf, P., Kurnikova, M.G., Coalson, R.D., Nitzan, A.: Comparison of dynamic lattice monte-carlo simulations and dielectric self energy Poisson-Nernst-Planck continuum theory for model ion channels. J. Phys. Chem. B 108, 2006–2015 (2004)

    Google Scholar 

  42. Liu, W., Wang, B.: Poisson-Nernst-Planck systems for narrow tubular-like membrane channels. J. Dyn. Diff. Equ. 22, 413–437 (2010)

    MathSciNet  MATH  Google Scholar 

  43. Mock, M.S.: An example of nonuniqueness of stationary solutions in device models. COMPEL 1, 165–174 (1982)

    Google Scholar 

  44. Mofidi, H., Eisenberg, B., Liu, W.: Effects of diffusion coefficients and permanent charge on reversal potentials in ionic channels. Entropy 22, 325 (2020)

    MathSciNet  Google Scholar 

  45. Rubinstein, I.: Electro-diffusion of ions. SIAM Studies in Applied Mathematics, SIAM, Philadelphia, PA (1990)

    Google Scholar 

  46. Saraniti, M., Aboud, S., Eisenberg, R.S.: The simulation of ionic charge transport in biological ion channels: an introduction to numerical methods. Rev. Comp. Chem. 22, 229–294 (2005)

    Google Scholar 

  47. 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)

    MathSciNet  MATH  Google Scholar 

  48. 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)

    MathSciNet  MATH  Google Scholar 

  49. 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)

    Google Scholar 

  50. 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)

    MathSciNet  MATH  Google Scholar 

  51. Zhang, M.: Boundary layer effects on ionic flows via classical Poisson-Nernst-Planck systems. Comput. Math. Biophys. 6, 14–27 (2018)

    MathSciNet  MATH  Google Scholar 

  52. Zheng, Q., Wei, G.W.: Poisson-Boltzmann-Nernst-Planck model. J. Chem. Phys. 134, 1–17 (2011)

    Google Scholar 

  53. Zhang, L., Liu, W.: Effects of large permanent charges on ionic flows via Poisson-Nernst-Planck models. SIAM J. Appl. Dyn. Syst. 19, 1993–2029 (2020)

    MathSciNet  MATH  Google Scholar 

  54. Rosenfeld, Y.: Free-energy model for the inhomogeneous hard-sphere fluid mixture and density-functional theory of freezing. Phys. Rev. Lett. 63, 980–983 (1989)

    Google Scholar 

  55. 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)

    Google Scholar 

  56. Aitbayev, R., Bates, P.W., Lu, H., Zhang, L., Zhang, M.: Mathematical studies of Poisson-Nernst-Planck systems: dynamics of ionic flows without electroneutrality conditions. J. Comput. Appl. Math. 362, 510–527 (2019)

    MathSciNet  MATH  Google Scholar 

  57. 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)

    MathSciNet  MATH  Google Scholar 

  58. 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)

    Google Scholar 

  59. 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)

    Google Scholar 

  60. 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)

    Google Scholar 

  61. Gillespie, D., Nonner, W., Eisenberg, R.S.: Crowded charge in biological ion channels. Nanotech. 3, 435–438 (2003)

    Google Scholar 

  62. 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)

    MathSciNet  MATH  Google Scholar 

  63. Hyon, Y., Liu, C., Eisenberg, B.: PNP equations with steric effects: a model of ion flow through channels. J. Phys. Chem. B 116, 11422–11441 (2012)

    Google Scholar 

  64. 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. Diff. Equ. 24, 955–983 (2012)

    MathSciNet  MATH  Google Scholar 

  65. 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. Ser. B 21, 1775–1802 (2016)

    MathSciNet  MATH  Google Scholar 

  66. Kilic, M.S., Bazant, M.Z., Ajdari, A.: Steric effects in the dynamics of electrolytes at large applied voltages II. Modified Poisson-Nernst-Planck equations. Phys. Rev. E. 75, 021503 (2007)

    Google Scholar 

  67. 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. Ser. B 23, 1623–1643 (2018)

    MathSciNet  MATH  Google Scholar 

  68. 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. Diff. Equ. 24, 985–1004 (2012)

    MathSciNet  MATH  Google Scholar 

  69. Sun, L., Liu, W.: Non-localness of excess potentials and boundary value problems of Poisson-Nernst-Planck systems for ionic flow: a case study. J. Dyn. Diff. Equ. 30, 779–797 (2018)

    MathSciNet  MATH  Google Scholar 

  70. Zhou, Z., Wang, Z., Li, B.: Mean-field description of ionic size effects with nonuniform ionic sizes: a numerical approach. Phy. Rev. E 84, 1–13 (2011)

    Google Scholar 

  71. Bikerman, J.J.: Structure and capacity of the electrical double layer. Philos. Mag. 33, 384 (1942)

    MATH  Google Scholar 

  72. Liu, J., Eisenberg, B.: Molecular mean-field theory of ionic solutions: a Poisson-Nernst-Planck-Bikerman model. Entropy 22, 550 (2020)

    MathSciNet  Google Scholar 

  73. Vera, J. H., Wilezek-Vera, G.: Classical thermodynamics of fluid systems: principles and applications, CRC Press, New York, NY, USA (2016)

  74. Fenichel, N.: Geometric singular perturbation theory for ordinary differential equations. J. Diff. Equ. 31, 53–98 (1979)

    MathSciNet  MATH  Google Scholar 

  75. Jones, C.: Geometric singular perturbation theory. Dynamical systems (Montecatini Terme, 1994). Lect. Notes in Math., vol. 1609, pp. 44-118. Springer, Berlin (1995)

  76. Jones, C., Kopell, N.: Tracking invariant manifolds with differential forms in singularly perturbed systems. J. Diff. Equ. 108, 64–88 (1994)

    MathSciNet  MATH  Google Scholar 

Download references

Funding

Jianing Chen and Mingji Zhang are supported by Simons Foundation No. 628308.

Author information

Authors and Affiliations

  1. Department of Mathematics, Zhejiang Normal University, Jinhua, Zhejiang, 321004, China

    Jianing Chen

  2. Department of Mathematics, New Mexico Institute of Mining and Technology, Socorro, NM, 87801, USA

    Mingji Zhang

Authors
  1. Jianing Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mingji Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

MZ is contributed to establishing the existence and local uniqueness of the boundary value problem and manuscript preparation, while JC is contributed to some detailed calculations related to the behaviors of ionic flows.

Corresponding author

Correspondence to Mingji Zhang.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Code Availability

Not applicable.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chen, J., Zhang, M. Geometric Singular Perturbation Approach to Poisson-Nernst-Planck Systems with Local Hard-Sphere Potential: Studies on Zero-Current Ionic Flows with Boundary Layers. Qual. Theory Dyn. Syst. 21, 139 (2022). https://doi.org/10.1007/s12346-022-00672-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s12346-022-00672-0

Keywords

Mathematics Subject Classification

Search

Navigation