Journal of Dynamics and Differential Equations

, Volume 24, Issue 4, pp 985–1004 | Cite as

Poisson–Nernst–Planck Systems for Ion Flow with Density Functional Theory for Hard-Sphere Potential: I–V Relations and Critical Potentials. Part II: Numerics

  • Weishi Liu
  • Xuemin Tu
  • Mingji Zhang


We consider a one-dimensional steady-state Poisson–Nernst–Planck type model for ionic flow through membrane channels. Improving the classical Poisson–Nernst–Planck models where ion species are treated as point charges, this model includes ionic interaction due to finite sizes of ion species modeled by hard sphere potential from the Density Functional Theory. The resulting problem is a singularly perturbed boundary value problem of an integro-differential system. We examine the problem and investigate the ion size effect on the current–voltage (I–V) relations numerically, focusing on the case where two oppositely charged ion species are involved and only the hard sphere components of the excess chemical potentials are included. Two numerical tasks are conducted. The first one is a numerical approach of solving the boundary value problem and obtaining I–V curves. This is accomplished through a numerical implementation of the analytical strategy introduced by Ji and Liu in [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. (to appear)]. The second task is to numerically detect two critical potential values V c and V c .The existence of these two critical values is first realized for a relatively simple setting and analytical approximations of V c and V c are obtained in the above mentioned reference. We propose an algorithm for numerical detection of V c and V c without using any analytical formulas but based on the defining properties and numerical I–V curves directly. For the setting in the above mentioned reference, our numerical values for V c and V c agree well with the analytical predictions. For a setting including a nonzero permanent charge in which case no analytic formula for the I–V relation is available now, our algorithms can still be applied to find V c and V c numerically.


Ion flow PNP–DFT Hard-sphere I–V relation Critical potentials 

Mathematics Subject Classification

34D15 45J05 65L10 78A35 92C35 


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  1. 1.
    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)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Aboud S., Marreiro D., Saraniti M., Eisenberg R.S.: A Poisson P3M force field scheme for particle-based simulations of ionic liquids. J. Comput. Electron. 3, 117–133 (2004)CrossRefGoogle Scholar
  3. 3.
    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)CrossRefGoogle Scholar
  4. 4.
    Barcilon V.: Ion flow through narrow membrane channels: Part I. SIAM J. Appl. Math. 52, 1391–1404 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Barcilon V., Chen D.-P., Eisenberg R.S.: Ion flow through narrow membrane channels: Part II. SIAM J. Appl. Math. 52, 1405–1425 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 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)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Boda D., Gillespie D., Nonner W., Henderson D., Eisenberg B.: Computing induced charges in inhomogeneous dielectric media: application in a Monte Carlo simulation of complex ionic systems. Phys. Rev. E 69(046702), 1–10 (2004)Google Scholar
  8. 8.
    Boda D., Busath D., Eisenberg B., Henderson D., Nonner W.: Monte Carlo simulations of ion selectivity in a biological Na+ channel: charge-space competition. Phys. Chem. Chem. Phys. 4, 5154–5160 (2002)CrossRefGoogle Scholar
  9. 9.
    Burger M., Eisenberg R.S., Engl H.W.: Inverse problems related to ion channel selectivity. SIAM J. Appl. Math. 67, 960–989 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    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)CrossRefGoogle Scholar
  11. 11.
    Chen D.P., Eisenberg R.S.: Charges, currents and potentials in ionic channels of one conformation. Biophys. J. 64, 1405–1421 (1993)CrossRefGoogle Scholar
  12. 12.
    Chung S., Kuyucak S.: Predicting channel function from channel structure using Brownian dynamics simulations. Clin. Exp. Pharmacol. Physiol. 28, 89–94 (2001)CrossRefGoogle Scholar
  13. 13.
    Coalson R.D.: Poisson–Nernst–Planck theory approach to the calculation of current through biological ion channels. IEEE Trans. Nanobiosci. 4, 81–93 (2005)CrossRefGoogle Scholar
  14. 14.
    Coalson R.D.: Discrete-state model of coupled ion permeation and fast gating in ClC chloride channels. J. Phys. A 41, 115001 (2009)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Coalson R., Kurnikova M.: Poisson–Nernst–Planck theory approach to the calculation of current through biological ion channels. IEEE Trans. NanoBiosci. 4, 81–93 (2005)CrossRefGoogle Scholar
  16. 16.
    Eisenberg B., Hyon Y., Liu C.: Energy variational analysis EnVarA of ions in water and channels: Field theory for primitive models of complex ionic fluids. J. Chem. Phys. 133, 104104 (2010)CrossRefGoogle Scholar
  17. 17.
    Eisenberg B., Hyon Y., Liu C.: Energy variational analysis EnVarA of ions in calcium and sodium channels: Field theory for primitive models of complex ionic fluids. Biophys. J. 98, 515a (2010)CrossRefGoogle Scholar
  18. 18.
    Eisenberg B., Liu W.: Poisson–Nernst–Planck systems for ion channels with permanent charges. SIAM J. Math. Anal. 38, 1932–1966 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Evans R.: The nature of the liquid-vapour interface and other topics in the statistical mechanics of non-uniform, classical fluids. Adv. Phys. 28, 143–200 (1979)CrossRefGoogle Scholar
  20. 20.
    Evans, R.: Density functionals in the theory of nonuniform fluids. In: Henderson, D. (eds.) Fundamentals of Inhomogeneous Fluids, pp. 85–176. Dekker, New York (1992)Google Scholar
  21. 21.
    Fischer J., Heinbuch U.: Relationship between free energy density functional, Born–Green–Yvon, and potential distribution approaches for inhomogeneous fluids. J. Chem. Phys. 88, 1909–1913 (1988)CrossRefGoogle Scholar
  22. 22.
    Gillespie, D. A singular perturbation analysis of the Poisson–Nernst–Planck system: Applications to Ionic Channels. Ph.D Dissertation, Rush University at Chicago (1999)Google Scholar
  23. 23.
    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)CrossRefGoogle Scholar
  24. 24.
    Gillespie D.: Energetics of divalent selectivity in a calcium channel: The Ryanodine receptor case study. Biophys. J. 94, 1169–1184 (2008)CrossRefGoogle Scholar
  25. 25.
    Gillespie D.: Intracellular calcium release channels mediate their own countercurrent: The Ryanodine receptor case study. Biophys. J. 95, 3706–3714 (2008)CrossRefGoogle Scholar
  26. 26.
    Gillespie D., Giri J., Fill M.: Reinterpreting the anomalous mole fraction effect: The Ryanodine receptor case study. Biophys. J. 97, 2212–2221 (2009)CrossRefGoogle Scholar
  27. 27.
    Gillespie D., Eisenberg R.S.: Modified Donnan potentials for ion transport through biological ion channels. Phys. Rev. E 63(061902), 1–8 (2001)Google Scholar
  28. 28.
    Gillespie D., Eisenberg R.S.: Physical descriptions of experimental selectivity measurements in ion channels. Eur. Biophys. J. 31, 454–466 (2002)CrossRefGoogle Scholar
  29. 29.
    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)CrossRefGoogle Scholar
  30. 30.
    Gillespie D., Nonner W., Eisenberg R.S.: Density functional theory of charged, hard-sphere fluids. Phys. Rev. E 68(0313503), 1–10 (2003)Google Scholar
  31. 31.
    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)CrossRefGoogle Scholar
  32. 32.
    Hollerbach U., Chen D.-P., Eisenberg R.S.: Two- and three-dimensional Poisson–Nernst–Planck simulations of current flow through Gramicidin-A. J. Comp. Sci. 16, 373–409 (2002)CrossRefGoogle Scholar
  33. 33.
    Hollerbach U., Chen D., Nonner W., Eisenberg B.: Three-dimensional Poisson–Nernst–Planck theory of open channels. Biophys. J. 76, A205 (1999)Google Scholar
  34. 34.
    Hyon Y., Eisenberg B., Liu C.: A mathematical model for the hard sphere repulsion in ionic solutions. Commun. Math. Sci. 9, 459–475 (2010)MathSciNetGoogle Scholar
  35. 35.
    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)CrossRefGoogle Scholar
  36. 36.
    Im W., Beglov D., Roux B.: Continuum solvation model: Electrostatic forces from numerical solutions to the Poisson–Bolztmann equation. Comp. Phys. Comm. 111, 59–75 (1998)zbMATHCrossRefGoogle Scholar
  37. 37.
    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)CrossRefGoogle Scholar
  38. 38.
    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. (to appear)Google Scholar
  39. 39.
    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), 11 (2007)Google Scholar
  40. 40.
    Kierzenka J., Shampine L.: A BVP solver based on residual control and the Matlab PSE. ACM Trans. Math. Software 27, 299–316 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  41. 41.
    Knepley M., karpeev D., Davidovits S., eisenberg R., Gillespie D.: An efficient algorithm for classical density functional theory in three dimensions: Ionic solutions. J. Chem. Phys 132, 124101 (2010)CrossRefGoogle Scholar
  42. 42.
    Kurnikova M.G., Coalson R.D., Graf P., Nitzan A.: A lattice relaxation algorithm for 3D Poisson–Nernst–Planck theory with application to ion transport through the Gramicidin A channel. Biophys. J. 76, 642–656 (1999)CrossRefGoogle Scholar
  43. 43.
    Liu W.: Geometric singular perturbation approach to steady-state Poisson–Nernst–Planck systems. SIAM J. Appl. Math. 65, 754–766 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  44. 44.
    Liu W.: One-dimensional steady-state Poisson–Nernst–Planck systems for ion channels with multiple ion species. J. Differ. Equ. 246, 428–451 (2009)zbMATHCrossRefGoogle Scholar
  45. 45.
    Mock M.S.: An example of nonuniqueness of stationary solutions in device models. COMPEL 1, 165–174 (1982)CrossRefGoogle Scholar
  46. 46.
    Nadler B., Hollerbach U., Eisenberg R.S.: Dielectric boundary force and its crucial role in gramicidin. Phys. Rev. E 68(021905), 1–9 (2003)Google Scholar
  47. 47.
    Nadler B., Schuss Z., Singer A., Eisenberg B.: Diffusion through protein channels: From molecular description to continuum equations. Nanotech 3, 439–442 (2003)Google Scholar
  48. 48.
    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)CrossRefGoogle Scholar
  49. 49.
    Noskov S.Y., Im W., Roux B.: Ion permeation through the α-Hemolysin channel: Theoretical studies based on Brownian dynamics and Poisson–Nernst–Planck electrodiffusion theory. Biophys. J. 87, 2299–2309 (2004)CrossRefGoogle Scholar
  50. 50.
    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)MathSciNetzbMATHCrossRefGoogle Scholar
  51. 51.
    Percus J.K.: Equilibrium state of a classical fluid of hard rods in an external field. J. Stat. Phys. 15, 505–511 (1976)MathSciNetCrossRefGoogle Scholar
  52. 52.
    Percus J.K.: Model grand potential for a nonuniform classical fluid. J. Chem. Phys. 75, 1316–1319 (1981)CrossRefGoogle Scholar
  53. 53.
    Robledo A., Varea C.: On the relationship between the density functional formalism and the potential distribution theory for nonuniform fluids. J. Stat. Phys. 26, 13–525 (1981)CrossRefGoogle Scholar
  54. 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)CrossRefGoogle Scholar
  55. 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)CrossRefGoogle Scholar
  56. 56.
    Roux B., Allen T.W., Berneche S., Im W.: Theoretical and computational models of biological ion channels. Quat. Rev. Biophys. 37, 15–103 (2004)CrossRefGoogle Scholar
  57. 57.
    Roux, B.: Theory of transport in ion channels: From molecular dynamics simulations to experiments. In: Goodefellow, J. (ed.) Computer Simulation in Molecular Biology, Chapter 6, pp. 133–169. VCH, Weinheim (1995)Google Scholar
  58. 58.
    Rubinstein I.: Multiple steady states in one-dimensional electrodiffusion with local electroneutrality. SIAM J. Appl. Math. 47, 1076–1093 (1987)MathSciNetCrossRefGoogle Scholar
  59. 59.
    Rubinstein I.: Electro-Diffusion of Ions. SIAM Studies in Applied Mathematics. SIAM, Philadelphia (1990)CrossRefGoogle Scholar
  60. 60.
    Saraniti M., Aboud S., Eisenberg R.: The simulation of ionic charge transport in biological ion channels: An introduction to numerical methods. Rev. Comp. Chem. 22, 229–294 (2005)Google Scholar
  61. 61.
    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)CrossRefGoogle Scholar
  62. 62.
    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)MathSciNetzbMATHCrossRefGoogle Scholar
  63. 63.
    Singer A., Gillespie D., Norbury J., Eisenberg R.S.: Singular perturbation analysis of the steady-state Poisson–Nernst–Planck system: Applications to ion channels. European J. Appl. Math. 19, 541–560 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  64. 64.
    Steinrück H.: Asymptotic analysis of the current-voltage curve of a pnpn semiconductor device. IMA J. Appl. Math. 43, 243–259 (1989)MathSciNetzbMATHCrossRefGoogle Scholar
  65. 65.
    Steinrück H.: A bifurcation analysis of the one-dimensional steady-state semiconductor device equations. SIAM J. Appl. Math. 49, 1102–1121 (1989)MathSciNetzbMATHCrossRefGoogle Scholar
  66. 66.
    van der Straaten T.A., Kathawala G., Eisenberg R.S., Ravaioli U.: BioMOCA—a Boltzmann transport Monte Carlo model for ion channel simulation. Mol. Simul. 31, 151–171 (2004)CrossRefGoogle Scholar

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© Springer Science+Business Media New York 2012

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of KansasLawrenceUSA

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