Dynamics of Classical Poisson–Nernst–Planck Systems with Multiple Cations and Boundary Layers

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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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 (2020). https://doi.org/10.1007/s10884-020-09861-4

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Keywords

  • Ionic flow
  • Individual flux
  • I–V relations
  • Competitions between cations
  • Boundary layer effects

Mathematics Subject Classification

  • 34A26
  • 34B16
  • 34D15
  • 37D10
  • 92C35