Solutions of Poisson-Nernst Planck Equations with Ion Interaction
- 1 School of Mathematics, Beijing University of Technology, Beijing, China
- 2 School of Mathematics, Beijing University of Technology, Beijing, China
- 3 School of Mathematics and Actuarial Sciences, Technical University of Kenya, Nairobi, Kenya
- 4 School of Mathematics, Beijing University of Technology, Beijing, China
Abstract
This paper develops a mixed Finite Element Method (mFEM) based on both classical rectangular elements (with equal nodal points for all degrees of freedom) and Taylor-hood elements to solve Poisson-Nernst-Planck (PNP) equations with steric effects. The Nernst-Planck (NP) equation for ion fluxes is modified to incorporate finite-size effects in terms of hard-sphere repulsion. The resultant modified NP and Poisson equation representing electrostatic potential is then coupled to form a system of the equation that describes a realistic transport phenomenon in an ion channel. Consequently, we apply mFEM based on both Taylor-hood and classical rectangular elements to discretize the 2D steady system of equations with iterative linearization for the nonlinear components. The numerical scheme is first validated using a 2D analytical solution for PNP equations, the steric components varied and the effects on concentration analyzed and compared against PNP and modified PNP for two monovalent ion species. Classical rectangular elements presented a better comparable approximate solution than Taylor-hood.
- Chen, D. and Wei, G.W. (2017) A Review of Mathematical Modeling, Simulation and Analysis of Membrane Channel Charge Transport. In: Reference Module in Life Sciences, Elsevier, Amsterdam, 1-42. https://doi.org/10.1016/B978-0-12-809633-8.12044-8
- Zheng, Q. and Wei, G.W. (2011) Poisson-Boltzmann-Nernst Planck Model. The Journal of Chemical Physics, 134, Article ID: 194101. https://doi.org/10.1063/1.3581031
- Eisenberg, B. (1998) Ionic Channels in Biological Membranes-Electrostatic Analysis of Natural Nanotube. Contemporary Physics, 39, 447-466. https://doi.org/10.1080/001075198181775
- Yang, Z.C., Van, T.A., Straaten, D., Ravaioli, U. and Liu, Y. (2005) A Coupled 3-D PNP/ECP Model for Ion Transport in Biological Ion Channels. Journal of Computational Electronics, 4, 167-170. https://doi.org/10.1007/s10825-005-7131-8
- Flavell, A., Machen, M., Eisenberg, B., Kabre, J., Liu, C. and Li, X.F. (2013) A Conservative Finite Difference Scheme for Poisson Nernst-Planck Equations. Journal of Computational Electronics, 13, 235-249. https://doi.org/10.1007/s10825-013-0506-3
- Meng, D., Zheng, B., Lin, G. and Sushko, M.L. (2014) Numerical Solution of 3D Poisson Nernst-Planck Equations Coupled with Classical Density Function Theory for Modeling Ion and Electron Transport in a Confined Environment. Communications in Computational Physics, 16, 1298-1322. https://doi.org/10.4208/cicp.040913.120514a
- Siddiqua, F., Wang, Z.M. and Zhou, S.G. (2017) A Modified Poisson-Nernst-Planck Model with Excluded Volume Effects: Theory and Numerical Implementation. Communications in Mathematical Sciences, 16, 26 p.
- Horng, T.L., Lin, T.C., Liu, C. and Eisenberg, B. (2012) PNP Equations with Steric Effects: A Model of Ion Flow through Channels. The Journal of Physical Chemistry B, 116, 11422-11441.
- Lu, B.Z. and Zhou, Y.C. (2011) Poisson Nernst Equations for Simulating Biomolecular Diffusion-Reaction Processes II: Size Effects on Ionic Distributions and Diffusion-Reaction Rates. Biophysical Journal, 100, 2475-2485. https://doi.org/10.1016/j.bpj.2011.03.059
- Hyon, Y., Eisenberg, B. and Liu, C. (2011) A Mathematical Model for Hard Sphere Repulsion in Ionic Solutions. Communications in Mathematical Sciences, 9, 459-475. https://doi.org/10.4310/CMS.2011.v9.n2.a5
- Qian, Y., Liu, X.J., Chen, M.X. and Lu, B.Z. (2016) A Local Approximation of Fundamental Measure Theory Incorporated into Three Dimensional Poisson-Nernst Planck Equations to Account for Hard Sphere Repulsions among Ions. Journal of Statistical Physics, 163, 156-174. https://doi.org/10.1007/s10955-016-1470-7