International Journal of Mathematics Trends and Technology
Volume 68 | Issue 4 | Year 2022 | Article Id. IJMTT-V68I4P515 | DOI : https://doi.org/10.14445/22315373/IJMTT-V68I4P515
| Received | Revised | Accepted |
|---|---|---|
| 19 Mar 2022 | 21 Apr 2022 | 26 Apr 2022 |
In this paper, Nernst-Planck(NP) equation for ion fluxes that uses Lennard Jones(LJ) potential to incorporate finite-size effects in terms of hard sphere repulsion was coupled with Poisson equations to form modified PNP(mPNP) system of equations. These coupled equations were then discretized using Galerkin finite element Method(GFEM) approach based on Taylor-hood elements with regular rectangular sub-domains. However, this method resulted into numerical oscillations in the approximate solution, necessitating stabilization to obtain desired results. Consequently, Stream-Upwind Petrov-Galerkin(SUPG)method which adds a mesh dependent term to the FEM together with iterative linearization was adopted resulting into a stable numerical scheme. The resulting linear system of equations was solved iteratively using preconditioned conjugate gradient(PCG) scheme to speed up convergence, where potential consistently updated the concentration components. Concentration profiles of two ion species under varied steric effects for mPNP and PNP were compared and analyzed.
[1] M. Abidha, W. Shu, and T. Onyango, Existence of Approximate Solution for Modified Poisson Nernst-Planck Equation Describing Ion Flow in Cell Membranes. Amer. Journ. of Comp. Math. 10 (2020) 473–484.
[2] M. Abidha, S. Wang, and T. Onyango, Numerical Approach for Determing Impact of Steric Effects in Biological Ion Channel, Journ. of Appl. and Comp. Math. 9(4) 2020.
[3] L. Benzhuo and Y. C. Zhou, Poisson Nernst Equations for Simulating Bio-Molecular Diffusion-Reaction Processes II: Size Effects on Ionic Distributions and Diffusion-Reaction Rates, Biophysica. J. 100(10) (2011) 2475–2485.
[4] T. Bin, Y. Xie, L. Zhang, and L. Benzhou, Stabilized Finite Element Methods to Simulate the Conductance of Ion Channels, Comp. Physics Commnctns. 188 (2015) 131–139.
[5] A. Brooke and T. Hughes, Streamline Upwind/Petrov-Galerkin Formulations for Convection Dominated Flows with Particular Emphasis on the Incompressible Navier-Stokes Equations, Comput. Methds. in Appl. Mech. and Engrg. 32(1-3) (1982) 199–259.
[6] M. Burger, B. Schlake, and M.-T. Wolfram, Nonlinear Poisson-Nernst-Planck Equations for Ion Flux Through Confined Geometries, Lond. Math. Soc. 25(4) (2012) 961–990.
[7] L. Chun, M. Metti, and X. Jinchao, Energetically Stable Discretizations for Charge Carrier Transport and Electrokinetic Models, J. of Compt. Phys. 306 (2016) 1–18.
[8] X. Dexuan and C. Zhen, A Finite Element Iterative Solver for a PNP Ion Channel Model with Neumann Boundary Condition and Membrane Surface Charge, Journ. of Comp. Phy. 423 (2020) 109915.
[9] B. Eisenberg, Ionic Channels in Biological Membranes-Electrostatic Analysis of Natural Nanotube, Contemp. Phys. 39(6) (1998) 447–466.
[10] L. Franca, H. Guillermo, and M. Arif, Revisiting Stabilized Finite Element Methods for Advective-Diffusive Equations, Comput. Methds. Appl. Mech. Engrg. 195 (2006) 1560–1572.
[11] L. Hailliang and W. Zhongming, A Free Satisfying Discontinous Galerkin Method for One Dimensional Poisson Nernst-Planck Systems, Journ. of Comp. Phy. 328 (2017) 413–437.
[12] L. Hongliang, W. Qin, L. Benzhou, and Z. Linbo, A Stabilized Finite Element Method for Poisson-Nernst-Planck Equations in Three Dimensional Ion Channel Simulations. Appl. Math. Lett. 111 (2021) 106652.
[13] D. Meng, Z. Bin, L. Guang, and L. S. Maria, Numerical Solution of 3d Poisson Nernst-Planck Equations Coupled with Classical Density Function Theory for Modelling Ion and Electron Transport in a Confined Environment, Commun. Comput. Phys. 16(5) (2014) 1298– 1322.
[14] J. Miller, W. Schilders, and S. Wang, Application of Finite Element Methods to the Simulation of Semiconductor Devices, Rep. on Prog. in Phys. 62(3) (1999) 277–353.
[15] L. Pichler, A. Masud, and L. Bergman, Numerical Solution of the Fokker-Planck Equations by Finite Difference and Finite Element Methods-A Comparative Study, Comput. Methds in Struc. Dyn. and Earthq. Engrg. 195 (2011) 1560–1572.
[16] Y. Qian, X. Liu, M. Chen, and L. Benzhou, A Local Approximation of Fundamental Measure Theory Incorporated Into Three Dimensional Poisson-Nernst Planck Equations to Account for Hard-Sphere Repulsions Among Ions, J.Stat.Physics. 163 (2016) 156–174.
[17] J. Rodrigo and V. Kovtunenko, Entropy Method for Generalized Poisson-Nernst-Planck Equations, Anal. and Math. Phys. 8(4) (2018) 603–619.
[18] H. Selcuk, The Finite Element Method Over a Simple Stabilized Grid Applied to Fluid Flow Problems, Phd Thesis Middle East Technical University. (2008).
[19] L. Tai-chia and B. Einsenberg, A New Approach to the Lennard-Jones Potential and a New Model: PNP-Steric Equations, Commctn. in Mathcal. Sci. 12(1) (2013) 149–173.
[20] J. Miller, W. Schilders, and S. Wang, Application of Finite Element Methods to the Simulation of Semiconductor Devices, Rep. on Prog. in Phys. 62(3) (1999) 277–353.
[21] Y. Xie, J. Cheng, L. Benzhou, and L. Zhang, Parallel Adaptive Finite Element Algorithms for Solving the Coupled Electro-Diffusion Equations, Mol. Based Math. Biol. 1 (2013) 90–108.
[22] T. Tezduyar, Stabilized Finite Element Formulations for Incompressible Flow Computrations, Adv. in Appl. Mech. 28 (1992) 1–44.
[23] H. Tzyy-Leng, L. Tai-chin, L. Chun, and B. Eisenberg, PNP Equations with Steric Effects, A Model of Ion Flow through Channels, J. Physica. Chem. 116(37) (2012) 11422– 11441.
[24] Y. Vladmir, M. Kiselev, I. Lobanov, D. Marenduzzo, and B. Goryachev, Lateral Dynamics of Charged Lipids and Peripheral Proteins in Spatially Heterogeous Membranes: Comparison of Continous and Monte Carlo Approaches, J. Chem. Physics. 135 (2011) 155103.
[25] H. Yunkyong, B. Eisenberg, and C. Liu, A Mathematical Model for Hard Sphere Repulsion in Ionic Solutions, Commun. Math. Sci. 9(2) (2011) 459–475.
[26] W. Zhongming, Z. Shenggao, D. Jie, and H. Sun, Computational Study on Hysteresis of Ion Channels: Multiple Solutions to Steady State Poisson-Nernst Planck Equations, Commun. Comput. Phys. 23 (2018) 1549–1572.
Abidha Monica Gwecho, Wang Shu, Onyango Thomas Mboya, "Stabilized Finite Element Method for Poisson Nernst-Planck Equations with Steric Effects for Ion Transport," International Journal of Mathematics Trends and Technology (IJMTT), vol. 68, no. 4, pp. 94-101, 2022. Crossref, https://doi.org/10.14445/22315373/IJMTT-V68I4P515
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