Existence of Approximate Solutions for Modified Poisson Nernst-Planck Describing Ion Flow in Cell Membranes
- 1 College of Applied Sciences, 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
Abstract
Dynamics of ions in biological ion channels has been classically analyzed using several types of Poisson-Nernst Planck (PNP) equations. However, due to complex interaction between individual ions and ions with the channel walls, minimal incorporation of these interaction factors in the models to describe the flow phenomena accurately has been done. In this paper, we aim at formulating a modified PNP equation which constitutes finite size effects to capture ions interactions in the channel using Lennard Jonnes (LJ) potential theory. Particularly, the study examines existence and uniqueness of the approximate analytical solutions of the mPNP equations, First, by obtaining the priori energy estimate and providing solution bounds, and finally constructing the approximate solutions and establishing its convergence in a finite dimensional subspace in L 2 , the approximate solution of the linearized mPNP equations was found to converge to the analytical solution, hence proof of existence.
- Karniadakis, G., Beskok, A. and Narayan, A. (2005) Microows and Nanoows: Fundamentals and Simulation. In: Antman, S.S., Marsden, J.E. and Sirovich, L., Eds., Springer-Verlag, Berlin, Vol. 29, 30-36.
- Qiao, Y., Bin, T. and Benzhuo, L. (2014) Ion Size Effects to Molecular Solvation Energy and to Ion Current across a Channel Resulted from the Non-Uniform SIZE Modified PNP System. The Journal of Chemical Physics, 140, Article No. 174102. https://doi.org/10.1063/1.4872330
- Yong, Y., Chun, L. and Zhang, T. (2016) A Generalized Poisson Nernst-Planck Navier Stokes Model on the UID with the Crowded Charged Particles; Derivation and Its Well Posedness. SIAM Journal on Mathematical Analysis, 48, 3191-3235. https://doi.org/10.1137/16M1055104
- Hsieh, C., Yunkyong, H., Hijin, L., Tai-Chia, L. and Chun, L. (2014) Transport of Charged Particles: Entropy Production and Maximum Dissipation Principle. Journal of Mathematical Analysis and Applications, 422, 31.
- Farjana, S., Zhongming, W. and Shenggao, Z. (2017) A Modified Poisson Nernst-Planck Model with Excluded Volume Effects; Theory and Numerical Implementation. Communications in Mathematical Sciences, 16, 26.
- Eisenberg, B. (2011) Mass Action in Ionic Solutions. Chemical Physics Letters, 511, 1-6. https://doi.org/10.1016/j.cplett.2011.05.037
- Eisenberg, B., Yunkyong, H. and Chun, L. (2010) Energy Variational Analysis of Ions in Water and Channels; Field Theory for Primitive Models of Complex Ionic UIDS. The Journal of Chemical Physics, 133, Article No. 104104. https://doi.org/10.1063/1.3476262
- Yunkyong, H., 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
- Hunter, J. (2014) Notes on Partial Differential Equations. 177-193.
- Tomastz, D. (2008) Certain Methods of Proving Existence and Uniqueness in Partial Differential Equations Theory. Silesian University, Poland, 18.
- Tanveer, S. (2017) Introduction to Energy Methods for Existence of Solution. Ohio state University, Columbus, 14.
- Brezis, H. (2010) Functional Analysis, Sobolev Spaces and Partial Diffrential Equations. Springer, New York, NY, 201-312. https://doi.org/10.1007/978-0-387-70914-7_8