Author(s)

Abidha Monica Gwecho¹, Shu Wang², Onyango Thomas Mboya³

¹College of Applied Sciences, Beijing University of Technology, Beijing, China.
²School of Mathematics, Beijing University of Technology, Beijing, China.
³School of Mathematics and Actuarial sciences, Technical University of Kenya, Nairobi, Kenya.

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 L², the approximate solution of the linearized mPNP equations was found to converge to the analytical solution, hence proof of existence.

Lennard Jonnes Potential, Finite Size Effects, Ion Channel, Modified PNP

Gwecho, A.M., Wang, S. and Mboya, O.T. (2020) Existence of Approximate Solutions for Modified Poisson Nernst-Planck Describing Ion Flow in Cell Membranes. American Journal of Computational Mathematics, 10, 473-484. doi: 10.4236/ajcm.2020.103027.