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 L2, the approximate solution of the linearized mPNP equations was found to converge to the analytical solution, hence proof of existence.
Highlights
Biological cells are composed of proteins arranged in folded chains of amino acids to form ionic channels that are nanoscopic water-filled pores to perform the role of controlling transport of ions in cell membranes
The study examines existence and uniqueness of the approximate analytical solutions of the modified Poisson Nernst Planck (mPNP) equations, First, by obtaining the priori energy estimate and providing solution bounds, and 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, proof of existence
Poisson Nernst Planck (PNP) equations for a long period of time has been adopted as a classical mathematical model and analysis tool of choice for studying ion flow
Summary
Biological cells are composed of proteins arranged in folded chains of amino acids to form ionic channels that are nanoscopic water-filled pores to perform the role of controlling transport of ions in cell membranes. These functions are varied and enabled by ability of the cells to carry strong and steeply varying distribution of permanent charges depending on combination of the nanotubes and prevalent physiological conditions. The major drawback of PNP model is that it neglects finite size effects in biological channel systems resulting into significant inaccuracies. We use the variational approach to derive the total energy for LJ repulsive potential which leads to generation of a system of equations that incorporates contribution of finite size effects. Analysis of the local existence of weak solutions of the resultant mPNP by constructing an approximate solution in a finite dimensional space in L2 is carried out
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