Existence of Approximate Solutions for Modified Poisson Nernst-Planck Describing Ion Flow in Cell Membranes

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 for-mulating 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.


Introduction
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 channel maintains correct ion composition and balance in cells that is crucial in their survival and numerous functions of propagating life such as, energy conversion, drug delivery, secretion, among others. 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.
Ion channels are characterized by their functioning, some are known to exhibit complex switching properties similar to electronic devices, others have the ability to selectively transport or block a particular ion species, while others have no selectivity, see [1]. To develop deeper understanding of the processes in the channel both analytical and empirical investigation are critical. Numerical experiments approximate transport through biological channels to determine amongst other things structures and conductance of ion channels as a means of minimizing cost and complementation of empirical findings, see [2].
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. The model couples electrostatics with diffusion process as a popular theoretical method to robustly simulate ion channel systems. However, the major drawback of PNP model is that it neglects finite size effects in biological channel systems resulting into significant inaccuracies.
Incorporating electrostatic interaction of ions and finite size effects particularly in narrow regions has been suggested, investigated and determined to impact in reduction of error in solution approximation. Energy Variational Approach was used by [3] to derive an accurate generalized Poisson Nernst Planck-Navier Stokes (PNP-NS) system which characterizes interactions of charged fluid and mutual friction between the crowded charged particles. A general method was thereafter developed to show that the system is globally asymptotically stable under small perturbation around a constant equilibrium.
Subsequently, [4] derived mPNP system which includes an extra dissipation due to effective velocity differences between ion species, then using Galerkins method and Schauders fixed point theorem local existence theorem of the classical solutions of the mPNP system was established.
Other forms of transformations though with inherent limitations to improve accuracy in the approximation of ion flow parameters in cell membranes have been suggested in several models such as steady state modified Poisson-Boltzmann (mPB) model. The mPB was later improved using Lambert-W special functions and the existence and uniqueness of its weak solution es- improved PNP equation. 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. Consequently, analysis of the local existence of weak solutions of the resultant mPNP by constructing an approximate solution in a finite dimensional space in L 2 is carried out.

Model Description
Deterministic mathematical model for simulating ion transport in bio cells are space and time dependent nonlinear PDEs posed as Initial value problems(IVP). Upon incorporating realistic and necessary physics for the flow phenomena the mPNP-IVP becomes complex deterring analytical determination of solution. In principle, to reduce intricacy in the approximation when fundamentally retaining accuracy in the estimation of the flow parameters, interaction of charged particles alone is declared sufficient. This leads to dropping inclusion of charges interaction with the channel walls and fluid which are assumed to have minimal contribution in the approximation results.

Modified Poisson Nernst-Planck (mPNP) Equation
The integral form of energy equation that integrates finite size effects and interactions between charged particles can be modelled as repulsive or attractive spherical particles in the energy term. The energy of these effects in the microscopic scale are summed to represent potential between the positive and negative charges [6] [7] [8] given by where the repulsion or attractive potential between two balls of ions i and j of radius i a , j a respectively situated at x and y in the two-dimensional spatial domain Ω given by ions, respectively as in Equations (3) and (4) Coupling Equations (3) and (4) with Poisson Equation (5) below we obtain the governing equation For brevity and without loss of generality, we assume non-standard values of the parameters to be  in Ω to the Cauchy problem described by Equation (7) subject to initial data

Consider a bounded domain
We specify the Dirichlet boundary conditions to represent fixed electrostatic potential at the boundaries as; and prescribe Neumann boundary condition describing null charge density fluxes at the boundaries by; where , i n p = ; and ν is the unit outward normal. Throughout the paper we assume electro-neutrality conditions at the boundaries, implying that the;

Existence of Approximate Solution of mPNP
In this section, energy method is used to prove existence of solution of the governing equation for ion transport through cell membrane. This will start by first defining the space in which the solution is estimated, describing the local existence, determining prior energy estimate and finally working out the discrete solution and its convergence to the defined bound.

Local Existence of Solution
The approach involves determining the priori estimates on Sobolev norms of concentration n c , p c . Galerkin method, see [9]  ∈ Ω of dimension k.
We project Equation (7) onto k b to obtain nk c , pk c which satisfies the equation upto a residual orthogonal to k b . This gives rise to a system of Ordinary Differential Equations (ODEs) in nk c , pk c which has a solution by standard ODE theory. The resultant solution nk c and pk c satisfies an energy estimate of the same form as a prior estimate for the solution of Equation (7). These estimates are uniform in k b , and permits us to impose the limit k → ∞ to obtain solution of Equation (7)

The Priori Energy Estimate
In general, it is demanding to solve mPNP analytically because of nonlinearity, thus derivation of some energy estimates for the solutions of the system of Equation (7) by assuming v and w are given functions becomes a possible way for studying and analysing the physical problem.