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 . 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 . 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  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,  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 established.  equally derived a simple and effective modified Poisson Nernst Planck (mPNP) based on spartial modification of the diffusion coefficients of ions.
The work presented here considers a mathematical model by Lennard Jones (LJ) in 2D. The model consists of nonlinear PDEs with transport properties assumed to vary in continuous and differentiable manner. Critically in this study we employ the LJ repulsive potential for the finite size effects to device a further 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 L2 is carried out.
2. 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.
2.1. 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    given by
where the repulsion or attractive potential between two balls of ions i and j of radius , respectively situated at and in the two-dimensional spatial domain given by
is the chosen energy coupling constant and , are the negative and positive charge densities, respectively.
Variation of Equation (1) is achieved by differentiating it with respect to charge density to result into flux due to the finite size effects of the charge densities, . When the fluxes are added to the Nernst Planck (NP) equation results into modified time dependent equations that incorporates finite size effects each representing rates of time change in concentration of the negative and positive ions, respectively as in Equations (3) and (4) below.
Coupling Equations (3) and (4) with Poisson Equation (5) below we obtain the governing equation
in which is the thermal energy, with the Boltzmann constant and T is the absolute temperature, is the charge density of the protein, and are the charge densities and diffusion coefficients for p and n ions respectively, is the electrostatic potential, is the dielectric coefficient, e is the unit charge, is the valence and N is the number of ions. The energy variational approach applied to the energy of LJ repulsive spheres ensures that the resulting mPNP system satisfies the energy dissipation law given by;
For brevity and without loss of generality, we assume non-standard values of the parameters to be , and , , and the flow to be two dimensional with a unit thickness. This simplification enables linearization of Equations (3), (4) and (5) to obtain governing equation of the flow through the convergent-divergent ion channel given by
2.2. Boundary and Initial Conditions
Consider a bounded domain , for , we seek the solutions in to the Cauchy problem described by Equation (7) subject to initial data
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 ; and is the unit outward normal. Throughout the paper we assume electro-neutrality conditions at the boundaries, implying that the;
3. 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.
3.1. Local Existence of Solution
The approach involves determining the priori estimates on Sobolev norms of concentration , . Galerkin method, see   is introduced to approximate the solution of mPNP equation by projection of the equation into finite dimensional subspace, . The fundamental objective of this method in the proof of existence is to approximate by functions which takes values in a finite dimensional subspace of dimension k.
We project Equation (7) onto to obtain , which satisfies the equation upto a residual orthogonal to . This gives rise to a system of Ordinary Differential Equations (ODEs) in , which has a solution by standard ODE theory. The resultant solution and satisfies an energy estimate of the same form as a prior estimate for the solution of Equation (7). These estimates are uniform in , and permits us to impose the limit to obtain solution of Equation (7) in a bounded domain for . Sobolev spaces , denotes spaces of continuously differentiable and denotes k times continuously differentiable functions,in addition denotes norm, denotes norm and denotes norm.
3.2. 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.
Lemma 1 According to , given , taking and , then if there exists a constant , such that for any
Proof Multiplying Equation (7a) by and integrate over the domain , then using young’s inequality to simplify we obtain
Integrating by parts and applying holder’s inequality, , in the last two terms of Equation (12) we get
then by substituting in Equation (13) followed by imposition of Cauchy’s inequality in Equation (12) results into
by further substituting to simplify right hand side of Equation (14) we finally get
Similarly multiplying Equation (7b) by and integrating by parts over the domain we obtain equivalence of Equation (12) as below
then following the same procedure as Equations (13) and (14) in Equation (16) we obtain
Applying Gronwall inequality in Equations (15) and (17) when taking and gives
then by continuous embedding of into , Equation (18) is simplified to obtain
To obtain the energy estimates for the derivatives we multiply Equation (7a) by and integrate by parts over the spatial domain we obtain
Re-evaluating two last terms of Equation (20) using Cauchy’s inequality give rise to Equations (21) and (22) below:
substituting Equations (21) and (22) in Equation (20) we have
and upon further simplification Equation (23) give rise to (24)
Similarly multiplying Equation (7b) by and following the same procedure as in Equations (21)-(23) we get
then adding Equations (24) and (25) we have
where the constants and .
Finally when Gronwall inequality is imposed on Equation (26) we obtain
ending proof of the lemma.
3.3. Galerkin Approximation
Next we use an ODE theory to get local in time solution to a finite dimensional approximation , of Equation (7) satisfying the same energy bounds as , . Using Galerkin approximation we first construct finite dimensional approximation to Equation (7). Let be smooth function and be an orthonormal basis for the functions in . Let be a finite dimensional subspace given by
We define a projection operator from to and if , then for . From the properties of orthogonal projection operator holds for .
Lemma 2 For any and ,
Lemma 3 If in , then as .
The proof of Lemma 2 and 3 is trivial see, .
Taking ; ; and in Equations (7a) and (7b), we define Galerkin approximation by seeking solution with finite representation given by
subject to initial conditions
Through transformation we obtain equivalence of Equation (30) given by
with initial conditions given by
The integral form of Equations (30) and (31) becomes
Theorem 1 For any integer n and real number there exists a unique solution to partial differential Equations (30) subject to initial condition (31) for also satisfying the ordinary differential Equations (32) with initial condition given by Equation (33).
Proof Considering equivalence of Equations (30) and (32) subject to respective initial conditions as above, proof of existence of local solution when taking any fixed for the system of Equations (30)-(31) in the interval to imply existence of solution to the Equations (32)-(33). Hence, we proceed to multiply Equation (32) by and integrate by parts in the domain to obtain
Following integration process as in Equation (13), the last two terms of Equation (35) reduces to
Substituting Equation (36) in (35) then using Cauchy’s inequality gives
then taking and applying continuous embedding of into ,  Equation (36) becomes
Consequently applying Gronwall inequality in Equation (38) gives
and from parseval’s inequality given in ,
we finally have that can be determined for any .
Theorem 2 For an integer n, the solution in Theorem 1 satisfies the same priori energy bounds as on s.
Proof; Using Lemma 2 to eliminate in the inner product of Equation (30) and similar procedure as in Equations (20)-(27) to find the time integral estimate of , we obtain the bounds of derivatives given by
To prove boundedness of the derivatives in we introduce the following lemma;
Lemma 4 For an integer and , the solution in Theorem 1 also satisfies
with E independent of n and T.
Proof Consider estimate of Equation (30) and noting that vanishes with respect to Lemma 2, we get
Evaluating the derivative of Equation (43) with respect to x gives
Therefore the derivatives of are uniformly bounded
It is imperative to remark that a uniform bounds on leads to uniform bounds on the derivatives. This uniform boundedness proves equicontinuity of . Thus by Arzela Ascoli theorem  there exists a subsequence which converges uniformly to s on the interval .
Theorem 3 Let is a sequence of functions which is equicontinous and bounded. Then there exists a subsequence which converges in the sup norm to s, where s satisfies the integral equation
implying that s also satisfies Equation (7).
Proof Since and its derivatives have bounds independent of t and n, it follows that forms an equicontinous family of functions which have a
subsequence that converges in the sup norm for each to s. Since it follows that as , as . Therefore each time integral
is bounded independent in t. For any given x, converges uniformly for as . Therefore from dominating convergence and using lemma 3, we obtain
hence and s satisfies Equation (7).
Having proved the existence and convergence of approximate solution, its important to note that the uniqueness of this solution conforms to proposition 3.9 on uniqueness found in Equations (3.39)-(3.40), see .
In this study mPNP differential equations which inco-orporate volume size effects is derived. The system of differential equations linearized and the proof of existence of its approximate analytical solution done. Lastly, a projection operator is used to map differential equation from L2 spaces into a finite dimensional subspace of dimension k generating equations whose solutions converge to that of mPNP equations. In the next study we will examine the existence and uniqueness of numerical approximate solution using Galerkin approach for which we shall conduct a numerical experiment for a 2D flow through ion channels.
 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.
 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.
 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., 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.
 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.