Numerical Approach for Determining Impact of Steric Effects in Biological Ion Channel

Abstract

Flow through biological ion channel is understandably complex to support numerous and vital processes that promote life. To account for the biological evolution, mathematical modelling that incorporates electrostatic interaction of ions and effects due to size exclusion has been studied, conceivably with element of difficulty and inaccuracy. In this paper the Nernst-Planck(NP) equation for ion fluxes that uses Lennard Jonnes(LJ) potential to incorporate finite size effects in terms of hard sphere repulsion is examined. To minimize emerging numerical intricacy, the LJ potential is modified by a band limit function with a cut-off length to eliminate troublesome high frequencies in the integral function. This process is achieved through Fourier transform to simplify and hence render the mPNP equation solvable with precision. The resultant modified NP and Poisson equation representing electrostatic potential are then coupled to form system of equation which describes a realistic transport phenomena in ion channel. Consequently, to discretize the 2D steady system of equations, mixed finite element approach based on Taylor hood eight node square referenced elements is adopted. In the method, Galerkin weighted residual approach help obtain sparse matrix and finally Picard Method applied to the nonlinear terms in the algebraic equations to linearize them and improve rate of convergence. Iterative solution for the system of equations then obtained and concentration profiles of ion species under varied steric effects for mPNP are computed and analysed