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 cha...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 <em>L</em><sup>2</sup>, the approximate solution of the linearized mPNP equations was found to converge to the analytical solution, hence proof of existence.展开更多
This paper develops a mixed Finite Element Method (mFEM) based on both classical rectangular elements (with equal nodal points for all degrees of freedom) and Taylor-hood elements to solve Poisson-Nernst-Planck (PNP) ...This paper develops a mixed Finite Element Method (mFEM) based on both classical rectangular elements (with equal nodal points for all degrees of freedom) and Taylor-hood elements to solve Poisson-Nernst-Planck (PNP) equations with steric effects. The Nernst-Planck (NP) equation for ion fluxes is modified to incorporate finite-size effects in terms of hard-sphere repulsion. The resultant modified NP and Poisson equation representing electrostatic potential is then coupled to form a system of the equation that describes a realistic transport phenomenon in an ion channel. Consequently, we apply mFEM based on both Taylor-hood and classical rectangular elements to discretize the 2D steady system of equations with iterative linearization for the nonlinear components. The numerical scheme is first validated using a 2D analytical solution for PNP equations, the steric components varied and the effects on concentration analyzed and compared against PNP and modified PNP for two monovalent ion species. Classical rectangular elements presented a better comparable approximate solution than Taylor-hood.展开更多
文摘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 <em>L</em><sup>2</sup>, the approximate solution of the linearized mPNP equations was found to converge to the analytical solution, hence proof of existence.
文摘This paper develops a mixed Finite Element Method (mFEM) based on both classical rectangular elements (with equal nodal points for all degrees of freedom) and Taylor-hood elements to solve Poisson-Nernst-Planck (PNP) equations with steric effects. The Nernst-Planck (NP) equation for ion fluxes is modified to incorporate finite-size effects in terms of hard-sphere repulsion. The resultant modified NP and Poisson equation representing electrostatic potential is then coupled to form a system of the equation that describes a realistic transport phenomenon in an ion channel. Consequently, we apply mFEM based on both Taylor-hood and classical rectangular elements to discretize the 2D steady system of equations with iterative linearization for the nonlinear components. The numerical scheme is first validated using a 2D analytical solution for PNP equations, the steric components varied and the effects on concentration analyzed and compared against PNP and modified PNP for two monovalent ion species. Classical rectangular elements presented a better comparable approximate solution than Taylor-hood.
