Modeling and Simulating Asymmetrical Conductance Changes in Gramicidin Pores

Abstract

Gramicidin A is a small and well characterized peptide that forms an ion channel in lipid membranes. An important feature of gramicidin A (gA) pore is that its conductance is a ected by the electric charges near the its entrance. This property has led to the application of gramicidin A as a biochemical sensor for monitoring and quantifying a number of chemical and enzymatic reactions. Here, a mathematical model of conductance changes of gramicidin A pores in response to the presence of electrical charges near its entrance, either on membrane surface or attached to gramicidin A itself, is presented. In this numerical simulation, a two dimensional computational domain is set to mimic the structure of a gramicidin A channel in the bilayer surrounded by electrolyte. The transport of ions through the channel is modeled by the Poisson-Nernst-Planck (PNP) equations that are solved by Finite Element Method (FEM). Preliminary numerical simulations of this mathematical model are in qualitative agreement with the experimental results in the literature. In addition to the model and simulations, we also present the analysis of the stability of the solution to the boundary conditions and the convergence of FEM method for the two dimensional PNP equations in our model.

Superconvergence analysis of finite element method for Poisson–Nernst–Planck equations

“…We refer the readers to finite difference method [10], finite volume method [11], FEMs [1,[12][13][14][15][16]. Especially, [1] discussed a priori error estimates of FEM in a completely accurate fashion comparing with [14,15]. However, for bilinear element, only suboptimal error estimates in L 2 -norm was derived theoretically, although the numerical tests are shown to be optimal.…”

Optimal Rate Convergence Analysis of a Second Order Numerical Scheme for the Poisson--Nernst--Planck System

“…Among the existing theoretical works on the convergence analysis, it is worthy of reviewing the following works. A semi-discrete scheme was analyzed in [29], with the spatial convergence estimate is given. A convergence proof was provided for certain class of fully discrete finite element schemes, while the convergence order has not been justified.…”

A decoupling two-grid method for the time-dependent Poisson-Nernst-Planck equations

“…Compared with the scheme in [13], this one is fully implicit nonlinear. It is considered that this implicit nonlinear scheme could preserve most of the properties of the PNP equations and has been commonly used in the computation of the PNP system [15,21,36]. The optimal error estimates in L 2 norm are obtained for both semi-and fully discrete finite element approximations.…”