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The Goldman-Hodgkin-Katz (GHK) equations have been widely applied to ionchannel studies, simulations, and model developments. However, they areconstructed under a constant electric field, causing them to have a low degreeof approximation in the prediction of ionic fluxes, electric currents, andmembrane potentials. In this paper, the equations are extended from theconstant electric field to the nonlinear electric field induced by charges froman ionic solution and an ion channel protein. Furthermore, a novel numericalquadrature scheme is developed to estimate one major parameter, called theextension parameter, of the extended GHK equations in terms of a set ofelectrostatic potential values. To this end, the extended GHK equations becomea bridge between the "macroscopic" ion channel kinetics and the "microscopic"electrostatic potential values across a cell membrane. To generate a set ofrequired electrostatic potential values, a nonlinear finite element iterativescheme for solving a one-dimensional Poisson-Nernst-Planck ion channel model isdeveloped and implemented as a Python software package. This package is thenused to do numerical studies on the extended GHK equations, the numericalquadrature scheme, and the nonlinear iterative scheme. Numerical resultsconfirm the importance of considering charge effects in the calculation ofionic fluxes. They also validate the high numerical accuracy of the numericalquadrature scheme, the fast convergence rate of the nonlinear iterative scheme,and the high performance of the software package.

An Extension of Goldman-Hodgkin-Katz Equations by Charges from Ionic Solution and Ion Channel Protein