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