Model reduction-based initialization methods for solving the Poisson-Nernst-Plank equations in three-dimensional ion channel simulations

Abstract The Poisson-Nernst-Planck (PNP) model is frequently-used in simulating ion transport through ion channel systems. Due to the nonlinear nature of the coupled system, choosing appropriate initial values is often crucial to helping obtain a convergent result or accelerate the convergence rate of finite element solution, especially for large channel protein systems. Continuation is an effective and commonly adopted strategy to provide good initial guesses for the solution procedure. However, this method needs multiple times to solve the whole system at different conditions. We utilize a reduced model describing a near- or partial-equilibrium state as an approximation of the original PNP system (describing a non-equilibrium process in general). Based on the reduced model, we design three initialization methods for the solution of PNP equations under general conditions. These methods provide the initial guess of the PNP system by solving a specifically designed Poisson-Boltzmann- or Smoluchowski-Poisson-Boltzmann-like equation(s). Numerical tests on two highly charged channel protein systems demonstrate that these methods can effectively reduce the number of Gummel iteration steps and/or the total CPU time in the solution of the PNP equations, and especially do not need the continuation approach anymore. The reason is that these initial guesses can approximate the PNP solution well in the channel region. Besides, our numerical experiments demonstrate that as one of the initialization methods, the “linear approximation method ” can even produce very close results such as current-voltage curves to that from the PNP model when the membrane potential is not high.