Georgia Institute of Technology College of Sciences

Ion channels are a class of proteins embedded in biological membranes, acting as biological devices or 'valves' for cells and playing a critical role in controlling various biological functions. To compute macroscopic ion channel kinetics, such as Gibbs free energy, electric currents, transport fluxes, membrane potential, and electrochemical potential, Poisson-Nernst-Planck ion channel (PNPIC) models have been developed as systems of nonlinear partial differential equations. However, they are difficult to solve numerically due to solution singularities, exponential nonlinearities, multiple physical domain issues, and the requirement of ionic concentration positivity. In this talk, I will present the recent progress we made in the development of finite element methods for solving PNPIC models. Specifically, I will introduce our improved PNPIC models and describe the mathematical and numerical techniques we utilized to develop efficient finite element iterative methods. Additionally, I will introduce the related software packages we developed for a voltage-dependent anion-channel protein and a mixture solution of multiple ionic species. Finally, I will present numerical results to demonstrate the fast convergence of our iterative methods and the high performance of our software package. This work was partially supported by the National Science Foundation through award number DMS-2153376 and the Simons Foundation through research award 711776.