Finite Element Ion Channel Modeling

Description

This repository contains the Python code for a series of numerical experiments regarding partial differential equations and ion-channel modeling:

• Manufactured-solution Poisson experiment: verifies a basic FEniCSx implementation against a known quadratic solution and measures how its error changes as the mesh is refined.
• Three-dimensional cube verification: compares a finite-element solution of the linearized Poisson-Boltzmann equation with Holst's screened single-charge formula after approximating the point charge with a Gaussian.
• Two-domain Poisson-Boltzmann experiment: models a low-permittivity molecular region inside a solvent region and studies how the computed potential changes across their interface and under mesh refinement.

Stack: Python | FEniCSx | Gmsh | UFL | MPI | NumPy | Matplotlib | PyVista

For further context, read the two papers in this Google Drive folder:

• The Poisson Equation and Its Role in Ion Channel Modeling: introduces the Poisson equation, numerical methods with FEniCSx, and the Poisson-Boltzmann and Poisson-Nernst-Planck equations used in ion-channel modeling.
• A Poisson Boltzmann Numerical Experiment: extends the earlier study with a numerical implementation of the linearized Poisson-Boltzmann equation, including the cube verification and two-domain experiment.

Motivation

The material in this repository is the result of an independent study conducted with Professor Zhen Chao of Western Washington University, whose research includes mathematical models for ion-channel proteins. Throughout the nearly year-long study, we started with the basics of partial differential equations and numerical methods, including finite differences, and continued until we reached the final deliverable: simulating the Poisson-Boltzmann equation, a PDE often used in ion-channel modeling, with the finite-element method in Python.

Quick Start

Create and activate the Conda environment:

conda env create -f environment.yml
conda activate finite-element-ion-channel-modeling

Run the introductory Poisson benchmark:

python poisson_fundamentals.py

Run the linearized Poisson-Boltzmann experiments:

python poisson_boltzmann.py

Generated meshes, plots, summaries, and solver output are written to results/.

Author

Built by Brandon Connely.