Poisson-Nernst-Planck Equations for Simulating Biomolecular Diffusion-Reaction Processes I: Finite Element Solutions.

Area of Science:

• Computational biology
• Biophysics
• Applied mathematics

Background:

• Electrodiffusion in solvated biomolecular systems is complex.
• Singular permanent charges in biomolecules pose computational challenges.

Purpose of the Study:

• To develop accurate finite element methods for 3-D Poisson-Nernst-Planck (PNP) equations.
• To handle singular permanent charges in biomolecular electrodiffusion.

Main Methods:

• Applied a stable regularization scheme to address singular electrostatic potentials.
• Utilized inexact-Newton and Adams-Bashforth-Crank-Nicolson methods for solving steady and unsteady PNP equations.
• Investigated stiffness matrix conditioning for Nernst-Planck equation formulations.

Main Results:

• Formulated regular, well-posed PNP equations by removing singularity.
• Demonstrated that a transformed Nernst-Planck formulation leads to ill-conditioned matrices.
• Observed significant net charge concentration near molecular surfaces.

Conclusions:

• The developed finite element methods are accurate and stable for 3-D PNP equations.
• These methods are applicable to large-scale biophysical electrodiffusion problems.
• Understanding charge distribution near biomolecular surfaces is critical.