An Unfitted Finite Element Poisson-Boltzmann Solver with Automatic Resolving of Curved Molecular Surface

Area of Science:

• Computational biology
• Biophysics
• Scientific computing

Background:

• Existing Poisson-Boltzmann (PB) solvers require pre-generated body-fitted meshes (molecular surface meshes) for accurate interface jump condition calculations.
• Biomolecular surface meshing is a challenging and laborious process, hindering the development and application of numerical methods in this field.
• Molecular surface meshes provide only low-order approximations of curved biomolecular surfaces.

Purpose of the Study:

• To propose an unfitted finite element method (IPFEM) for solving the Poisson-Boltzmann equation (PBE) without requiring user-generated molecular surface meshes.
• To utilize Gaussian molecular surfaces for high-order approximation of biomolecular surfaces within the IPFEM framework.
• To demonstrate the stability and accuracy of the IPFEM for calculating biomolecular electrostatic solvation energy.

Main Methods:

• Developed and implemented an interface-penalty finite element method (IPFEM), an unfitted finite element approach.
• Employed Gaussian molecular surfaces for automatic and high-order approximation of biomolecular surfaces.
• Conducted theoretical convergence rate analysis for the linear PBE and validated on a benchmark problem with an analytical solution.

Main Results:

• The IPFEM successfully solves the PBE without the need for generating molecular surface meshes.
• Theoretical convergence rates for the linear PBE were established and numerically validated.
• Numerical results showed the IPFEM is stable and accurate for calculating electrostatic solvation energy across various biomolecule sizes.
• Similar convergence rates were observed for the nonlinear PBE.

Conclusions:

• The proposed IPFEM offers a robust and efficient alternative to traditional PB solvers that rely on complex meshing.
• This method simplifies the computational workflow for biomolecular electrostatics, enabling broader applications.
• The IPFEM provides accurate, high-order approximations of molecular surfaces and reliable electrostatic solvation energy calculations.