A stabilized finite volume element method for solving Poisson-Nernst-Planck equations.

One difficulty in solving the Poisson-Nernst-Planck (PNP) equations used for studying the ion transport in channel proteins is the possible convection-dominant problem in the Nernst-Planck equations. In this paper, to overcome this issue, considering the general mixed boundary conditions of concentration functions on the interface, a novel stabilized finite volume element method based on the standard weak formulation to solve the steady-state PNP equations is proposed and analyzed. Numerical tests on four ion-channel proteins served as benchmark with varying boundary conditions in a certain range show that the new stabilized technique not only improves the robustness of the new PNP solver, but also makes the computed (especially the maximal) concentration values much more reasonable.

An accelerated nonlocal Poisson-Boltzmann equation solver for electrostatics of biomolecule.

The nonlocal modified Poisson-Boltzmann equation (NMPBE) is one important variant of a commonly used dielectric continuum model, the Poisson-Boltzmann equation (PBE), for computing electrostatics of biomolecules. In this paper, an accelerated NMPBE solver is constructed by finite element and finite difference hybrid techniques. It is then programmed as a software package for computing electrostatic solvation and binding free energies for a protein in a symmetric 1:1 ionic solvent. Numerical results validate the new solver and its numerical stability. They also demonstrate that the new solver has much better performance than the corresponding finite element solver in terms of computer CPU time. Furthermore, they show that the binding free energies produced by NMPBE can match chemical experiment data better than those by PBE.

SMPBS: Web server for computing biomolecular electrostatics using finite element solvers of size modified Poisson-Boltzmann equation.

SMPBS (Size Modified Poisson-Boltzmann Solvers) is a web server for computing biomolecular electrostatics using finite element solvers of the size modified Poisson-Boltzmann equation (SMPBE). SMPBE not only reflects ionic size effects but also includes the classic Poisson-Boltzmann equation (PBE) as a special case. Thus, its web server is expected to have a broader range of applications than a PBE web server. SMPBS is designed with a dynamic, mobile-friendly user interface, and features easily accessible help text, asynchronous data submission, and an interactive, hardware-accelerated molecular visualization viewer based on the 3Dmol.js library. In particular, the viewer allows computed electrostatics to be directly mapped onto an irregular triangular mesh of a molecular surface. Due to this functionality and the fast SMPBE finite element solvers, the web server is very efficient in the calculation and visualization of electrostatics. In addition, SMPBE is reconstructed using a new objective electrostatic free energy, clearly showing that the electrostatics and ionic concentrations predicted by SMPBE are optimal in the sense of minimizing the objective electrostatic free energy. SMPBS is available at the URL: smpbs.math.uwm.edu © 2017 Wiley Periodicals, Inc.

Analytical solutions of nonlocal Poisson dielectric models with multiple point charges inside a dielectric sphere.

The nonlocal dielectric approach has led to new models and solvers for predicting electrostatics of proteins (or other biomolecules), but how to validate and compare them remains a challenge. To promote such a study, in this paper, two typical nonlocal dielectric models are revisited. Their analytical solutions are then found in the expressions of simple series for a dielectric sphere containing any number of point charges. As a special case, the analytical solution of the corresponding Poisson dielectric model is also derived in simple series, which significantly improves the well known Kirkwood's double series expansion. Furthermore, a convolution of one nonlocal dielectric solution with a commonly used nonlocal kernel function is obtained, along with the reaction parts of these local and nonlocal solutions. To turn these new series solutions into a valuable research tool, they are programed as a free fortran software package, which can input point charge data directly from a protein data bank file. Consequently, different validation tests can be quickly done on different proteins. Finally, a test example for a protein with 488 atomic charges is reported to demonstrate the differences between the local and nonlocal models as well as the importance of using the reaction parts to develop local and nonlocal dielectric solvers.