Poisson–Boltzmann model for protein–surface electrostatic interactions and grid-convergence study using the PyGBe code

Abstract

Interactions between surfaces and proteins occur in many vital processes and are crucial in biotechnology: the ability to control specific interactions is essential in fields like biomaterials, biomedical implants and biosensors. In the latter case, biosensor sensitivity hinges on ligand proteins adsorbing on bioactive surfaces with a favorable orientation, exposing reaction sites to target molecules. Protein adsorption, being a free-energy-driven process, is difficult to study experimentally. This paper develops and evaluates a computational model to study electrostatic interactions of proteins and charged nanosurfaces, via the Poisson–Boltzmann equation. We extended the implicit-solvent model used in the open-source code PyGBe to include surfaces of imposed charge or potential. This code solves the boundary integral formulation of the Poisson–Boltzmann equation, discretized with surface elements. PyGBe has at its core a treecode-accelerated Krylov iterative solver, resulting in O(NlogN) scaling, with further acceleration on hardware via multi-threaded execution on gpus. It computes solvation and surface free energies, providing a framework for studying the effect of electrostatics on adsorption. We derived an analytical solution for a spherical charged surface interacting with a spherical dielectric cavity, and used it in a grid-convergence study to build evidence on the correctness of our approach. The study showed the error decaying with the average area of the boundary elements, i.e., the method is O(1/N), which is consistent with our previous verification studies using PyGBe. We also studied grid-convergence using a real molecular geometry (protein GB1D4′), in this case using Richardson extrapolation (in the absence of an analytical solution) and confirmed the O(1/N) scaling. With this work, we can now access a completely new family of problems, which no other major bioelectrostatics solver, e.g. APBS, is capable of dealing with. PyGBe is open-source under an MIT license and is hosted under version control at https://github.com/barbagroup/pygbe. To supplement this paper, we prepared “reproducibility packages” consisting of running and post-processing scripts in Python for replicating the grid-convergence studies, all the way to generating the final plots, with a single command.