Electrostatic Potential Maps on Molecular Surfaces Using Least Squares Reconstruction

Biomolecular surface electrostatic potential maps (EPMs) are routinely used to infer biological function, binding sites, and qualitative guidance (and subsequent quantitative assessment) of the geometric and electrostatic properties of receptor-drug complexes. A major challenge in Poisson-Boltzmann (PB) modeling and generation of EPMs is accurately accounting for the dielectric discontinuity at the molecular surface and associated jump in normal electrostatic field. One option for accommodating such solutions is to develop a surface conforming grid such that no element or mesh edge intersects the molecular surface. Given the complex shapes of biomolecules, unstructured tetrahedral grids are usually necessary to develop such grids. Here an alternate approach is presented that first develops an adaptive Cartesian grid with no attempt to align it with the molecular surface. Next each mesh edge that intersects the surface is processed by identifying the intersection point and then using a least squares-based reconstruction (LSR) method to estimate the surface potentials and their gradients. The LSR incorporates the analytical surface matching constraints, satisfies the PBE locally, accounts for surface curvature, and is combined with the finite difference scheme used at off-surface points to solve the PB equation. This results in a global set of equations that solves the governing equations at all interior and exterior points, and couples then at the edge intersection points. The LSR is implemented in the adaptive Cartesian grid PB solver (from herein called CPB) and applied to a variety of biomolecular systems ranging from small to very large-scale (e.g., viruses) assemblies to compute electrostatic energies, forces, and high-quality estimates of the surface EPM. Using LSR within CPB allows additional memory savings and proper visualization and rationalization of electrostatic complementarity/recognition features of protein-drug and nucleic acid-ligand complexes, with the latter requiring inclusion of nonlinear screening effects. In addition to detailing the LSR method, the article provides several recommendations for producing accurate, reliable, and reproducible biomolecular EPMs. The importance of performing grid convergence tests to ensure converged predictions is emphasized along with the need to properly account for nonlinear effects in highly charged systems and provide sufficient details of the biomolecular geometry, grid size, environmental conditions, and choice of contour levels to facilitate comparison against other EPM generation software.