A Fast and Robust Poisson-Boltzmann Solver Based on Adaptive Cartesian Grids

摘要

An adaptive Cartesian grid (ACG) concept is presented for the fast and robust numerical solution of the 3D Poisson-Boltzmann equation (PBE) governing the electrostatic interactions of large-scale biomolecules and highly charged biomolecularassemblies such as ribosomes and viruses. The ACG offers numerous advantages over competing grid topologies such as regular3D lattices and unstructured grids. For very large biological molecules and their assemblies, the total number of grid pointsis several orders of magnitude less than that required in a conventional lattice grid used in the current PBE solvers, thusallowing the end user to obtain accurate and stable nonlinear PBE solutions on a desktop computer. Compared to tetrahedral-based unstructured grids, ACG offers a simpler hierarchical grid structure, which is naturally suited to multigrid, relievesindirect addressing requirements, and uses fewer neighboring nodes in the finite difference stencils. Construction of the ACGand determination of the dielectric/ionic maps are straightforward and fast and require minimal user intervention. Chargesingularities are eliminated by reformulating the problem to produce the reaction field potential in the molecular interiorand the total electrostatic potential in the exterior ionic solvent region. This approach minimizes grid dependency andalleviates the need for fine grid spacing near atomic charge sites. The technical portion of this paper contains three parts.First, the ACG and its construction for general biomolecular geometries are described. Next, a discrete approximation to thePBE upon this mesh is derived. Finally, the overall solution procedure and multigrid implementation are summarized. Resultsobtained with the ACG-based PBE solver are presented for (i) a low dielectric spherical cavity, containing interior pointcharges, embedded in a high dielectric ionic solvent-analytical solutions are available for this case, thus allowing rigorousassessment of the solution accuracy; (ii) a pair of low dielectric charged spheres embedded in an ionic solvent to computeelectrostatic interaction free energies as a function of the distance between sphere centers; (iii) surface potentials ofproteins, nucleic acids, and their larger-scale assemblies such as ribosomes; and (iv) electrostatic solvation free energiesand their salt sensitivities-obtained with both linear and nonlinear Poisson-Boltzmann equations-for a large set of proteins.These latter results along with timings can serve as benchmarks for comparing the performance of different PBE solvers.