AQUASOL: An efficient solver for the dipolar Poisson–Boltzmann–Langevin equation

Abstract

The Poisson-Boltzmann ͑PB͒ formalism is among the most popular approaches to modeling the solvation of molecules. It assumes a continuum model for water, leading to a dielectric permittivity that only depends on position in space. In contrast, the dipolar Poisson-Boltzmann-Langevin ͑DPBL͒ formalism represents the solvent as a collection of orientable dipoles with nonuniform concentration; this leads to a nonlinear permittivity function that depends both on the position and on the local electric field at that position. The differences in the assumptions underlying these two models lead to significant differences in the equations they generate. The PB equation is a second order, elliptic, nonlinear partial differential equation ͑PDE͒. Its response coefficients correspond to the dielectric permittivity and are therefore constant within each subdomain of the system considered ͑i.e., inside and outside of the molecules considered͒. While the DPBL equation is also a second order, elliptic, nonlinear PDE, its response coefficients are nonlinear functions of the electrostatic potential. Many solvers have been developed for the PB equation; to our knowledge, none of these can be directly applied to the DPBL equation. The methods they use may adapt to the difference; their implementations however are PBE specific. We adapted the PBE solver originally developed by Holst and Saied ͓J. Comput. Chem. 16, 337 ͑1995͔͒ to the problem of solving the DPBL equation. This solver uses a truncated Newton method with a multigrid preconditioner. Numerical evidences suggest that it converges for the DPBL equation and that the convergence is superlinear. It is found however to be slow and greedy in memory requirement for problems commonly encountered in computational biology and computational chemistry. To circumvent these problems, we propose two variants, a quasi-Newton solver based on a simplified, inexact Jacobian and an iterative self-consistent solver that is based directly on the PBE solver. While both methods are not guaranteed to converge, numerical evidences suggest that they do and that their convergence is also superlinear. Both variants are significantly faster than the solver based on the exact Jacobian, with a much smaller memory footprint. All three methods have been implemented in a new code named AQUASOL, which is freely available.

Beyond Poisson–Boltzmann: fluctuations and fluid structure in a self-consistent theory

“…This is done first within the point dipole limit, which allows the formulation local to be kept local. The properties of these theories within mean-field treatments and their application to protein electrostatics have been discussed in a series of papers in recent years [12,13,[31][32][33][34][35][36][37][38], see also the review [39].…”

Adapting Poisson-Boltzmann to the self-consistent mean field theory: Application to protein side-chain modeling

“…[54][55][56][57] Such a model has proved useful for characterizing dielectric permittivity profiles in the vicinity of charged surfaces 54,57 as well as for characterizing the hydration layer in the vicinity of proteins and nucleic acids. 45 We believe that the structuring of water molecules and ions in the neighborhood of the surface of a protein plays a role in defining the conformations of exposed side chains. We are, therefore, currently working on adapting the SCMF-PB formalism to incorporate these modified PB equations.…”

“…The computing time required to solve the PB equation is linear in the total number of vertices of the 3D grid considered. 45 Once the electrostatic potential maps are known, we compute in step (3) the corresponding solvation free energy that is subsequently added to the internal energy and the conformational entropy to get the full variational free energy (Eq. (2)).…”

“…The solvent, dielectric, and fixed charge density maps are computed over all vertices in that grid, following the methods implemented in AQUASOL. 45 In particular, we use the sphere charging model 44 to project charges in the grid. Note that these maps are probabilistic, as described above, and need to be computed at each cycle, as they depend on the weights P (i, j ).…”

Charge density distributions derived from smoothed electrostatic potential functions: design of protein reduced point charge models

“…The solvent can also be described using a dipolar lattice as detailed in [17]. A generalized procedure, such as the dipolar Poisson-Boltzmann-Langevin equation [18] allows to model the solvent as a set of interacting dipoles with a non-uniform dielectric constant at the solute/solvent interface that depends on the location, the electrostatic potential, and the electric field. The design of reduced, or coarse-grained (CG), descriptions is also a current field of research in the modeling of large biomolecules [19,20].…”