A Boundary‐Integral Approach for the Poisson–Boltzmann Equation with Polarizable Force Fields

Abstract

Implicit‐solvent models are widely used to study the electrostatics in dissolved biomolecules, which are parameterized using force fields. Standard force fields treat the charge distribution with point charges; however, other force fields have emerged which offer a more realistic description by considering polarizability. In this work, we present the implementation of the polarizable and multipolar force field atomic multipole optimized energetics for biomolecular applications (AMOEBA), in the boundary integral Poisson–Boltzmann solver PyGBe. Previous work from other researchers coupled AMOEBA with the finite‐difference solver APBS, and found difficulties to effectively transfer the multipolar charge description to the mesh. A boundary integral formulation treats the charge distribution analytically, overlooking such limitations. This becomes particularly important in simulations that need high accuracy, for example, when the quantity of interest is the difference between solvation energies obtained from separate calculations, like happens for binding energy. We present verification and validation results of our software, compare it with the implementation on APBS, and assess the efficiency of AMOEBA and classical point‐charge force fields in a Poisson–Boltzmann solver. We found that a boundary integral approach performs similarly to a volumetric method on CPU. Also, we present a GPU implementation of our solver. Moreover, with a boundary element method, the mesh density to correctly resolve the electrostatic potential is the same for standard point‐charge and multipolar force fields. Finally, we saw that for binding energy calculations, a boundary integral approach presents more consistent results than a finite difference approximation for multipolar force fields. © 2019 Wiley Periodicals, Inc.

Recent Developments in Free Energy Calculations for Drug Discovery

“…Integration of a Gaussian based model for molecular volume and surface area determination with the Gaussian dielectric distribution removes sharp surfaces separating the solute and solvent for a surface free approach to MM-PBSA calculation ( Chakravorty et al, 2019 ). Electronic polarization effects can be incorporated through the use of polarizable force fields such as AMOEBA, this is implemented in the boundary integral PBE solver PyGBe ( Cooper, 2019 ). Combination of the polarizable Drude oscillator force field with PBSA lowers RMSE from 2.5 kcal/mol with the standard CHARMM36 force field to 0.8 kcal/mol in calculation of solvation free energies for 70 molecules in addition to reducing errors in alanine scanning ( Aleksandrov et al, 2018 ).…”

ddX: Polarizable Continuum Solvation from Small Molecules to Proteins

“…In the context of biophysics, on the other hand, modeling ionic solutions is paramount, which makes the (linearized) Poisson-Boltzmann ((L)PB) model [27][28][29] the de facto standard. Many numerical realizations of such a model have been implemented and are available to the community in popular software that solve the (L)PB equation using a variety of methods that include finite differences [30][31][32][33] , finite elements [34][35][36] , and boundary elements [37][38][39][40][41][42][43][44][45][46][47] . As the modeled systems are typically large biomolecules described with a classical force field (i.e., point charges, or sometimes more sophisticated treatments including distributed multipoles and polarizabilities) and not with quantum mechanics, the treatment of solvation can become a cost-dominating factor in such computations.…”

A generalized Kirkwood implicit solvent for the polarizable AMOEBA protein model

“…The total implicit solvent potential of mean force can be divided into polar (electrostatic) and non-polar terms. The polar term can be calculated numerically using Poisson-Boltzmann (PB) solvers such as the adaptive Poisson-Boltzmann solver (APBS) 6 and PyGBe 7,8 . Alternatively, the popular generalized Born (GB) [9][10][11] model for fixed partial charges or the generalized Kirkwood (GK) 12 model for polarizable multipoles offer efficient analytic approximations.…”