Efficient formulation of polarizable Gaussian multipole electrostatics for biomolecular simulations

Abstract

Molecular dynamics simulations of biomolecules have been widely adopted in biomedical studies. As classical point-charge models continue to be used in routine biomolecular applications, there have been growing demands on developing polarizable force fields for handling more complicated biomolecular processes. Here, we focus on a recently proposed polarizable Gaussian Multipole (pGM) model for biomolecular simulations. A key benefit of pGM is its screening of all short-range electrostatic interactions in a physically consistent manner, which is critical for stable charge-fitting and is needed to reproduce molecular anisotropy. Another advantage of pGM is that each atom's multipoles are represented by a single Gaussian function or its derivatives, allowing for more efficient electrostatics than other Gaussian-based models. In this study, we present an efficient formulation for the pGM model defined with respect to a local frame formed with a set of covalent basis vectors. The covalent basis vectors are chosen to be along each atom's covalent bonding directions. The new local frame can better accommodate the fact that permanent dipoles are primarily aligned along covalent bonds due to the differences in electronegativity of bonded atoms. It also allows molecular flexibility during molecular simulations and facilitates an efficient formulation of analytical electrostatic forces without explicit torque computation. Subsequent numerical tests show that analytical atomic forces agree excellently with numerical finite-difference forces for the tested system. Finally, the new pGM electrostatics algorithm is interfaced with the particle mesh Ewald (PME) implementation in Amber for molecular simulations under the periodic boundary conditions. To validate the overall pGM/PME electrostatics, we conducted an NVE simulation for a small water box of 512 water molecules. Our results show that to achieve energy conservation in the polarizable model, it is important to ensure enough accuracy on both PME and induction iteration. It is hoped that the reformulated pGM model will facilitate the development of future force fields based on the pGM electrostatics for applications in biomolecular systems and processes where polarization plays crucial roles.

FIG. 1.

Definition of covalent basis vectors for atoms in the water molecule. (a) For covalent dipoles centered at A (oxygen), the two basis vectors are ${\stackrel{\rightharpoonup }{e}}^{BA}$ and ${\stackrel{\rightharpoonup }{e}}^{CA}$, which are defined as unit vectors along its two O–H bonds whose two covalent dipoles are the same due to symmetry. (b) For covalent dipoles centered at C (hydrogen), the two basis vectors are ${\stackrel{\rightharpoonup }{e}}^{AC}$, the unit vector along the H–O bond, and ${\stackrel{\rightharpoonup }{e}}^{BC}$, the unit vector along the H–H virtual bond. The covalent dipoles centered at the other hydrogen atom B can be defined similarly.

FIG. 2.

Total energy vs the simulation time for the 512-water box simulation. Here, all the simulations are performed with a 5 × 10⁻⁵ PME accuracy, but with different induction iteration tolerances (1 × 10⁻⁴–1 × 10⁻⁶).

FIG. 3.

Total energy vs the simulation time for the 512-water box simulation. Here, both simulations were performed with a 1 × 10⁻⁶ induction iteration tolerance, but with different PME accuracies, low for 5 × 10⁻⁴ and high for 5 × 10⁻⁵.