Towards an accurate representation of electrostatics in classical force fields: Efficient implementation of multipolar interactions in biomolecular simulations

Abstract

The accurate simulation of biologically active macromolecules faces serious limitations that originate in the treatment of electrostatics in the empirical force fields. The current use of "partial charges" is a significant source of errors, since these vary widely with different conformations. By contrast, the molecular electrostatic potential (MEP) obtained through the use of a distributed multipole moment description, has been shown to converge to the quantum MEP outside the van der Waals surface, when higher order multipoles are used. However, in spite of the considerable improvement to the representation of the electronic cloud, higher order multipoles are not part of current classical biomolecular force fields due to the excessive computational cost. In this paper we present an efficient formalism for the treatment of higher order multipoles in Cartesian tensor formalism. The Ewald "direct sum" is evaluated through a McMurchie-Davidson formalism [L. McMurchie and E. Davidson, J. Comput. Phys. 26, 218 (1978)]. The "reciprocal sum" has been implemented in three different ways: using an Ewald scheme, a particle mesh Ewald (PME) method, and a multigrid-based approach. We find that even though the use of the McMurchie-Davidson formalism considerably reduces the cost of the calculation with respect to the standard matrix implementation of multipole interactions, the calculation in direct space remains expensive. When most of the calculation is moved to reciprocal space via the PME method, the cost of a calculation where all multipolar interactions (up to hexadecapole-hexadecapole) are included is only about 8.5 times more expensive than a regular AMBER 7 [D. A. Pearlman et al., Comput. Phys. Commun. 91, 1 (1995)] implementation with only charge-charge interactions. The multigrid implementation is slower but shows very promising results for parallelization. It provides a natural way to interface with continuous, Gaussian-based electrostatics in the future. It is hoped that this new formalism will facilitate the systematic implementation of higher order multipoles in classical biomolecular force fields.

Generalization of the Gaussian electrostatic model: Extension to arbitrary angular momentum, distributed multipoles, and speedup with reciprocal space methods

“…The calculations, however, generalize to arbitrary levels of angular momentum in a manner similar to Ref. 36.…”

Notes on “Ewald summation of electrostatic multipole interactions up to quadrupolar level” [J. Chem. Phys. 119, 7471 (2003)]

“…It is worth mentioning that few authors 6,7 have proposed an alternative approach for an efficient evaluation of the real space terms of the multipolar Ewald sum based on tree-code algorithms 7 or through a McMurchie-Davidson formalism. 6,8 All the self-interaction correction terms for the energy as well as for the potential, electric field, and electric field gradients are properly derived from the work of Aguado and Madden 1 ͓see Eqs. ͑50͒-͑62͒ therein͔.…”

Transferable kriging machine learning models for the multipolar electrostatics of helical deca-alanine

“…The use of multipolar electrostatics is becoming increasingly common, and several groups [9][10][11][12][13][14] focus on accurately describing molecular and atomic electronic properties without relying on the fitting and parameterisation of point charges. Multipole moments also continue to appear in molecular dynamics simulations [15][16][17][18] and geometry optimisations [18][19][20].…”