A regularized Matched Interface and Boundary Method (MIB) for Solving Polarizable Multipole Poisson-Boltzmann model

TL;DR

This work presents a regularized Matched Interface and Boundary (MIB) method for solving the Polarizable Multipole Poisson-Boltzmann (PMPB) model, pairing AMOEBA-derived polarizable multipoles with a linear PB equation in implicit solvent. By decomposing the potential into a regularization component and a Green's-function-based Coulomb term, the method analytically handles singular sources and complex interfaces while preserving second-order accuracy on Cartesian grids. The PMPB formulation accounts for both vacuum and solvated-phase polarization, including direct and mutual induction and reaction-field effects, enabling accurate computation of electrostatic energies in biomolecular systems. Validation against Kirkwood-sphere analytic solutions and AMOEBA-parameterized proteins demonstrates robust convergence and improved solvation energies when using multipole sources, with provided software to facilitate replication and broader use in biomolecular electrostatics.

Abstract

To accurately model the electron density and polarization, a polarizable multipole (PM) model using the AMOEBA force field has been introduced \cite{Ren:2003, Shi:2013} recently. In the AMOEBA force field, the traditional point atomic representation is updated with permanent multipoles including additional dipoles and quadrupoles at atom centers in terms of derivatives of delta functions. Meanwhile, the polarization of the solute is considered by the introduction of induced dipoles. The AMOEBA forcefield thus shows significantly better agreement with experimental and high-level {\it ab initio} results. Moreover, the AMOEBA force field keeps the simple atomic structure, so that it can conviniently replace the traditional partial charge model. In this paper, we address the numerical challenges associated with the Polarizable Multipole Poisson--Boltzamnnn (PM-PB) model, which couples the AMOEBA force field with a linear Poisson-Boltzmann equation for implicit solvent and polarization modeling. To solve the PM-PB model, we designed a regularized Matched Interface and Boundary (MIB) method to analytically regularizes the singular source term in the PMPB model while maintains 2nd order accuracy by rigorously treating the interface conditions. The accuracy of the method is validated on Kirkwood sphere with available analytical solutions and on proteins whose charge distribution are assigned using AMOEBA force field.

Figures (1)

• Figure 1: The illustration of PMPB Model: (a) Domains of the PB model with $\Omega^-$: a charged solute molecule, $\Omega^+$: solvent with mobile ions, and $\Gamma$: molecular surface; (b) An partial charge (a circled plus or minus sign) in the PB model is replaced by a multipole consisting of a monopole (a dot), a dipole (a solid-lined ellipse), and a quadrupole (two cross-intercepted ellipses), as well as an induced dipole (an dash-lined ellipse). For solvated molecules, besides direct induction and mutual induction, induced dipoles are also subject to polarization produced by the reaction field. In the AMOEBA force field, all multipoles are defined at atom centers. Here, components of the multipole are placed off-centers for illustration purpose. (c) An 2-d illustration of the MIB schemes in which fictitious points (red/yellow) near the molecular surface are marked.