Image Charge Methods for a Three-Dielectric-Layer Hybrid Solvation Model of Biomolecules

Abstract

This paper introduces a three dielectric layer hybrid solvation model for treating electrostatic interactions of biomolecules in solvents using the Poisson-Boltzmann equation. In this model, an interior spherical cavity will contain the solute and some explicit solvent molecules, and an intermediate buffer layer and an exterior layer contain the bulk solvent. A special dielectric permittivity profile is used to achieve a continuous dielectric transition from the interior cavity to the exterior layer. The selection of this special profile using a harmonic interpolation allows an analytical solution of the model by generalizing the classical Kirkwood series expansion. Discrete image charges are used to speed up calculations for the electrostatic potential within the interior and buffer layer regions. Semi-analytical and least squares methods are used to construct an accurate discrete image approximation for the reaction field due to solvent with or without salt effects. In particular, the image charges obtained by the least squares method provide accurate approximations to the reaction field independent of the ionic concentration of the solvent. Numerical results are presented to validate the accuracy and effectiveness of the image charge methods.

Figure 1

Schematic of a point charge q inside the interior sphere of the three-layer dielectric model.

Figure 2

The intermediate layer (a<r <b) is divided into several thin shells. The dielectric permittivity ε(r) in each shell can be considered as a constant.

Figure 3

Linear (dot line) and ${(\alpha +\frac{\beta }{r})}^2$ (solid line) dielectric profiles in the intermediate layer.

Figure 4

Semi-analytical vs. least squares image charge methods. Accuracy of the reaction field for the source charge located r_s = (r_s, 0, 0) with discrete image charges number M+1 = 2 and 10 including the Kelvin image. Left: h=0.1; right: h=0.01.

Figure 5

Semi-analytical vs. least squares image charge methods. Left: accuracy with the thickness of the buffer layer h = b − a with the source location r_s = 0.4, using 2 and 10 discrete image charges; right: accuracy with the number of discrete image charges.

Figure 6

Spatial distribution of the errors by using the least squares approximation with 2 discrete image charges. Left: 21 observation points on the x-axis; right: 21 observation points on the y-axis.

Figure 7

Relative L² error of various models with different dielectric profiles in the buffer zone compared to the no-buffer zone model. For r_s =0.4, the harmonic profile; and linear profile using n piecewise dielectric constants.

Figure 8

Spatial distribution of the errors by using the least squares approximation with 2 discrete image charges for the three-layer model. Left: 21 observation points on the x-axis; right: 21 observation points on the y-axis.

Figure 9

Relative error of the least squares approximation compared with the exact solution for the three-layer model. Left: accuracy vs. source charge location r_s for different dielectric constants ε_o of bulk solvents; right: accuracy vs. the number of discrete image charges.

Figure 10

Error of the least squares approximation. Left: accuracy vs. the thickness of the of the buffer layer using 2 discrete image charges, right: accuracy vs. the ionic strength u.