Generalized Born model with a simple, robust molecular volume correction

Abstract

Generalized Born (GB) models provide a computationally efficient means of representing the electrostatic effects of solvent and are widely used, especially in molecular dynamics (MD). A class of particularly fast GB models is based on integration over an interior volume approximated as a pairwise union of atom spheres-effectively, the interior is defined by a van der Waals rather than Lee-Richards molecular surface. The approximation is computationally efficient, but if uncorrected, allows for high dielectric (water) regions smaller than a water molecule between atoms, leading to decreased accuracy. Here, an earlier pairwise GB model is extended by a simple analytic correction term that largely alleviates the problem by correctly describing the solvent-excluded volume of each pair of atoms. The correction term introduces a free energy barrier to the separation of non-bonded atoms. This free energy barrier is seen in explicit solvent and Lee-Richards molecular surface implicit solvent calculations, but has been absent from earlier pairwise GB models. When used in MD, the correction term yields protein hydrogen bond length distributions and polypeptide conformational ensembles that are in better agreement with explicit solvent results than earlier pairwise models. The robustness and simplicity of the correction preserves the efficiency of the pairwise GB models while making them a better approximation to reality.

Figure 1

The neck region (shaded) is defined by the radius of atom 1, R₁, the radius of atom 2, R₂, the distance that separates them, d, and the radius of the solvent molecule, R_w. The coordinate system used for performing integration is also illustrated (see Appendix I).

Figure 2

Values of numerical integration over the neck region (black) and analytical approximation (red) as a function of distance between atoms in angstroms. Left to right, top to bottom, radii (in angstroms) for atoms 1 and 2, respectively are 1.2 and 1.2; 1.2 and 1.7; 1.7 and 1.2; 1.7 and 1.7.

Figure 3

Scatter plot comparison of inverse effective radii calculated by the current GB neck model (red +) and earlier OBC GB model (black X) to inverse “perfect” PB radii for thioredoxin (PDB code 2TRX). Diagonal line indicates perfect agreement.

Figure 4

Relative deviation from PB solvation energy for GBn and OBC GB for a series of snapshots from a denaturation trajectory of protein A. GBn has a tighter clustering of points, indicating less random error than OBC GB (stdev 6.4 vs 7.2 kcal/mol), while maintaining a similar native state bias (trend of points across the plot). Average errors of −9.2 (OBC GB) and 68.9 (GBn) kcal/mol removed to facilitate comparison.

Figure 5

Potentials of mean force for hydrogen bonding systems not included in the objective function, calculated with three implicit solvent methods. Two protonated aspartic acids and two alanines (β-sheet model) are used as examples. Potential includes electrostatic and van der Waals energies.

Figure 6

Potentials of mean force for hydrogen bonding and salt bridge systems included in the objective function, calculated with three implicit solvent methods. The hydrogen bonding systems are asparagine and asparagine; aspartate and serine; arginine and aspartate. Potential includes electrostatic and van der Waals energies.

Figure 7

RMSD of alpha carbons from crystal structure over the course of 10 ns of molecular dynamics of ubiquitin (left) and thioredoxin (right).

Figure 8

Ubiquitin backbone hydrogen bond length data collected over 10 ns of MD for TIP3P explicit solvent, OBC GB and GBn. Plots represent difference between implicit and explicit solvent bond length distribution mean (left) and standard deviation (right) as a function of mean explicit solvent bond length. The zero line represents an exact match between the explicit and implicit solvent results. Hydrogen bond lengths under the GBn model are generally in better agreement with explicit solvent results.

Figure 9

Free energy surfaces at 300K for the backbone conformation of Ala5 in the Ala10 peptide calculated from 100ns of REMD. Energies are in kcal/mol, with the lowest free energy assigned a value of 0. TIP3P and GBn result in similar free energies for the α, β and polyproline II basins, while OBC GB shows a strong preference for α-helix.

Figure 10

End to end distance distributions of deca-alanine at 300K for 3 solvent models. Profiles from GBn and TIP3P explicit water are in good agreement, with a relatively broad distribution slightly peaked near 15–20 Å. However, OBC GB significantly differs from the other models, with a strong peak at 10 Å.

Figure 11

Three cases of neck regions (shaded) formed by atoms (solid circles) at varying separations. Dotted lines represent the surface of the solvent sphere. The leftmost vertex of the dashed triangle in (i) describes the angle A′ referenced in equation 9. Although this figure shows two atoms with the same radius, neck regions may also be formed between atoms with unequal radii.