Level-Set Variational Implicit-Solvent Modeling of Biomolecules with the Coulomb-Field Approximation

Abstract

Central in the variational implicit-solvent model (VISM) [Dzubiella, Swanson, and McCammon Phys. Rev. Lett.2006, 96, 087802 and J. Chem. Phys.2006, 124, 084905] of molecular solvation is a mean-field free-energy functional of all possible solute-solvent interfaces or dielectric boundaries. Such a functional can be minimized numerically by a level-set method to determine stable equilibrium conformations and solvation free energies. Applications to nonpolar systems have shown that the level-set VISM is efficient and leads to qualitatively and often quantitatively correct results. In particular, it is capable of capturing capillary evaporation in hydrophobic confinement and corresponding multiple equilibrium states as found in molecular dynamics (MD) simulations. In this work, we introduce into the VISM the Coulomb-field approximation of the electrostatic free energy. Such an approximation is a volume integral over an arbitrary shaped solvent region, requiring no solutions to any partial differential equations. With this approximation, we obtain the effective boundary force and use it as the "normal velocity" in the level-set relaxation. We test the new approach by calculating solvation free energies and potentials of mean force for small and large molecules, including the two-domain protein BphC. Our results reveal the importance of coupling polar and nonpolar interactions in the underlying molecular systems. In particular, dehydration near the domain interface of BphC subunits is found to be highly sensitive to local electrostatic potentials as seen in previous MD simulations. This is a first step toward capturing the complex protein dehydration process by an implicit-solvent approach.

Figure 1

The geometry of a solvation system with an implicit solvent. The solute region, solvent region, and solute–solvent interface are denoted by Ω_m, Ω_w, and Γ, respectively.

Figure 2

Different contributions to the PMF of the two-atom system vs the center-to-center distance d for different values of charges. (a) The geometrical part G_geom^pmf. (b) The vdW part G_vdW^pmf. The solute–solute vdW interaction is excluded in the curves in the main frame but included in those in the inset. (c) The electrostatic part G_elec^pmf. The Coulomb law of the charge–charge interaction is also plotted for comparison. (d) The total PMF G_tot^pmf. The solute–solute vdW interaction is excluded in the curves in the main frame but included in those in the inset.

Figure 3

2D cuts through the center of the 3D stable equilibrium solute–solvent interfaces around the two plates at d = 10 Å with tight or loose initial surfaces for different charge combinations.

Figure 4

Stable 3D equilibrium solute–solvent surfaces of the two-plate system obtained by the level-set VISM calculations with loose initials at d = 10 Å. From left to right: atomic charges (q₁,q₂) = (0e,0e), (+0.2e,+0.2e), and (+0.2e,–0.2e). The color code represents the mean curvature being convex (red), flat (green), and concave (blue).

Figure 5

Different components of the full PMF vs separation distance d between the two plates for different charge combinations (q₁,q₂) (see legend) obtained by the level-set VISM with loose initial surfaces.

Figure 6

Different components of the full PMF vs separation distance d between the two plates for different charge combinations (q₁,q₂) (see legend) obtained by the level-set VISM with tight initial surfaces.

Figure 7

Comparison of the two branches of PMF corresponding to the wet and dry states for the two plates carrying different charges (q₁,q₂) (see legend).

Figure 8

The stable equilibrium solute–solvent interfaces of BphC at three different domain separations, obtained by the level-set VISM with loose initial surfaces. The top row is with atomic partial charges, and the bottom row is without partial charges.

Figure 9

Different parts of the PMF of BphC with respect to the domain separations, with or without partial charges, and with loose and tight initial surfaces.