Coupling the Level-Set Method with Molecular Mechanics for Variational Implicit Solvation of Nonpolar Molecules

Abstract

We construct a variational explicit-solute implicit-solvent model for the solvation of molecules. Central in this model is an effective solvation free-energy functional that depends solely on the position of solute-solvent interface and solute atoms. The total free energy couples altogether the volume and interface energies of solutes, the solute-solvent van der Waals interactions, and the solute-solute mechanical interactions. A curvature dependent surface tension is incorporated through the so-called Tolman length which serves as the only fitting parameter in the model. Our approach extends the original variational implicit-solvent model of Dzubiella, Swanson, and McCammon [Phys. Rev. Lett. 2006, 96, 087802 and J. Chem. Phys. 2006, 124, 084905] to include the solute molecular mechanics. We also develop a novel computational method that combines the level-set technique with optimization algorithms to determine numerically the equilibrium conformation of nonpolar molecules. Numerical results demonstrate that our new model and methods can capture essential properties of nonpolar molecules and their interactions with the solvent. In particular, with a suitable choice of the Tolman length for the curvature correction to the surface tension, we obtain the solvation free energy for a benzene molecule in a good agreement with experimental results.

Fig. 1

The geometry of a solvation system with an implicit solvent. The free energy depends on the position of solute-solvent interface Γ and solute atomic positions x₁, …,x_N.

Fig. 2

The level-set optimization of a two-atom system. The initial solute-solvent interface consists of two separated spheres. Order of snapshots: from left to right and from top to bottom.

Fig. 3

The free energy (kcal/mol) vs. the computational step in a level-set optimization for the two-atom system with the initial solute-solvent interface consisting of two separated spheres, cf. Figure 2.

Fig. 4

The level-set optimization of a two-atom system. The two atoms are initially close and then move apart from each other. Order of snapshots: from left to right and from top to bottom.

Fig. 5

The free energy (kcal/mol) vs. the computational step in a level-set optimization for the two-atom system with the initial solute-solvent interface consisting of a single surface containing the two atoms, cf. Figure 4.

Fig. 6

Snapshots of a relaxing system of two non-interacting particles. Order: left to right and top to bottom.

Fig. 7

Two non-interacting particles relax with increasing surface area.

Fig. 8

Left: Initial positions of the solute-solvent interface and solute atoms. Right: The relaxed positions of the solute-solvent interface and solute atoms.

Fig. 9

The level-set relaxation of the ethane molecule. Order of snapshots: from first row to second and from left to right in each row.

Fig. 10

A plot of the total free energy (kcal/mol) vs. computational step in the level-set optimization for the ethane molecule.

Fig. 11

The level-set relaxation of the benzene molecule. Order of snapshots: from first row to second and from left to right in each row.

Fig. 12

A plot of the total free energy (kcal/mol) vs. computational step in the level-set optimization for the benzene molecule.

Fig. 13

The sum of the surface energy and benzene-water interaction energy vs. the Tolman length τ.