Reference state for the generalized Yvon-Born-Green theory: application for coarse-grained model of hydrophobic hydration

Abstract

Coarse-grained (CG) models provide a computationally efficient means for investigating phenomena that remain beyond the scope of atomically detailed models. Although CG models are often parametrized to reproduce the results of atomistic simulations, it is highly desirable to determine accurate CG models from experimental data. Recently, we have introduced a generalized Yvon-Born-Green (g-YBG) theory for directly (i.e., noniteratively) determining variationally optimized CG potentials from structural correlation functions. In principle, these correlation functions can be determined from experiment. In the present work, we introduce a reference state potential into the g-YBG framework. The reference state defines a fixed contribution to the CG potential. The remaining terms in the potential are then determined, such that the combined potential provides an optimal approximation to the many-body potential of mean force. By specifying a fixed contribution to the potential, the reference state significantly reduces the computational complexity and structural information necessary for determining the remaining potentials. We also validate the quantitative accuracy of the proposed method and numerically demonstrate that the reference state provides a convenient framework for transferring CG potentials from neat liquids to more complex systems. The resulting CG model provides a surprisingly accurate description of the two- and three-particle solvation structures of a hydrophobic solute in methanol. This work represents a significant step in developing the g-YBG theory as a useful computational framework for determining accurate CG models from limited experimental data.

Figure 1

Comparison of force functions for a united atom CG model of butane calculated from atomistic forces via the MS-CG method (solid black curves) and from structures via the g-YBG method (dashed red curves). All calculations were performed without a reference potential. (a)–(c) correspond to the intermolecular pair forces between CH3–CH3, CH3–CH2, and CH2–CH2 pairs, respectively. (d)–(f) correspond to the CH3–CH2 bond stretch, CH3–CH2–CH2 bond angle, and CH3–CH2–CH2–CH3 dihedral angle force functions, respectively. The solid green curves in (d)–(f) correspond to the smoothed force functions employed in defining the intramolecular reference potential. The force functions in (a)–(d) are presented in units of kJ∕mol nm. The force functions in (e) and (f) are presented in units of kJ∕mol rad.

Figure 2

Differences between the force functions calculated via the g-YBG method with a reference potential and the force functions calculated via the g-YBG method without a reference potential (i.e., the force functions presented in the corresponding panels of Fig. 1). (a)–(c) present the differences in the intermolecular pair forces when using the intramolecular force functions [presented as the solid green curves in Figs. 1d, 1e, 1f] as a reference force field. (d)–(f) present the differences in the intramolecular force functions when using the intermolecular pair forces [presented as the dashed red curves in Figs. 1a, 1b, 1c] as a reference force field. The errors in (a)–(d) are presented in units of kJ∕mol nm. The errors in (e)–(f) are presented in units of kJ∕mol rad.

Figure 3

Pair potentials employed in a CG model for hydrophobic solvation in methanol. (a)–(c) present the methanol intermolecular site-site potentials for CGC-CGC, CGO-CGO, and CGC-CGO site pairs, respectively. (d) and (e) present the site-site potentials for the hydrophobic solute site (SPH) with methanol CGC and CGO sites, respectively. (d) and (e) also present the WCA potentials employed to model the interaction of the hydrophobic sphere with the C, HC, and O atoms in all-atom simulations of the solute in methanol. All potentials are presented in units of kJ/mol.

Figure 4

Comparison of site-site RDFs calculated from atomistic (solid black) and CG (dashed green) simulations of a single hydrophobic sphere in methanol. (a)–(c) present the CGC-CGC, CGO-CGO, and CGC-CGO RDFs, respectively. (d) and (e) present the SPH-CGC and SPH-CGO RDFs, respectively.

Figure 5

Illustration of the geometry defining the angle θ used to characterize the orientation of methanol molecules about the hydrophobic sphere. This image was made with VMD (Ref. 93).

Figure 6

Analysis of methanol orientation at different distances from the hydrophobic solute in atomistic and CG simulations. Molecules were partitioned into different spherical shells based on the distance of the CGC site from the solute. The angle θ corresponds to the angle formed by the two vectors illustrated in Fig. 5. Each panel presents distributions of cos θ calculated from atomistic (solid) and CG (dashed) simulations in two successive shells with the distributions from the nearer (farther) shell represented by the black (green) curve.