Potential of mean force of association of large hydrophobic particles: toward the nanoscale limit

Abstract

The potentials of mean force (PMFs) were determined, in both water with the TIP3P water model and in vacuo, for systems involving formation of nonpolar dimers composed of bicyclooctane, adamantane (both an all-atom model and a sphere with the radius of 3.4 A representing adamantane), and fullerene, respectively. A series of umbrella-sampling molecular dynamics simulations with the AMBER force field were carried out for each pair under both environmental conditions. The PMFs were calculated by using the weighted histogram analysis method. The results were compared with our previously determined PMF for neopentane. The shape of the PMFs for dimers of all four nonpolar molecules is characteristic of hydrophobic interactions with contact and solvent-separated minima and desolvation maxima. The positions of all these minima and maxima change with the size of the nonpolar molecule; for larger molecules they shift toward larger distances. Comparison of the PMFs of the bicyclooctane, adamantane, and fullerene dimers in water and in vacuo shows that hydrophobic interactions in each dimer are different from that for the dimer of neopentane. Interactions in the bicyclooctane, adamantane, and fullerene dimers are stronger in vacuo than in water. These dimers cannot be treated as classical, spherical, hydrophobic objects. The solvent contribution to the PMF was also computed by subtracting the PMF determined in vacuo from that in explicit solvent. The solvent contribution to the PMFs of bicyclooctane, adamantane, and fullerene is positive, as opposed to that of neopentane. The water molecules in the first solvation sphere of both adamantane and neopentane dimers are more ordered as compared to bulk water, with their dipole moments pointing away from the surface of the dimers. The average number of hydrogen bonds per water molecule in the first hydration shell of adamantane is smaller compared to that in bulk water, but this shell is thicker for all-atom adamantane than for neopentane or a spherical model of adamantane. In the second hydration shell, the average number of hydrogen bonds is greater compared to that in bulk water only for neopentane and a spherical model of adamantane but not for the all-atom model. The strength of the hydrophobic interactions shows a linear dependence on the number of carbon atoms both in water and in vacuo. Smaller nonpolar particles interact more strongly in water than in vacuo. For larger molecules, such as bicyclooctane, adamantane and fullerene, the reversed tendency is observed.

Figure 1

The molecules studied in this work: (a) neopentane; (b) bicyclooctane; (c) adamantane, and (d) fullerene. The restraints during the MD simulations were imposed on the distances between the atoms labeled with stars.

Figure 2

PMF curves for the fullerene dimer for two different numbers of configurations, obtained from the umbrella-sampling/WHAM method using the TIP3P water model at temperature 298K. The simulations were carried out at a temperature of 298 K. The dotted and solid lines refer to 50% (25,000 configurations), and 100% (50,000 configurations), respectively, of the total number of generated configurations.

Figure 3

Schematic illustration of the definitions for the cylindrical distributions around (a) a pair of solutes at contact distance, (b) a single solute molecule. In (a), the cylindrical axis h in part links the solute molecules, and r is an axis perpendicular to it and passing through the middle of the dimer. In (b), the h axis is an arbitrary direction in space, and r is an axis perpendicular to it and passing through the center of the solute. The distribution functions of water molecules are averaged over the azimuthal angle θ. A water molecule is presented (at an arbitrary orientation) to define the angle ω between the r axis and the dipole moment vector of the water molecule. The H–O· · ·O angle α used to define a hydrogen bond is presented using two water molecules in the hydrogen-bonded dimer geometry.

Figure 3

Schematic illustration of the definitions for the cylindrical distributions around (a) a pair of solutes at contact distance, (b) a single solute molecule. In (a), the cylindrical axis h in part links the solute molecules, and r is an axis perpendicular to it and passing through the middle of the dimer. In (b), the h axis is an arbitrary direction in space, and r is an axis perpendicular to it and passing through the center of the solute. The distribution functions of water molecules are averaged over the azimuthal angle θ. A water molecule is presented (at an arbitrary orientation) to define the angle ω between the r axis and the dipole moment vector of the water molecule. The H–O· · ·O angle α used to define a hydrogen bond is presented using two water molecules in the hydrogen-bonded dimer geometry.

Figure 4

PMF curves for the dimer systems investigated in this study, determined at temperature 298K by using the TIP3P model of water (solid lines) and obtained in vacuo (dotted lines): neopentane (a); bicyclooctane (b); adamantane (c); fullerene (d).

Figure 4

PMF curves for the dimer systems investigated in this study, determined at temperature 298K by using the TIP3P model of water (solid lines) and obtained in vacuo (dotted lines): neopentane (a); bicyclooctane (b); adamantane (c); fullerene (d).

Figure 5

The solvent contribution to the PMF determined at temperature 298K as a difference between the PMFs in solvent and in vacuo: neopentane (solid line); bicyclooctane (dashed line); adamantane (dot-dashed line); fullerene (dotted line).

Figure 6

Linear regressions of the depth of the contact minima on the PMF curves in vacuo (solid line and full-circles) and in water (dotted line and open circles) vs. number of carbon atoms in a molecule. The values of the depth of the contact minima on PMF for methane, ethane, propane, isobutane and neopentane were taken from ref. , and for bicyclooctane and adamantane were obtained in this work. Fullerene is omitted from the plot because the depth of contact minimum is too large to fit on this scale.

Figure 7

Normalized distribution functions of the water molecule density in the vicinity of the adamantane dimer (Figures a-c) and neopentane (Figures d-f) at monomer-separation distances Δh (a) 6.8 Å, (b) 8.8 Å, (c) 10.2 Å, (d) 5.8 Å, (e) 7.85 Å and (f) 9.2 Å, which correspond to the contact minima (a and d), the desolvation barrier (b and e), and the solvent-separated minimum configurations (c and f), respectively. The color scale is shown above the panels; and the bulk water density is displayed in white. The solute is in grey, the space between the solute and the first hydration layer is in violet, and the first hydration layer is in blue plus light blue (a-c) and green, red and yellow (d-f).

Figure 8

Distribution of the average number of hydrogen bonds between water molecules in the vicinity of the adamantane dimer (Figures a-c) and neopentane (Figures d-f) at monomer-separation distances Δh (a) 6.8 Å, (b) 8.8 Å, (c) 10.2 Å, (d) 5.8 Å, (e) 7.85 Å and (f) 9.2 Å, which correspond to the contact minimum, the desolvation barrier, and the solvent-separated minimum configurations, respectively. The color scale is shown above the panels, and the average number of hydrogen bonds for bulk water density is displayed in white. The identification of the colored layers is described in Fig. 7.

Figure 9

Distribution of the average number of nearest water neighbors (a and d), calculated by using the same oxygen⋯oxygen distance cutoff (3.5Å) to define a hydrogen bond as for the average number of calculated hydrogen bonds presented in Figure 7; (b and e) the average water-water interaction energy (kcal/mol) between nearest-neighbor water molecules; (c and f) the average water-water pair energy (kcal/mol) calculated by dividing the average water-water interaction energy by the average number of nearest water neighbors, i.e., normalized per water-water contact. Distributions in Figures a-c are calculated around the adamantane dimer at 6.8 Å distance, and the distributions in Figures d-f are calculated around the neopentane dimer at 5.8 Å distance. The color scale is shown above the panels, and the average values for bulk water are displayed in white. The identification of the colored layers is described in Fig. 7

Figure 10

Distribution of the cosine of the angle ω (see Figure 3) between the vector normal to the axis of the dimer and the water dipole moment vector in the vicinity of the adamantane dimer (Figures a-c), and the neopentane dimer (Figures d-f) at monomer-separation distances Δh (a) 6.8 Å, (b) 8.8 Å, (c) 10.2 Å, (d) 5.8 Å, (e) 7.85 Å, (f) 9.2 Å, respectively. The color scale is shown above the panels, and the value of the average cosine of the angle ω of bulk water is displayed in white. The identification of the colored layers is described in Fig. 7

Figure 11

PMF curves for the dimers of adamantane (solid line) and two spheres (dotted line) with the radius 3.4 Å, determined by using the TIP3P model of water. The identification of the colored layers is described in Fig. 7

Figure 12

Normalized distribution functions of the water molecule density around the single molecule of adamantane (a) and a single sphere with the radius 3.4 Å (b). The color scale is shown above the panels, and the bulk water density is displayed in white. The solute is in grey. The identification of the colored layers is described in Fig. 7

Figure 13

Distribution of the average number of hydrogen bonds between water molecules around the single adamantane molecule (a) and a single sphere with a radius of 3.4 Å (b). The color scale is shown above the panels, and the average number of hydrogen bonds for bulk water density is displayed in white. The identification of the colored layers is described in Fig. 7