The Potential of Mean Force Calculated by Classical Force Field

In our calculation we used umbrella sampling with eleven energy windows to compute the free energy of the association of two methanes in water. The parameters for Lennard-Jones potentials are listed in Table 2.

Table 2: The parameters for Lennard-Jones potential calculations [4].

	$\epsilon$(kcal/mol)	$r_{m}/2$ (Å)
C4 (united atom methane)	-0.294	1.865
H (TIP3P hydrogen)	-0.046	0.2245
O (TIP3P oxygen)	-0.1521	1.7682

The methane-methane distance was restrained using a weighting function with $k_{w}=4.0$ kcal/mol. Two methane molecules and $213$ water molecules are enclosed in a cubic unit cell with box length $18.856$ Å. Time step is $1$ fs. The simulation is run with CHARMM [4] at the room temperature $300$ K. From an initial configuration, the system is equilibrated during the first $12$ ps run. We prepared 11 different initial configurations in this manner; the methane-methane separation spans from $3$ to $9$ Å in $0.5$ Å increment. All simulations are done with a $20$ ps equilibration followed by a $380$ ps production run. During production runs, the methane-methane pair separation was saved every $10$ steps for later analysis.

Figure 13: The methane-methane PMF shows that contact configuration is more favorable than solvent separated configuration.

Figure 14: The methane-methane PMF shows different configurations.

The calculated methane-methane PMF is shown in Fig. 13. It shows a contact pair minimum at $3.9$ Å and a solvent separated potential minimum at about $7.4$ Å. Contact configuration is more favorable than the solvent separation configuration. The configurations and the PMF are juxtaposed in Fig. 14.

From the argument of the hydrophobic effect, contact configuration has less contact surface area than solvent-separated configuration and is thus more stable, by about $0.4$ kcal/mol. There are many factors influencing the shape and the depth of the PMF, including hydrogen-bond network, temperature, methane-methane and water-methane interactions [22,32,18]. In particular, the strongly repulsive short-range methane-methane interaction dominates the potential curve shorter than $3.9$ Å. Since the van der Waals radius of methane is about 2 Å, there is not enough space for water molecules to squeeze in-between. For larger distances, water-methane and water-water interactions are the main factors to the PMF.

Figure 15: The PMF calculated with different water models. This picture is token from Ref. [32].

Different water models influence the water-methane and water-water interactions but not methane-methane interactions. In Lennard-Jones potentials, molecules act as hard spheres and give rise the clear peak in the PMF. Because of the hard repulsive force at short distances, there is no leeway between molecules and results in the large energy barrier at $5.8$ Å. As an analogy, imagine two volleyballs surrounded by many baseballs. It is far harder to move from the contact configuration to baseball-separated configuration than when baseballs are replaced by cotton candies.