Quantifying density fluctuations in volumes of all shapes and sizes using indirect umbrella sampling

Abstract

Water density fluctuations are an important statistical mechanical observable that is related to many-body correlations, as well as hydrophobic hydration and interactions. Local water density fluctuations at a solid-water surface have also been proposed as a measure of it's hydrophobicity. These fluctuations can be quantified by calculating the probability, P(v)(N), of observing N waters in a probe volume of interest v. When v is large, calculating P(v)(N) using molecular dynamics simulations is challenging, as the probability of observing very few waters is exponentially small, and the standard procedure for overcoming this problem (umbrella sampling in N) leads to undesirable impulsive forces. Patel et al. [J. Phys. Chem. B, 114, 1632 (2010)] have recently developed an indirect umbrella sampling (INDUS) method, that samples a coarse-grained particle number to obtain P(v)(N) in cuboidal volumes. Here, we present and demonstrate an extension of that approach to volumes of other basic shapes, like spheres and cylinders, as well as to collections of such volumes. We further describe the implementation of INDUS in the NPT ensemble and calculate P(v)(N) distributions over a broad range of pressures. Our method may be of particular interest in characterizing the hydrophobicity of interfaces of proteins, nanotubes and related systems.

Fig. 1

The coarse-graining function, φ(α), as defined in Eq. 3b, for α_c = 2σ.

Fig. 2

The functions h_α (α_i), h̃_α (α_i) and its derivative, ${\stackrel{\sim }{h}}_{\alpha }^{\prime }({\alpha }_i)$, for coordinates that have (a) two (α → x), (b) one (α → r) or (c) zero (α → θ) boundaries.

Fig. 3

(a) log P_v as a function of $(N-\langle N\rangle )/\sqrt{\langle \delta N^2\rangle }$ for volumes of four different shapes: a sphere of radius 0.6 nm, a cube of side 0.9 nm, a ylinder of radius 0.3 nm and length 3 nm, and a thin cuboid of dimensions 0.3 nm × 1.6 nm × 1.6 nm. (b) The ratio of μ^ex to surface area A, as a function of A/v for the four different shapes. The dashed line represents the surface tension, γ_∞, of a vapor-liquid interface of SPC/E water [18].

Fig. 4

P_v(N) obtained by umbrella sampling a probe volume that spells, ‘I N D U S’. The volume is composed of 156 cubic subvolumes of side 0.25 nm. The inset shows a superposition of five independent configurations, taken from an MD simulation with a strong biasing potential that empties the probe volume. The red spheres represent water oxygens. The letter ‘I’ in the inset is 0.5 nm wide and 2.0 nm tall.

Fig. 5

Comparing P_v(N) for a cubic v of side 0.9 nm, obtained using NPT ensemble simulations ( = 1bar), with that obtained from NVT ensemble simulations having a buffering vapor-liquid interface located far from v.

Fig. 6

(a) log P_v as a function of $(N-\langle N\rangle )/\sqrt{\langle \delta N^2\rangle }$ for a cube of side 1.2 nm, calculated in the NPT ensemble, over a range of pressures at T = 300 K. (b) Free energy, μ^ex, of the same cube as a function of pressure. A linear fit yields the excess volume of the cavity, v^ex ≈ 0.67v.