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Register a new userQuantifying density fluctuations in volumes of all shapes and sizes using indirect umbrella sampling
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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 its 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 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.
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Hydrophobic effects drive diverse aqueous assemblies, such as micelle formation or protein folding, wherein the solvent plays an important role. Consequently, characterizing the free energetics of solvent density fluctuations can lead to important insights into these processes. Although techniques such as the indirect umbrella sampling (INDUS) method (Patel et al. J. Stat. Phys. 2011, 145, 265-275) can be used to characterize solvent fluctuations in static observation volumes of various sizes and shapes, characterizing how the solvent mediates inherently dynamic processes, such as self-assembly or conformational change, remains a challenge. In this work, we generalize the INDUS method to facilitate the enhanced sampling of solvent fluctuations in dynamical observation volumes, whose positions and shapes can evolve. We illustrate the usefulness of this generalization by characterizing water density fluctuations in dynamic volumes pertaining to the hydration of flexible solutes, the assembly of small hydrophobes, and conformational transitions in a model peptide. We also use the method to probe the dynamics of hard spheres.
The Self-Healing Umbrella Sampling (SHUS) algorithm is an adaptive biasing algorithm which has been proposed to efficiently sample a multimodal probability measure. We show that this method can be seen as a variant of the well-known Wang-Landau algorithm. Adapting results on the convergence of the Wang-Landau algorithm, we prove the convergence of the SHUS algorithm. We also compare the two methods in terms of efficiency. We finally propose a modification of the SHUS algorithm in order to increase its efficiency, and exhibit some similarities of SHUS with the well-tempered metadynamics method.
We introduce an accurate and efficient method for characterizing surface wetting and interfacial properties, such as the contact angle made by a liquid droplet on a solid surface, and the vapor-liquid surface tension of a fluid. The method makes use of molecular simulations in conjunction with the indirect umbrella sampling technique to systematically wet the surface and estimate the corresponding free energy. To illustrate the method, we study the wetting of a family of Lennard-Jones surfaces by water. We estimate contact angles for surfaces with a wide range of attractions for water by using our method and also by using droplet shapes. Notably, as surface-water attractions are increased, our method is able to capture the transition from partial to complete wetting. Finally, the method is straightforward to implement and computationally efficient, providing accurate contact angle estimates in roughly 5 nanoseconds of simulation time.
Biochemical reactions are fundamentally noisy at a molecular scale. This limits the precision of reaction networks, but also allows fluctuation measurements which may reveal the structure and dynamics of the underlying biochemical network. Here, we study non-equilibrium reaction cycles, such as the mechanochemical cycle of molecular motors, the phosphorylation cycle of circadian clock proteins, or the transition state cycle of enzymes. Fluctuations in such cycles may be measured using either of two classical definitions of the randomness parameter, which we show to be equivalent in general microscopically reversible cycles. We define a stochastic period for reversible cycles and present analytical solutions for its moments. Furthermore, we associate the two forms of the randomness parameter with the thermodynamic uncertainty relation, which sets limits on the timing precision of the cycle in terms of thermodynamic quantities. Our results should prove useful also for the study of temporal fluctuations in more general networks.
Anomalous transport in a circular comb is considered. The circular motion takes place for a fixed radius, while radii are continuously distributed along the circle. Two scenarios of the anomalous transport, related to the reflecting and periodic angular boundary conditions, are studied. The first scenario with the reflection boundary conditions for the circular diffusion corresponds to the conformal mapping of a 2D comb Fokker-Planck equation on the circular comb. This topologically constraint motion is named umbrella comb model. In this case, the reflecting boundary conditions are imposed on the circular (rotator) motion, while the radial motion corresponds to geometric Brownian motion with vanishing to zero boundary conditions on infinity. The radial diffusion is described by the log-normal distribution, which corresponds to exponentially fast motion with the mean squared displacement (MSD) of the order of $e^t$. The second scenario corresponds to the circular diffusion with periodic boundary conditions and the outward radial diffusion with vanishing to zero boundary conditions at infinity. In this case the radial motion corresponds to normal diffusion. The circular motion in both scenarios is a superposition of cosine functions that results in the stationary Bernoulli polynomials for the probability distributions.
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