Bayesian ensemble refinement by replica simulations and reweighting

We describe different Bayesian ensemble refinement methods, examine their interrelation, and discuss their practical application. With ensemble refinement, the properties of dynamic and partially disordered (bio)molecular structures can be characterized by integrating a wide range of experimental data, including measurements of ensemble-averaged observables. We start from a Bayesian formulation in which the posterior is a functional that ranks different configuration space distributions. By maximizing this posterior, we derive an optimal Bayesian ensemble distribution. For discrete configurations, this optimal distribution is identical to that obtained by the maximum entropy "ensemble refinement of SAXS" (EROS) formulation. Bayesian replica ensemble refinement enhances the sampling of relevant configurations by imposing restraints on averages of observables in coupled replica molecular dynamics simulations. We show that the strength of the restraint should scale linearly with the number of replicas to ensure convergence to the optimal Bayesian result in the limit of infinitely many replicas. In the "Bayesian inference of ensembles" (BioEn) method, we combine the replica and EROS approaches to accelerate the convergence. An adaptive algorithm can be used to sample directly from the optimal ensemble, without replicas. We discuss the incorporation of single-molecule measurements and dynamic observables such as relaxation parameters. The theoretical analysis of different Bayesian ensemble refinement approaches provides a basis for practical applications and a starting point for further investigations.Comment: Paper submitted to The Journal of Chemical Physics (15 pages, 4 figures); updated references; expanded discussions of related formalisms, error treatment, and ensemble refinement with and without replicas; appendi

Coarse Molecular Dynamics of a Peptide Fragment: Free Energy, Kinetics, and Long-Time Dynamics Computations

We present a ``coarse molecular dynamics'' approach and apply it to studying the kinetics and thermodynamics of a peptide fragment dissolved in water. Short bursts of appropriately initialized simulations are used to infer the deterministic and stochastic components of the peptide motion parametrized by an appropriate set of coarse variables. Techniques from traditional numerical analysis (Newton-Raphson, coarse projective integration) are thus enabled; these techniques help analyze important features of the free-energy landscape (coarse transition states, eigenvalues and eigenvectors, transition rates, etc.). Reverse integration of (irreversible) expected coarse variables backward in time can assist escape from free energy minima and trace low-dimensional free energy surfaces. To illustrate the ``coarse molecular dynamics'' approach, we combine multiple short (0.5-ps) replica simulations to map the free energy surface of the ``alanine dipeptide'' in water, and to determine the ~ 1/(1000 ps) rate of interconversion between the two stable configurational basins corresponding to the alpha-helical and extended minima.Comment: The article has been submitted to "The Journal of Chemical Physics.

Hydrodynamics of Diffusion in Lipid Membrane Simulations

By performing molecular dynamics simulations with up to 132 million coarse-grained particles in half-micron sized boxes, we show that hydrodynamics quantitatively explains the finite-size effects on diffusion of lipids, proteins, and carbon nanotubes in membranes. The resulting Oseen correction allows us to extract infinite-system diffusion coefficients and membrane surface viscosities from membrane simulations despite the logarithmic divergence of apparent diffusivities with increasing box width. The hydrodynamic theory of diffusion applies also to membranes with asymmetric leaflets and embedded proteins, and to a complex plasma-membrane mimetic

Pair diffusion, hydrodynamic interactions, and available volume in dense fluids

We calculate the pair diffusion coefficient D(r) as a function of the distance r between two hard-sphere particles in a dense monodisperse suspension. The distance-dependent pair diffusion coefficient describes the hydrodynamic interactions between particles in a fluid that are central to theories of polymer and colloid dynamics. We determine D(r) from the propagators (Green's functions) of particle pairs obtained from discontinuous molecular dynamics simulations. At distances exceeding 3 molecular diameters, the calculated pair diffusion coefficients are in excellent agreement with predictions from exact macroscopic hydrodynamic theory for large Brownian particles suspended in a solvent bath, as well as the Oseen approximation. However, the asymptotic 1/r distance dependence of D(r) associated with hydrodynamic effects emerges only after the pair distance dynamics has been followed for relatively long times, indicating non-negligible memory effects in the pair diffusion at short times. Deviations of the calculated D(r) from the hydrodynamic models at short distances r reflect the underlying many-body fluid structure, and are found to be correlated to differences in the local available volume. The procedure used here to determine the pair diffusion coefficients can also be used for single-particle diffusion in confinement with spherical symmetry.Comment: 6 pages, 5 figure

Vibrational and optical properties of MoS2_2: from monolayer to bulk

Molybdenum disulfide, MoS2, has recently gained considerable attention as a layered material where neighboring layers are only weakly interacting and can easily slide against each other. Therefore, mechanical exfoliation allows the fabrication of single and multi-layers and opens the possibility to generate atomically thin crystals with outstanding properties. In contrast to graphene, it has an optical gap of 1.9 eV. This makes it a prominent candidate for transistor and opto-electronic applications. Single-layer MoS$_2$ exhibits remarkably different physical properties compared to bulk MoS$_2$ due to the absence of interlayer hybridization. For instance, while the band gap of bulk and multi-layer MoS$_2$ is indirect, it becomes direct with decreasing number of layers. In this review, we analyze from a theoretical point of view the electronic, optical, and vibrational properties of single-layer, few-layer and bulk MoS$_2$. In particular, we focus on the effects of spin-orbit interaction, number of layers, and applied tensile strain on the vibrational and optical properties. We examine the results obtained by different methodologies, mainly ab initio approaches. We also discuss which approximations are suitable for MoS$_2$ and layered materials. The effect of external strain on the band gap of single-layer MoS$_2$ and the crossover from indirect to direct band gap is investigated. We analyze the excitonic effects on the absorption spectra. The main features, such as the double peak at the absorption threshold and the high-energy exciton are presented. Furthermore, we report on the phonon dispersion relations of single-layer, few-layer and bulk MoS$_2$. Based on the latter, we explain the behavior of the Raman-active $A_{1g}$ and $E^1_{2g}$ modes as a function of the number of layers