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

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

CADISHI: Fast parallel calculation of particle-pair distance histograms on CPUs and GPUs

We report on the design, implementation, optimization, and performance of the CADISHI software package, which calculates histograms of pair-distances of ensembles of particles on CPUs and GPUs. These histograms represent 2-point spatial correlation functions and are routinely calculated from simulations of soft and condensed matter, where they are referred to as radial distribution functions, and in the analysis of the spatial distributions of galaxies and galaxy clusters. Although conceptually simple, the calculation of radial distribution functions via distance binning requires the evaluation of $\mathcal{O}(N^2)$ particle-pair distances where $N$ is the number of particles under consideration. CADISHI provides fast parallel implementations of the distance histogram algorithm for the CPU and the GPU, written in templated C++ and CUDA. Orthorhombic and general triclinic periodic boxes are supported, in addition to the non-periodic case. The CPU kernels feature cache-blocking, vectorization and thread-parallelization to obtain high performance. The GPU kernels are tuned to exploit the memory and processor features of current GPUs, demonstrating histogramming rates of up to a factor 40 higher than on a high-end multi-core CPU. To enable high-throughput analyses of molecular dynamics trajectories, the compute kernels are driven by the Python-based CADISHI engine. It implements a producer-consumer data processing pattern and thereby enables the complete utilization of all the CPU and GPU resources available on a specific computer, independent of special libraries such as MPI, covering commodity systems up to high-end HPC nodes. Data input and output are performed efficiently via HDF5. (...) The CADISHI software is freely available under the MIT license.Comment: 19 page

Nanoporous Membranes of Densely Packed Carbon Nanotubes Formed by Lipid-Mediated Self-Assembly

Nanofiltration technology faces the competing challenges of achieving high fluid flux through uniformly narrow pores of a mechanically and chemically stable filter. Supported dense-packed 2D-crystals of single-walled carbon nanotube (CNT) porins with ∼1 nm wide pores could, in principle, meet these challenges. However, such CNT membranes cannot currently be synthesized at high pore density. Here, we use computer simulations to explore lipid-mediated self-assembly as a route toward densely packed CNT membranes, motivated by the analogy to membrane-protein 2D crystallization. In large-scale coarse-grained molecular dynamics (MD) simulations, we find that CNTs in lipid membranes readily self-assemble into large clusters. Lipids trapped between the CNTs lubricate CNT repacking upon collisions of diffusing clusters, thereby facilitating the formation of large ordered structures. Cluster diffusion follows the Saffman-Delbrück law and its generalization by Hughes, Pailthorpe, and White. On longer time scales, we expect the formation of close-packed CNT structures by depletion of the intervening shared annular lipid shell, depending on the relative strength of CNT-CNT and CNT-lipid interactions. Our simulations identify CNT length, diameter, and end functionalization as major factors for the self-assembly of CNT membranes

Water in nanopores

Wir untersuchen das Verhalten von Wasser in engen, unpolaren Poren, deren geringer Durchmesser die Wassermoleküle zwingt, sich hintereinander anzuordnen. Aufgrund dieser fast eindimensionalen räumlichen Einschränkung der Moleküle zeigt sich neuartiges physikalisches Verhalten im Vergleich zu Wasser in makroskopischen Volumina. Um diese Eigenschaften, die unter anderem von der Länge der Pore abhängen, zu untersuchen, entwickeln wir ein eindimensionales Dipol-Gittermodell, für welches wir drei mathematisch äquivalente Darstellungen ableiten. Diese bilden die Grundlage für unsere theoretischen Überlegungen und ermöglichen es uns, Poren von nanoskopischen bis makroskopischen Längen zu simulieren. Das Dipolmodell, welches wir mit Resultaten von molekularen Simulationen parametrisieren, beschreibt die freie Energetik und die Struktur von Wasser in Nanoporen quantitativ. Wir untersuchen das Füllverhalten von Kohlenstoffnanoröhren und die Ordnung der sich bildenden Wasserketten, in welchen die Moleküle durch Wasserstoffbrücken miteinander verbunden sind. Es stellt sich heraus, dass eine enge Kohlenstoffröhre in Kontakt mit einem Wasserbad bei Raumtemperatur und unter atmosphärischem Druck vollständig mit einer praktisch ununterbrochenen Wasserkette gefüllt ist, die bis zu einer makroskopischen Länge von ~ 0.1 mm dipolar geordnet ist. Um die Konsequenzen dieser Ordnungseigenschaften für das dielektrische Verhalten zu untersuchen, erweitern wir das Gittermodell um die Kinetik von Orientierungsdefekten und bestimmen die lineare Antwort von Wasserketten auf ein zeitlich veränderliches elektrisches Feld in Richtung der Röhrenachse. Die Ketten zeigen für alle Längen Debye-Relaxation, deren Ursache die Diffusion von nahezu unkorrelierten Defekten ist. Aus diesem Verhalten leiten wir einfache Ausdrücke für die statische Suszeptibilität und die Relaxationszeit für die Grenzfälle von kurzen, geordneten und langen, ungeordneten Ketten ab. Diese Ausdrücke ermöglichen es, die Ordnungseigenschaften mit Impedanzspektroskopie experimentell zu bestimmen und die grundlegenden Größen von Wasser in Nanoröhren zu messen.We investigate the behavior of single-file water in narrow nanopores. The quasi one-dimensional confinement changes the dynamical and structural properties of nanopore water compared to bulk water and new properties emerge. To explore these properties, which depend on the length of the pore, we develop a one-dimensional dipole lattice model and derive three mathematically equivalent representations. These pictures form the basis of our theoretical considerations and allow the simulation of pores from nanoscopic to macroscopic lengths. Parameterized with results from atomically detailed simulations, this model reproduces the free energetics and structure of nanopore water quantitatively. We investigate the filling transition of carbon nanotubes and explore the order properties of hydrogen bonded chains of water molecules within the pore. We find that narrow carbon nanotubes, which are in contact with a water bath at room temperature and atmospheric pressure, fill completely with an essentially continuous chain of water molecules, that is predominately dipole ordered up to a tube length of ~ 0.1 mm. We explore the consequences of these order properties for the dielectric behavior by determining the linear response of a single chain of water molecules to a time-dependent electric field in direction of the tube axis. To this end, we include the kinetics of orientational defects in the dipole lattice model. At all chain lengths, nanopore water shows Debye relaxation due to the diffusion of essentially uncorrelated defects. We derive simple expressions for the static dielectric susceptibility and the relaxation time in the limits of short, ordered and long, disordered chains and suggest how dielectric loss spectroscopy can be used to determine the order properties and to measure the fundamental quantities that determine the behavior of nanopore water

Phase behaviour of a symmetrical binary fluid mixture

We have investigated the phase behaviour of a symmetrical binary fluid mixture for the situation where the chemical potentials $\mu_1$ and $\mu_2$ of the two species differ. Attention is focused on the set of interparticle interaction strengths for which, when $\mu_1=\mu_2$, the phase diagram exhibits both a liquid-vapor critical point and a tricritical point. The corresponding phase behaviour for the case $\mu_1\ne\mu_2$ is investigated via integral-equation theory calculations within the mean spherical approximation (MSA), and grand canonical Monte Carlo (GCMC) simulations. We find that two possible subtypes of phase behaviour can occur, these being distinguished by the relationship between the critical lines in the full phase diagram in the space of temperature, density, and concentration. We present the detailed form of the phase diagram for both subtypes and compare with the results from GCMC simulations, finding good overall agreement. The scenario via which one subtype evolves into the other, is also studied, revealing interesting features.Comment: 22 pages, 13 figure

A one-dimensional dipole lattice model for water in narrow nanopores

We present a recently developed one-dimensional dipole lattice model that accurately captures the key properties of water in narrow nanopores. For this model, we derive three equivalent representations of the Hamiltonian that together yield a transparent physical picture of the energetics of the water chain and permit efficient computer simulations. In the charge representation, the Hamiltonian consists of nearest-neighbor interactions and Coulomb-like interactions of effective charges at the ends of dipole ordered segments. Approximations based on the charge picture shed light on the influence of the Coulomb-like interactions on the structure of nanopore water. We use Monte Carlo simulations to study the system behavior of the full Hamiltonian and its approximations as a function of chemical potential and system size and investigate the bimodal character of the density distribution occurring at small system sizes

Global Structure of the Intrinsically Disordered Protein Tau Emerges from Its Local Structure

The paradigmatic disordered protein tau plays an important role in neuronal function and neurodegenerative diseases. To disentangle the factors controlling the balance between functional and disease-associated conformational states, we build a structural ensemble of the tau K18 fragment containing the four pseudorepeat domains involved in both microtubule binding and amyloid fibril formation. We assemble 129-residue-long tau K18 chains with atomic detail from an extensive fragment library constructed with molecular dynamics simulations. We introduce a reweighted hierarchical chain growth (RHCG) algorithm that integrates experimental data reporting on the local structure into the assembly process in a systematic manner. By combining Bayesian ensemble refinement with importance sampling, we obtain well-defined ensembles and overcome the problem of exponentially varying weights in the integrative modeling of long-chain polymeric molecules. The resulting tau K18 ensembles capture nuclear magnetic resonance (NMR) chemical shift and J-coupling measurements. Without further fitting, we achieve very good agreement with measurements of NMR residual dipolar couplings. The good agreement with experimental measures of global structure such as single-molecule Förster resonance energy transfer (FRET) efficiencies is improved further by ensemble refinement. By comparing wild-type and mutant ensembles, we show that pathogenic single-point P301L, P301S, and P301T mutations shift the population from the turn-like conformations of the functional microtubule-bound state to the extended conformations of disease-associated tau fibrils. RHCG thus provides us with an atomically detailed view of the population equilibrium between functional and aggregation-prone states of tau K18, and demonstrates that global structural characteristics of this intrinsically disordered protein emerge from its local structure