The amounts of thermal vibrations and static disorder in protein X-ray crystallographic B-factors.

Crystallographic B-factors provide direct dynamical information on the internal mobility of proteins that is closely linked to function, and are also widely used as a benchmark in assessing elastic network models. A significant question in the field is: what is the exact amount of thermal vibrations in protein crystallographic B-factors? This work sets out to answer this question. First, we carry out a thorough, statistically sound analysis of crystallographic B-factors of over 10 000 structures. Second, by employing a highly accurate all-atom model based on the well-known CHARMM force field, we obtain computationally the magnitudes of thermal vibrations of nearly 1000 structures. Our key findings are: (i) the magnitude of thermal vibrations, surprisingly, is nearly protein-independent, as a corollary to the universality for the vibrational spectra of globular proteins established earlier; (ii) the magnitude of thermal vibrations is small, less than 0.1 Å2 at 100 K; (iii) the percentage of thermal vibrations in B-factors is the lowest at low resolution and low temperature (<10%) but increases to as high as 60% for structures determined at high resolution and at room temperature. The significance of this work is that it provides for the first time, using an extremely large dataset, a thorough analysis of B-factors and their thermal and static disorder components. The results clearly demonstrate that structures determined at high resolution and at room temperature have the richest dynamics information. Since such structures are relatively rare in the PDB database, the work naturally calls for more such structures to be determined experimentally.

Memory effects in a random walk description of protein structure ensembles.

In this paper, we show that ensembles of well-structured and unstructured proteins can be distinguished by borrowing concepts from non-equilibrium statistical mechanics. For this purpose, we represent proteins by two different polymer models and interpret the resulting polymer configurations as random walks of a diffusing particle in space. The first model is the trace of the Cα-atoms along the protein main chain, and the second is their projections onto the protein axis. The resulting trajectories are subsequently analyzed using the theory of the generalized Langevin equation. Velocities are replaced by displacements relating consecutive points on the discrete protein axes and equilibrium ensemble averages by averages over appropriate protein structure ensembles. The resulting displacement autocorrelation functions resemble those of the velocity autocorrelation functions of simple liquids and display a minimum, which can be related to the lengths of secondary structure elements. This minimum is clearly more pronounced for well-structured proteins than for unstructured ones, and the corresponding memory function displays a slower decay, indicating a stronger "folding memory."

Construction and validation of an atomic model for bacterial TSPO from electron microscopy density, evolutionary constraints, and biochemical and biophysical data.

The 18 kDa protein TSPO is a highly conserved transmembrane protein found in bacteria, yeast, animals and plants. TSPO is involved in a wide range of physiological functions, among which the transport of several molecules. The atomic structure of monomeric ligand-bound mouse TSPO in detergent has been published recently. A previously published low-resolution structure of Rhodobacter sphaeroides TSPO, obtained from tubular crystals with lipids and observed in cryo-electron microscopy, revealed an oligomeric structure without any ligand. We analyze this electron microscopy density in view of available biochemical and biophysical data, building a matching atomic model for the monomer and then the entire crystal. We compare its intra- and inter-molecular contacts with those predicted by amino acid covariation in TSPO proteins from evolutionary sequence analysis. The arrangement of the five transmembrane helices in a monomer of our model is different from that observed for the mouse TSPO. We analyze possible ligand binding sites for protoporphyrin, for the high-affinity ligand PK 11195, and for cholesterol in TSPO monomers and/or oligomers, and we discuss possible functional implications.

Communication: A multiscale Bayesian inference approach to analyzing subdiffusion in particle trajectories.

Anomalous diffusion is characterized by its asymptotic behavior for t → ∞. This makes it difficult to detect and describe in particle trajectories from experiments or computer simulations, which are necessarily of finite length. We propose a new approach using Bayesian inference applied directly to the observed trajectories sampled at different time scales. We illustrate the performance of this approach using random trajectories with known statistical properties and then use it for analyzing the motion of lipid molecules in the plane of a lipid bilayer.

Protein secondary-structure description with a coarse-grained model.

A coarse-grained geometrical model for protein secondary-structure description and analysis is presented which uses only the positions of the C(α) atoms. A space curve connecting these positions by piecewise polynomial interpolation is constructed and the folding of the protein backbone is described by a succession of screw motions linking the Frenet frames at consecutive C(α) positions. Using the ASTRAL subset of the SCOPe database of protein structures, thresholds are derived for the screw parameters of secondary-structure elements and demonstrate that the latter can be reliably assigned on the basis of a C(α) model. For this purpose, a comparative study with the widely used DSSP (Define Secondary Structure of Proteins) algorithm was performed and it was shown that the parameter distribution corresponding to the ensemble of all pure C(α) structures in the RCSB Protein Data Bank matches that of the ASTRAL database. It is expected that this approach will be useful in the development of structure-refinement techniques for low-resolution data.

MOSAIC: a data model and file formats for molecular simulations.

The MOlecular SimulAtion Interchange Conventions (MOSAIC) consist of a data model for molecular simulations and of concrete implementations of this data model in the form of file formats. MOSAIC is designed as a modular set of specifications, of which the initial version covers molecular structure and configurations. A reference implementation in the Python language facilitates the development of simulation software based on MOSAIC.

Model-free simulation approach to molecular diffusion tensors.

In the present work, we propose a simple model-free approach for the computation of molecular diffusion tensors from molecular dynamics trajectories. The method uses a rigid body trajectory of the molecule under consideration, which is constructed a posteriori by an accumulation of quaternion-based superposition fits of consecutive conformations. From the rigid body trajectory, we compute the translational and angular velocities of the molecule and by integration of the latter also the corresponding angular trajectory. All quantities can be referred to the laboratory frame and a molecule-fixed frame. The 6 × 6 diffusion tensor is computed from the asymptotic slope of the tensorial mean square displacement and, for comparison, also from the Kubo integral of the velocity correlation tensor. The method is illustrated for two simple model systems - a water molecule and a lysozyme molecule in bulk water. We give estimations of the statistical accuracy of the calculations.

A comparison of reduced coordinate sets for describing protein structure.

In all-atom molecular simulation studies of proteins, each atom in the protein is represented by a point mass and interactions are defined in terms of the atomic positions. In recent years, various simplified approaches have been proposed. These approaches aim to improve computational efficiency and to provide a better physical insight. The simplified models can differ widely in their description of the geometry and the interactions inside the protein. This study explores the most fundamental choice in the simplified protein models: the choice of a coordinate set defining the protein structure. A simplified model can use fewer point masses than the all-atom model and/or eliminate some of the internal coordinates of the molecule by setting them to an average or ideal value. We look at the implications of such choices for the overall protein structure. We find that care must be taken for angular coordinates, where even very small variations can lead to significant changes in the positions of far away atoms. In particular, we show that the φ/ψ torsion angles are not a sufficient coordinate set, whereas another coordinate set with two degrees of freedom per residue, virtual Cα backbone bond, and torsion angles performs satisfactorily.

nMoldyn 3: using task farming for a parallel spectroscopy-oriented analysis of molecular dynamics simulations.

We present a new version of the program package nMoldyn, which has been originally developed for a neutron-scattering oriented analysis of molecular dynamics simulations of macromolecular systems (Kneller et al., Comput. Phys. Commun. 1995, 91, 191) and was later rewritten to include in-depth time series analyses and a graphical user interface (Rog et al., J. Comput. Chem. 2003, 24, 657). The main improvement in this new version and the focus of this article are the parallelization of all the analysis algorithms for use on multicore desktop computers as well as distributed-memory computing clusters. The parallelization is based on a task farming approach which maintains a simple program structure permitting easy modification and extension of the code to integrate new analysis methods.

Communication: a minimal model for the diffusion-relaxation backbone dynamics of proteins.

We present a model for the local diffusion-relaxation dynamics of the C(α)-atoms in proteins describing both the diffusive short-time dynamics and the asymptotic long-time relaxation of the position autocorrelation functions. The relaxation rate spectra of the latter are represented by shifted gamma distributions, where the standard gamma distribution describes anomalous slow relaxation in macromolecular systems of infinite size and the shift accounts for a smallest local relaxation rate in macromolecules of finite size. The resulting autocorrelation functions are analytic for any time t ≥ 0. Using results from a molecular dynamics simulation of lysozyme, we demonstrate that the model fits the position autocorrelation functions of the C(α)-atoms exceptionally well and reveals moreover a strong correlation between the residue's solvent-accessible surface and the fitted model parameters.

A path-integral Langevin equation treatment of low-temperature doped helium clusters.

We present an implementation of path integral molecular dynamics for sampling low temperature properties of doped helium clusters using Langevin dynamics. The robustness of the path integral Langevin equation and white-noise Langevin equation [M. Ceriotti, M. Parrinello, T. E. Markland, and D. E. Manolopoulos, J. Chem. Phys. 133, 124104 (2010)] sampling methods are considered for those weakly bound systems with comparison to path integral Monte Carlo (PIMC) in terms of efficiency and accuracy. Using these techniques, convergence studies are performed to confirm the systematic error reduction introduced by increasing the number of discretization steps of the path integral. We comment on the structural and energetic evolution of He(N)-CO(2) clusters from N = 1 to 20. To quantify the importance of both rotations and exchange in our simulations, we present a chemical potential and calculated band origin shifts as a function of cluster size utilizing PIMC sampling that includes these effects. This work also serves to showcase the implementation of path integral simulation techniques within the molecular modelling toolkit [K. Hinsen, J. Comp. Chem. 21, 79 (2000)], an open-source molecular simulation package.

Least constraint approach to the extraction of internal motions from molecular dynamics trajectories of flexible macromolecules.

We propose a rigorous method for removing rigid-body motions from a given molecular dynamics trajectory of a flexible macromolecule. The method becomes exact in the limit of an infinitesimally small sampling step for the input trajectory. In a recent paper [G. Kneller, J. Chem. Phys. 128, 194101 (2008)], one of us showed that virtual internal atomic displacements for small time increments can be derived from Gauss' principle of least constraint, which leads to a rotational superposition problem for the atomic coordinates in two consecutive time frames of the input trajectory. Here, we demonstrate that the accumulation of these displacements in a molecular-fixed frame, which evolves in time according to the virtual rigid-body motions, leads to the desired trajectory for internal motions. The atomic coordinates in the input and output trajectory are related by a roto-translation, which guarantees that the internal energy of the molecule is left invariant. We present a convenient implementation of our method, in which the accumulation of the internal displacements is performed implicitly. Two numerical examples illustrate the difference to the classical approach for removing macromolecular rigid-body motions, which consists of aligning its configurations in the input trajectory with a fixed reference structure.

Structural flexibility in proteins: impact of the crystal environment.

MOTIVATION: In the study of the structural flexibility of proteins, crystallographic Debye-Waller factors are the most important experimental information used in the calibration and validation of computational models, such as the very successful elastic network models (ENMs). However, these models are applied to single protein molecules, whereas the experiments are performed on crystals. Moreover, the energy scale in standard ENMs is undefined and must be obtained by fitting to the same data that the ENM is trying to predict, reducing the predictive power of the model. RESULTS: We develop an elastic network model for the whole protein crystal in order to study the influence of crystal packing and lattice vibrations on the thermal fluctuations of the atom positions. We use experimental values for the compressibility of the crystal to establish the energy scale of our model. We predict the elastic constants of the crystal and compare with experimental data. Our main findings are (1) crystal packing modifies the atomic fluctuations considerably and (2) thermal fluctuations are not the dominant contribution to crystallographic Debye-Waller factors. AVAILABILITY: The programs developed for this work are available as supplementary material at Bioinformatics Online.

Quantitative model for the heterogeneity of atomic position fluctuations in proteins: a simulation study.

We propose a simple analytical model for the elastic incoherent structure factor of proteins measured by neutron scattering, which allows extracting the distribution of atomic position fluctuations from a fit of the model to the experimental data. The method is validated by applying it to elastic incoherent structure factors of lysozyme which have been obtained by molecular dynamics simulation and by normal mode analysis, respectively, and for which distributions of the atomic position fluctuations can be generated numerically for direct comparison with the predictions of the model. The comparison shows a remarkable agreement, in particular, concerning the lower limit for the position fluctuations, which is pronounced in the numerical data.

Solvent effects in the slow dynamics of proteins.

The influence of solvent on the slow internal dynamics of proteins is studied by comparing molecular dynamics simulations of solvated and unsolvated lysozyme. The dynamical trajectories are projected onto the protein's normal modes in order to obtain a separate analysis for each of the associated time scales. The results show that solvent effects are important for the slowest motions (below approximately 1 ps(-1)) but negligible for faster motions. The damping effects seen in the latter show that the principal source of friction in protein dynamics is not the solvent, but the protein itself.