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The paper deals with a model-free approach to the analysis of quasielastic neutron scattering intensities from anomalously diffusing quantum particles. All quantities are inferred from the asymptotic form of their time-dependent mean square displacements which grow ∝t α, with 0 ≤ α < 2. Confined diffusion (α = 0) is here explicitly included. We discuss in particular the intermediate scattering function for long times and the Fourier spectrum of the velocity autocorrelation function for small frequencies. Quantum effects enter in both cases through the general symmetry properties of quantum time correlation functions. It is shown that the fractional diffusion constant can be expressed by a Green-Kubo type relation involving the real part of the velocity autocorrelation function. The theory is exact in the diffusive regime and at moderate momentum transfers.
Anomalous diffusion processes are usually detected by analyzing the time-dependent mean square displacement of the diffusing particles. The latter evolves asymptotically as W(t) approximately 2Dalphat(alpha), where Dalpha is the fractional diffusion constant and 0 < alpha < 2. In this article we show that both Dalpha and alpha can also be extracted from the low-frequency Fourier spectrum of the corresponding velocity autocorrelation function. This offers a simple method for the interpretation of quasielastic neutron scattering spectra from complex (bio)molecular systems, in which subdiffusive transport is frequently encountered. The approach is illustrated and validated by analyzing molecular dynamics simulations of molecular diffusion in a lipid POPC bilayer.
A coarse-grained geometrical model for protein secondary-structure description and analysis is presented which uses only the positions of the C(alpha) 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(alpha) 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(alpha) 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(alpha) 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.
This contribution gives a short introduction into the theory of anomalous diffusion and relaxation with illustrations from computer simulations of biomolecular systems. The theory is presented from the perspective of the non-equilibrium statistical physics, confronting stochastic models with exact results which have been recently obtained on the basis of asymptotic analysis. In this context, conditions for anomalous diffusion will be discussed and the Kubo relations for the fractional diffusion and relaxation constant will be derived.
In this work, we study dynamical properties of an extremophilic protein, Initiation Factor 6 (IF6), produced by the archeabacterium Methanocaldococcus jannascii, which thrives close to deep-sea hydrothermal vents where temperatures reach 80 °C and the pressure is up to 750 bar. Molecular dynamics simulations (MD) and quasi-elastic neutron scattering (QENS) measurements give new insights into the dynamical properties of this protein with respect to its eukaryotic and mesophilic homologue. Results obtained by MD are supported by QENS data and are interpreted within the framework of a fractional Brownian dynamics model for the characterization of protein relaxation dynamics. IF6 from M. jannaschii at high temperature and pressure shares similar flexibility with its eukaryotic homologue from S. cerevisieae under ambient conditions. This work shows for the first time, to our knowledge, that the very common pattern of corresponding states for thermophilic protein adaptation can be extended to thermo-barophilic proteins. A detailed analysis of dynamic properties and of local structural fluctuations reveals a complex pattern for ?corresponding ? structural flexibilities. In particular, in the case of IF6, the latter seems to be strongly related to the entropic contribution given by an additional, C-terminal, 20 amino-acid tail which is evolutionary conserved in all mesophilic IF6s.
The paper presents a rigorous derivation of the velocity autocorrelation function for an anomalously diffusing slow solute particle in a bath of fast solvent molecules. The result is obtained within the framework of the generalized Langevin equation and uses only scaling arguments and identities which are based on asymptotic analysis. It agrees with the velocity autocorrelation function of an anomalously diffusing Rayleigh particle whose dynamics is described by a fractional Ornstein-Uhlenbeck process in velocity space. A simple semi-analytical example illustrates under which conditions the latter model is appropriate.
In the present article we study the lateral molecular di-usion in homogenous lipid (POPC) bilayers by molecular dynamics simulations with the all-atom OPLS and the coarse-grained MARTINI force fields. On the statistically relevant time scales the center-of-mass mean-square displacement exhibits in both cases the subdiffusive asymptotic form W (t) 2Dαtα , with α ≈ 0.67 and α ≈ 0.57 , respectively. The diffusive dynamics obtained by the MARTINI force field is, however, faster by a factor of about 3. The subdiffusive characteristics of the di-usion process is confirmed by comparing the integral of the center-of-mass velocity autocorrelation function with its analytical long-time tail. The agreement is particularly good for the MARTINI force field, which permits to extend the simulation length and the system size considerably. Our results are in agreement with experimental observations of subdi-usion in lipid bilayers on longer time scales and do not support the finding of some authors that the latter should be considered as a transient phenomenon.
We study the dynamical transition of human acetylcholinesterase by analyzing elastic neutron scat- tering data with a simulation gauged analytical model that goes beyond the standard Gaussian ap- proximation for the elastic incoherent structure factor [G. R. Kneller and K. Hinsen, J. Chem. Phys. 131, 045104 (2009)]. The model exploits the whole available momentum transfer range in the ex- perimental data and yields not only a neutron-weighted average of the atomic mean square position fluctuations, but also an estimation for their distribution. Applied to the neutron scattering data from human acetylcholinesterase, it reveals a strong increase of the motional heterogeneity at the two transition temperatures T = 150 K and T = 220 K, respectively, which can be located with less am- biguity than with the Gaussian model. We find that the first transition is essentially characterized by a change in the form of the elastic scattering profile and the second by a homogeneous increase of all motional amplitudes. These results are in agreement with previous combined experimental and simulation studies of protein dynamics, which attribute the first transition to an onset of methyl rotations and the second to more unspecific diffusion processes involving large amplitude motions.
In the present work, we propose a simple model-free approach for the computation of molecular dif- fusion 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 cor- responding 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 cor- relation 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.
This paper addresses the question to which extent anisotropic atomic motions in proteins impact angular-averaged incoherent neutron scattering intensities, which are typically recorded for powder samples. For this purpose, the relevant correlation functions are represented as multipole series in which each term corresponds to a different degree of intrinsic motional anisotropy. The approach is illustrated by a simple analytical model and by a simulation-based example for lysozyme, considering in both cases the elastic incoherent structure factor. The second example shows that the motional anisotropy of the protein atoms is considerable and contributes significantly to the scattering intensity.
We present a new application of the ScrewFit algorithm (Acta Cryst. D 62, p.
302-11 (2006)) which adds the detection of protein secondary structure elements to
their detailed geometrical description in terms of a curve with intrinsic torsion. The
extension is based on con-dence and persistence criteria for the ScrewFit parameters
which are established by analyzing the structural
uctuations of standard motifs in the
SCOP fold classes. The agreement with the widely used DSSP method is comparable
with the general consensus among other methods in literature. The combination of
secondary structure detection and analysis is illustrated for the enzyme Adenylate
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.
We present a model for the local diffusion-relaxation dynamics of the Cα-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 Cα-atoms exceptionally well and reveals moreover a strong correlation between the residue’s solvent-accessible surface and the fitted model parameters.
Protein function often requires large-scale domain motion. An exciting new development in the experimental characterization of domain motions in proteins is the application of neutron spin-echo spectroscopy (NSE). NSE directly probes coherent (i.e., pair correlated) scattering on the ∼1–100 ns timescale. Here, we report on all-atom molecular-dynamics (MD) simulation of a protein, phosphoglycerate kinase, from which we calculate small-angle neutron scattering (SANS) and NSE scattering properties. The simulation-derived and experimental-solution SANS results are in excellent agreement. The contributions of translational and rotational whole-molecule diffusion to the simulation-derived NSE and potential problems in their estimation are examined. Principal component analysis identifies types of domain motion that dominate the internal motion’s contribution to the NSE signal, with the largest being classic hinge bending. The associated free-energy profiles are quasiharmonic and the frictional properties correspond to highly overdamped motion. The amplitudes of the motions derived by MD are smaller than those derived from the experimental analysis, and possible reasons for this difference are discussed. The MD results confirm that a significant component of the NSE arises from internal dynamics. They also demonstrate that the combination of NSE with MD is potentially useful for determining the forms, potentials of mean force, and time dependence of functional domain motions in proteins.
In a recent simulation study [J. Chem. Phys. 2010, 133, 145101], it has been shown that the time correlation functions probed by nuclear magnetic resonance (NMR) relaxation spectroscopy of proteins are well described by a fractional Brownian dynamics model, which accounts for the wide spectrum of relaxation rates characterizing then. internal dynamics. Here, we perform numerical experiments to explore the possibility of using this model directly in the analysis of experimental NMR relaxation data. Starting from a molecular dynamics simulation of the 266 residue protein 6PGL in explicit water, we construct virtual (15)N R(1), R(2), and NOE relaxation rates at two different magnetic fields, including artificial noise, and test how far the parameters obtained from a fit of the model to the virtual experimental data coincide with those obtained from an analysis of the MD time correlation functions that have been used to construct these data. We show that in most cases, close agreement is found. Acceptance or rejection of parameter values obtained from relaxation rates are discussed on a physical basis, therefore avoiding overfitting.
This communication presents a molecular dynamics simulation study of a bilayer consisting of 128 dioleoyl-sn-glycero-3-phosphocholine molecules, which focusses on the center-of-mass diffusion of the lipid molecules parallel to the membrane plane. The analysis of the simulation results is performed within the framework of the generalized Langevin equation and leads to a consistent picture of subdiffusion. The mean square displacement of the lipid molecules evolves as proportional to t(alpha), with alpha between 0.5 and 0.6, and the fractional diffusion coefficient is close to the experimental value for a similar system obtained by fluorescence correlation spectroscopy. We show that the long-time tails of the lateral velocity autocorrelation function and the associated memory function agree well with exact results which have been recently derived by asymptotic analysis [G. Kneller, J. Chem. Phys. 134, 224106 (2011)]. In this context, we define characteristic time scales for these two quantities. (C) 2011 American Institute of Physics. [doi : 10.1063/1.3651800]
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 ï¿1/2 1/ps) 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.
The paper describes preliminary results of a molecular dynamics simulation study on the influence of non-denaturing hydrostatic pressure on the structure and the relaxation dynamics of lysozyme. The overall compression and the structural changes are in agreement with results from recent nuclear magnetic resonance experiments. We find that moderate hydrostatic pressure reduces essentially the amplitudes of the atomic motions. but does not change the characteristics of the slow internal dynamics. The latter is well described by a fractional Ornstein-Uhlenbeck process, concerning both single particle and collective motions. (c) 2006 Elsevier B.V. All rights reserved.
Correlation functions describing relaxation processes in proteins and other complex molecular systems are known to exhibit a nonexponential decay. The simulation study presented here shows that fractional Brownian dynamics is a good model for the internal dynamics of a lysozyme molecule in solution. We show that both the dynamic structure factor and the associated memory function fit well the corresponding analytical functions calculated from the model. The numerical analysis is based on autoregressive modeling of time series. (C) 2004 American Institute of Physics.
We present a new implementation of the program nMoldyn, which has been developed for the computation and decomposition of neutron scattering intensities from Molecular Dynamics trajectories (Comp. Phys. Commun 1995, 91, 191-214). The new implementation extends the functionality of the original version, provides a much more convenient user interface (both graphical/interactive and batch), and can be used as a tool set for implementing new analysis modules. This was made possible by the use of a high-level language, Python, and of modern object-oriented programming techniques.
Using autoregressive modeling of discrete signals, we investigate the influence of mass and size on the memory function of a tracer particle immersed in a Lennard-Jones liquid. We find that the memory function of the tracer particle scales with the inverse reduced mass of the simulated system. Increasing the particle’s mass leads rapidly to a slow exponential decay of the velocity autocorrelation function, whereas the memory function changes just its amplitude. This effect is the more pronounced the smaller and the heavier the tracer particle is. (C) 2003 American Institute of Physics.
Recent analyses of molecular dynamics simulations of hydrated C-phycocyanin suggest that the internal single-particle dynamics of this protein can be decomposed into four almost decoupled motion types : (1) diffusion of residues ("beads") in an effective harmonic potential, (2) corresponding vibrations in a local potential well, (3) purely rotational rigid side-chain diffusion, and (4) residue deformations, Each residue bead is represented by the corresponding C-alpha carbon atom on the main chain.
We propose a new method to compute reliable estimates for memory functions of dynamical variables from molecular dynamics simulations. The key point is that the dynamical variable under consideration, which we take to be the velocity of a fluid particle, is modeled as an autoregressive stochastic process. The parameters of this stochastic process can be determined from molecular dynamics trajectories using efficient algorithms that are well established in signal processing. The procedure is also referred to as the maximum entropy method. From the autoregressive model of the velocity autocorrelation function we compute the one-sided z transform of the discretized memory function and the memory function itself. Using liquid argon as a simple model system, we demonstrate that the autocorrelation function and its power spectrum can be approximated to almost arbitrary precision. The same is therefore true for the memory function, which is calculated within the same stochastic model. (C) 2001 American Institute of Physics.
In studies of macromolecular dynamics it is often desirable to analyze complex motions in terms of a small number of coordinates. Only for simple types of motion, e.g., rigid-body motions, these coordinates can be easily constructed from the Cartesian atomic coordinates. This article presents an approach that is applicable to infinitesimal or approximately infinitesimal motions, e.g., Cartesian velocities, normal modes, or atomic fluctuations. The basic idea is to characterize the subspace of interesting motions by a set of (possibly linearly dependent) vectors describing elementary displacements, and then project the dynamics onto this subspace. Often the elementary displacements can be found by physical intuition. The restriction to small displacements facilitates the study of complicated coupled motions and permits the construction of collective-motion subspaces that do not correspond to any set of generalized coordinates. As an example for this technique, we analyze the low-frequency normal modes of proteins up to approximate to 20 THz (600 cm(-1)) in order to see what kinds of motions occupy which frequency range. This kind of analysis is useful for the interpretation of spectroscopic measurements on proteins, e.g., inelastic neutron scattering experiments.
Radiation damage in DNA is caused mainly by hydroxyl radicals which are generated by ionizing radiation in water and removing hydrogen atoms from the DNA chain. This damage affects certain nucleotide sequences more than others due to differences in the local structure of the DNA chains. This sequence dependence has been analyzed experimentally and calculated theoretically for a rigid DNA model. In this paper we take into account the flexibility of the DNA chain and show how it modifies the strand breakage probabilities. We use a simple harmonic model for DNA flexibility which permits the study of a long (68 base pair) fragment with modest computational effort. The essential influence of flexibility is an increased breakage probability towards the ends of the fragment, which can also be identified in the experimental data.
The slow dynamics of proteins around its native folded state is usually described by diffusion in a strongly anharmonic potential. In this paper, we try to understand the form and origin of the anharmonicities, with the principal aim of gaining a better understanding of the principal motion types, but also in order to develop more efficient numerical methods for simulating neutron scattering spectra of large proteins.
The empirical force fields used for protein simulations contain short-ranged terms (chemical bond structure, steric effects, van der Waals interactions) and long-ranged electrostatic contributions. It is well known that both components are important for determining the structure of a protein. We show that the dynamics around a stable equilibrium state can be described by a much simpler midrange force field made up of the chemical bond structure terms plus unspecific harmonic terms with a distance-dependent force constant. A normal mode analysis of such a model can reproduce the experimental density of states as well as a conventional molecular dynamics simulation using a standard force field with long-range electrostatic terms. This finding is consistent with a recent observation that effective Coulomb interactions are short ranged for systems with a sufficiently homogeneous charge distribution. (C) 1999 American Institute of Physics. [S0021-9606(99)52348-9].
Starting from the N-body friction matrix of an unconstrained system of N rigid particles immersed in a viscous liquid, we derive rigorous expressions for the corresponding friction and mobility matrices of a geometrically constrained dynamical system. Our method is based on the fact that geometrical constraints in a dynamical system can be cast in the form of linear constraints for the Cartesian translational and angular velocities of its constituents. Corresponding equations of motion for Molecular Dynamics simulations have been derived recently .
We describe a numerical method for calculating hydrodynamic interactions between spherical particles efficiently and accurately, both for particles immersed in an infinite liquid and for systems with periodic boundary conditions. Our method is based on a multipole expansion in Cartesian tensors. We then show how to solve the equations of motion for translational and rotational motion of suspended particles at large Peclet numbers. As an example we study the sedimentation of an array of spheres with and without periodic boundary conditions. We also study the effect of perturbations on the stability of the trajectories.
Based on the equations of motion for linked rigid bodies that we derived recently [G. Kneller and K. Hinsen, Phys Rev. E 50, 1559 (1994)], we develop a technique for the simulation of molecular systems with constraints. We apply it to analyze the importance of the-various degrees of freedom of a polypeptide chain for its dynamics. We find that keeping the peptide planes rigid does not change the dynamics much, but that the bending degrees of freedom of the alpha-carbon bond geometry are essential for large-amplitude backbone motions. This means that the phi and psi angles commonly used to characterize protein conformations and protein backbone dynamics do not constitute a sufficient set of variables to perform dynamical simulations.
We derive the equations of motion for linked rigid bodies from Lagrange mechanics and from Gauss’s principle of least constraint. The rotational motion of the subunits is described in terms of quaternion parameters and angular velocities. Different types of joints can be incorporated via axis constraints for the angular velocities. The resulting equations of motion are generalizations of the Euler equations of motion for a single rotor.
Professeur , Responsable de groupe thématique , Biophysique théorique, simulation moléculaire et calcul scientifique