Functional domain motions in proteins on the ~1-100 ns timescale: comparison of neutron spin-echo spectroscopy of phosphoglycerate kinase with molecular-dynamics simulation.

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.

Self-similar dynamics of proteins under hydrostatic pressure-Computer simulations and experiments.

Different experimental techniques, such as kinetic studies of ligand binding and fluorescence correlation spectroscopy, have revealed that the diffusive, internal dynamics of proteins exhibits autosimilarity on the time scale from microseconds to hours. Computer simulations have demonstrated that this type of dynamics is already established on the much shorter nanosecond time scale, which is also covered by quasielastic neutron scattering experiments. The autosimilarity of protein dynamics is reflected in long-time memory effects in the underlying diffusion processes, which lead to a non-exponential decay of the observed time correlation functions. Fractional Brownian dynamics is an empirical model which is able to capture the essential aspects of internal protein dynamics. Here we give a brief introduction into the theory and show how the model can be used to interpret neutron scattering experiments and molecular dynamics simulation of proteins in solution under hydrostatic pressure.

Dynamical heterogeneity of specific amino acids in bacteriorhodopsin.

Components of biological macromolecules, complexes and membranes are animated by motions occurring over a wide range of time and length scales, the synergy of which is at the basis of biological activity. Understanding biological function thus requires a detailed analysis of the underlying dynamical heterogeneity. Neutron scattering, using specific isotope labeling, and molecular dynamics simulations were combined in order to study the dynamics of specific amino acid types in bacteriorhodopsin within the purple membrane (PM) of Halobacterium salinarum. Motions of leucine, isoleucine and tyrosine residues on the pico- to nanosecond time scale were examined separately as a function of temperature from 20 to 300 K. The dynamics of the three residue types displayed different temperature dependence: isoleucine residues have larger displacements compared to the global PM above 120 K; leucine residues have displacements similar to that of PM in the entire temperature range studied; and tyrosine residues have displacements smaller than that of the average membrane in an intermediate temperature range. Experimental features were mostly well reproduced by molecular dynamics simulations performed at five temperatures, which allowed the dynamical characterisation of the amino acids under study as a function of local environment. The resulting dynamical map of bacteriorhodopsin revealed that movements of a specific residue are determined by both its environment and its residue type.

Hamiltonian formalism for semiflexible molecules in Cartesian coordinates.

The article gives a concise description of Hamiltonian dynamics and thermal averages of semiflexible molecules in Cartesian coordinates. Using the concept of constrained inverse matrices introduced by Bott and Duffin [Trans. Am. Math. Soc. 74, 99 (1953)] explicit expressions are derived for the constrained Hamiltonian, the corresponding equations of motion, and the momentum partition function. In this context Fixman-type corrections of constrained configurational averages are derived for different forms of the constraints. It is shown that the use of mass-weighted coordinates leads to a nonbiased sampling of constrained configurational averages in Cartesian coordinates. The formalism allows moreover to define and to calculate effective masses arising in thermal velocity averages of atoms in semiflexible molecules. These effective masses are identical to the corresponding Sachs-Teller recoil masses, which are here generalized to the case of only partially rigid molecules.

Comment on "Using quaternions to calculate RMSD" [J. Comp. Chem. 25, 1849 (2004)].

Coutsias et al. have recently published a method to find the optimal rotational superposition of two molecular structures, which is based on a representation of rotations by quaternions (J. Comp. Chem. 25(15), 1849 (2004)). The method, which has been suggested by other authors before, is compared to the one by Kabsch, where the elements of the rotation matrix are directly used as variables of the optimization problem. The statement that the two methods are equivalent is misleading in the sense that the Kabsch method may yield an improper optimal rotation, which must be explicitly checked for, whereas the quaternion method does not mix proper and improper rotations. Nevertheless, both types of solutions can be considered by solving the same eigenvector problem. The relation between the two types of solutions is briefly discussed and bounds for the eigenvalues are given.

Fractional Brownian dynamics in proteins.

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.

Scaling of the memory function and Brownian motion.

It has been recently shown that the velocity autocorrelation function of a tracer particle immersed in a simple liquid scales approximately with the inverse of its mass. With increasing mass the amplitude is systematically reduced and the velocity autocorrelation function tends to a slowly decaying exponential, which is characteristic for Brownian motion. We give here an analytical proof for this behavior and comment on the usual explanation for Brownian dynamics which is based on the assumption that the memory function is proportional to a Dirac distribution. We also derive conditions for Brownian dynamics of a tracer particle which are entirely based on properties of its memory function.

nMoldyn: a program package for a neutron scattering oriented analysis of molecular dynamics simulations.

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. The quantities that can be calculated by nMoldyn are the mean-square displacement, the velocity autocorrelation function as well as its Fourier transform (the density of states) and its memory function, the angular velocity autocorrelation function and its Fourier transform, the reorientational correlation function, and several functions specific to neutron scattering: the coherent and incoherent intermediate scattering functions with their Fourier transforms, the memory function of the coherent scattering function, and the elastic incoherent structure factor. The possibility to compute memory function is a new and powerful feature that allows to relate simulation results to theoretical studies.

Liquid-like side-chain dynamics in myoglobin.

At temperatures above approximately 200 K the motions of atoms in globular proteins contain a non-vibrational component that gives rise to characteristic elastic and quasi-elastic neutron scattering profiles. There is evidence that the non-vibrational dynamics is required for protein function. Here we show by analysing a molecular dynamics simulation of myoglobin that the neutron scattering results from liquid-like rigid-body motion of the protein side-chains.

Picosecond timescale rigid-helix and side-chain motions in deoxymyoglobin.

The contribution of rigid-body motions to the atomic trajectories in a 100 ps molecular dynamics simulation of deoxymyoglobin is examined. Two types of rigid-body motions are considered: one in which the helices are rigid units and one in which the side-chains are rigid units. Using a quaternion-based algorithm, fits of the rigid reference structures are made to each time frame of the simulation to derive trajectories of the rigid-body motions. The fitted trajectories are analysed in terms of atomic position fluctuations, mean-square displacements as a function of time, velocity autocorrelation functions and densities of states. The results are compared with the corresponding quantities calculated from the full trajectory. The relative contribution of the rigid helix motions to the helix atom dynamics depends on which quantity is examined and on which subset of atoms is chosen; rigid-helix motions contribute 86% of the rms helix backbone atomic position fluctuations, but 30% of the helix atom (backbone and side-chain) mean square displacements and only 1.1% of total kinetic energy. Only very low-frequency motions contribute to the rigid-helix dynamics; the rigid-body analysis allows characteristic rigid-helix vibrations to be identified and described. Treating the side-chains as rigid bodies is found to be an excellent approximation to both their diffusive and vibrational mean-square displacements: 96% of side-chain atom mean-square displacements originate from rigid side-chain motions. However, the errors in the side-chain atomic positional fits are not always small. An analysis is made of factors contributing to the positional error for different types of side-chain.