Determination of Dynamical Heterogeneity from Dynamic Neutron Scattering of Proteins.

Motional displacements of hydrogen (H) in proteins can be measured using incoherent neutron-scattering methods. These displacements can also be calculated numerically using data from molecular dynamics simulations. An enormous amount of data on the average mean-square motional displacement (MSD) of H as a function of protein temperature, hydration, and other conditions has been collected. H resides in a wide spectrum of sites in a protein. Some H are tightly bound to molecular chains, and the H motion is dictated by that of the chain. Other H are quite independent. As a result, there is a distribution of motions and MSDs of H within a protein that is denoted dynamical heterogeneity. The goal of this paper is to incorporate a distribution of MSDs into models of the H incoherent intermediate scattering function, I(Q,t), that is calculated and observed. The aim is to contribute information on the distribution as well as on the average MSD from comparison of the models with simulations and experiment. For example, we find that simulations of I(Q,t) in lysozyme are well reproduced if the distribution of MSDs is bimodal with two broad peaks rather than a single broad peak.

Excitations in Quantum Liquids.

Progress made in measuring and interpreting the elementary excitations of superfluid and normal liquid \4he in the past 25 years is reviewed. The goal is to bring up to date the data, calculations and our understanding of the excitations since the books and reviews of the early 1990s. Only bulk liquid \4he is considered. Reference to liquid \3he, mixtures, reduced dimensions (films and confined helium) is made where useful to enhance interpretation. The focus is on the excitations as measured by inelastic neutron scattering methods. The review covers the dynamic response of liquid \4he from the collective excitations at low energy and long wavelength (i.e. phonon-roton modes) to the single particle excitations at high energy from which the atomic momentum distribution and Bose-Einstein condensate fraction are determined. A goal is to show the interplay of these excitations with other spectacular properties such as superfluidity and the test of fundamental calculations of quantum liquids that is possible. The role of Bose-Einstein condensation in determining the nature of the \pr~ mode and particularly it's temperature dependence is emphasized. The similarity of normal liquid \4he with other quantum and classical liquids is discussed. .

Anomalous and anisotropic nanoscale diffusion of hydration water molecules in fluid lipid membranes.

We have studied nanoscale diffusion of membrane hydration water in fluid-phase lipid bilayers made of 1,2-dimyristoyl-3-phosphocholine (DMPC) using incoherent quasi-elastic neutron scattering. Dynamics were fit directly in the energy domain using the Fourier transform of a stretched exponential. By using large, 2-dimensional detectors, lateral motions of water molecules and motions perpendicular to the membranes could be studied simultaneously, resulting in 2-dimensional maps of relaxation time, τ, and stretching exponent, ß. We present experimental evidence for anomalous (sub-diffusive) and anisotropic diffusion of membrane hydration water molecules over nanometer distances. By combining molecular dynamics and Brownian dynamics simulations, the potential microscopic origins for the anomaly and anisotropy of hydration water were investigated. Bulk water was found to show intrinsic sub-diffusive motion at time scales of several picoseconds, likely related to caging effects. In membrane hydration water, however, the anisotropy of confinement and local dynamical environments leads to an anisotropy of relaxation times and stretched exponents, indicative of anomalous dynamics.

Motional displacements in proteins: The origin of wave-vector-dependent values.

The average mean-square displacement, 〈r(2)〉, of H atoms in a protein is frequently determined using incoherent neutron-scattering experiments. 〈r(2)〉 is obtained from the observed elastic incoherent dynamic structure factor, S(i)(Q,ω=0), assuming the form S(i)(Q,ω=0) =exp(-Q(2)〈r(2)〉/3). This is often referred to as the Gaussian approximation (GA) to S(i)(Q,ω=0). 〈r(2)〉 obtained in this way depends on the value of the wave vector, Q considered. Equivalently, the observed S(i)(Q,ω=0) deviates from the GA. We investigate the origin of the Q dependence of 〈r(2)〉 by evaluating the scattering functions in different approximations using molecular dynamics (MD) simulation of the protein lysozyme. We find that keeping only the Gaussian term in a cumulant expansion of S(Q,ω) is an accurate approximation and is not the origin of the Q dependence of 〈r(2)〉. This is demonstrated by showing that the term beyond the Gaussian is negligible and that the GA is valid for an individual atom in the protein. Rather, the Q dependence (deviation from the GA) arises from the dynamical heterogeneity of the H in the protein. Specifically it arises from representing, in the analysis of data, this diverse dynamics by a single average scattering center that has a single, average 〈r(2)〉. The observed Q dependence of 〈r(2)〉 can be used to provide information on the dynamical heterogeneity in proteins.

Long-time mean-square displacements in proteins.

We propose a method for obtaining the intrinsic, long-time mean square displacement (MSD) of atoms and molecules in proteins from finite-time molecular dynamics (MD) simulations. Typical data from simulations are limited to times of 1 to 10 ns, and over this time period the calculated MSD continues to increase without a clear limiting value. The proposed method consists of fitting a model to MD simulation-derived values of the incoherent intermediate neutron scattering function, I(inc)(Q,t), for finite times. The infinite-time MSD, , appears as a parameter in the model and is determined by fits of the model to the finite-time I(inc)(Q,t). Specifically, the is defined in the usual way in terms of the Debye-Waller factor as I(Q,t=∞)=exp(-Q(2)/3). The method is illustrated by obtaining the intrinsic MSD of hydrated lysozyme powder (h=0.4 g water/g protein) over a wide temperature range. The intrinsic obtained from data out to 1 and to 10 ns is found to be the same. The intrinsic is approximately twice the value of the MSD that is reached in simulations after times of 1 ns which correspond to those observed using neutron instruments that have an energy resolution width of 1 µeV.

Intrinsic mean-square displacements in proteins.

The thermal mean-square displacement (MSD) of hydrogen in proteins and its associated hydration water is measured by neutron scattering experiments and used an indicator of protein function. The observed MSD as currently determined depends on the energy resolution width of the neutron scattering instrument employed. We propose a method for obtaining the intrinsic MSD of H in the proteins, one that is independent of the instrument resolution width. The intrinsic MSD is defined as the infinite time value of (r(2)) that appears in the Debye-Waller factor. The method consists of fitting a model to the resolution broadened elastic incoherent structure factor or to the resolution dependent MSD. The model contains the intrinsic MSD, the instrument resolution width, and a rate constant characterizing the motions of H in the protein. The method is illustrated by obtaining the intrinsic MSD (r(2)) of heparan sulphate (HS-0.4), ribonuclease A, and staphysloccal nuclase (SNase) from data in the literature.

Vibrational dynamics of hydrogen in proteins.

Biological macromolecules expand with increasing temperature and this dynamic expansion is associated with the onset of function. The expansion is typically characterized by the mean square vibrational displacement (MSD), of specific constituents such as hydrogen within the macromolecules. The increases with increasing temperature and the slope of versus temperature can increase significantly at a temperature T{D} identified as a dynamical transition. We illustrate that the observed expansion and change in slope of with temperature at T{D} can be reproduced within a simple model of the vibration, an atom in an anharmonic potential, V(u). Given V(u), only the temperature is varied in the model. A simple Gaussian potential or a potential containing a hard wall is particularly effective is reproducing the observed change in the slope of with temperature around T{D}.

Phonon-roton modes and localized Bose-Einstein condensation in liquid helium under pressure in nanoporous media.

We show, using inelastic neutron scattering, that liquid helium in porous media, two gelsils and MCM-41, supports a phonon-roton mode up to a pressure of 36-37 bars only. Modes having the highest energy ("maxons") broaden and become unobservable at the lowest pressures (p approximately 26 bars) while rotons survive to the highest pressure. By comparing with the superfluid density observed by Yamamoto and co-workers in gelsil, we propose that there is a Bose glass phase containing islands of BEC surrounding the superfluid phase.

Phonon-roton excitations in liquid 4He at negative pressures.

We present neutron scattering measurements of the phonon-roton excitations of superfluid 4He held at negative pressures from zero to -5 bar. The liquid was stretched to negative pressures by immersing it in the porous medium MCM-41. In the wave vector range 0.35< or =Q< or =1.55 A(-1) and temperature T=0.4 K investigated, the phonon and maxon energies decrease systematically below bulk values as the negative pressure is increased. The energies are consistent with extrapolation of positive pressure values from which the negative internal pressure can be estimated. The maximum negative pressure realized is consistent with surface tension arguments and the MCM-41 pore diameter of 47 A.