The form of an internal reorientational correlation function

A non-Markovian, non-linearly coupled quantum stochastic dynamics model was used to investigate the reorientational correlation function of the internal motion of a methyl group attached to a large molecule. The values for the parameters used were determined by comparing angular momentum correlation functions calculated from the stochastic model to results of a molecular dynamics simulation.The results show that the correlation function has a rapid short time decay due to inertial effects, followed by a slower, exponential tail. Thus if there is significant inertial motion it is necessary to have at least two order parameters as well as two correlation times, one each for the overall and the internal motion, in order to accurately describe the system.The model can also explain data of Lee and McClung that cannot be explained with a Markovian model.     

Molecular motions of tactic poly(2-hydroxyethyl methacrylate) in solutions studied by 13C-NMR relaxation measurement

The spin-lattice relaxation time and the nuclear Overhauser enhancement were measured using Bruker AM 300 spectrometer operating at 75.5 MHz for ¹³C to investigate the molecular motional characteristics and its tacticity effect for tactic poly(2-hydroxyethyl methacrylate) (PHEMA) as a function of temperature in dimethyl sulfoxide and methanol solvents. The observed relaxation data have been analyzed for both backbone motion and methyl internal rotation according to the log-χ² distribution model and the diamond-lattice model. The correlation times thus obtained for the molecular motions show that isotactic PHEMA is more flexible than syndiotactic counterpart. The syndiotactic PHEMA seems to have intramolecular hydrogen bonding which restricts the motion of C-2 carbon at temperatures below 35°C, whereas the isotactic one indicated no hydrogen bonding at all temperatures examined in this study. The methyl group of isotactic PHEMA shows a greater degree of freedom for the internal rotation than that of syndiotactic one. From the temperature dependence of correlation times, the activation energies for both backbone segmental motion and methyl internal rotation are obtained. The activation energies, 20 kJ/mol for backbone motion and 19 kJ/mol for methyl internal rotation, for isotactic PHEMA are substantially lower than the corresponding activation energies of 30 and 32 kJ/mol obtained for syndiotactic one. An examination of these energies indicates that methyl side group and backbone motions in tactic PHEMA are linked together.     

General framework for studying the dynamics of folded and nonfolded proteins by NMR relaxation spectroscopy and MD simulation

A general framework is presented for the interpretation of NMR relaxation data of proteins. The method, termed isotropic reorientational eigenmode dynamics (iRED), relies on a principal component analysis of the isotropically averaged covariance matrix of the lattice functions of the spin interactions responsible for spin relaxation. The covariance matrix, which is evaluated using a molecular dynamics (MD) simulation, is diagonalized yielding reorientational eigenmodes and amplitudes that reveal detailed information about correlated protein dynamics. The eigenvalue distribution allows one to quantitatively assess whether overall and internal motions are statistically separable. To each eigenmode belongs a correlation time that can be adjusted to optimally reproduce experimental relaxation parameters. A key feature of the method is that it does not require separability of overall tumbling and internal motions, which makes it applicable to a wide range of systems, such as folded, partially folded, and unfolded biomolecular systems and other macromolecules in solution. The approach was applied to NMR relaxation data of ubiquitin collected at multiple magnetic fields in the native form and in the partially folded A-state using MD trajectories with lengths of 6 and 70 ns. The relaxation data of native ubiquitin are well reproduced after adjustment of the correlation times of the 10 largest eigenmodes. For this state, a high degree of separability between internal and overall motions is present as is reflected in large amplitude and collectivity gaps between internal and overall reorientational modes. In contrast, no such separability exists for the A-state. Residual overall tumbling motion involving the N-terminal beta-sheet and the central helix is observed for two of the largest modes only. By adjusting the correlation times of the 10 largest modes, a high degree of consistency between the experimental relaxation data and the iRED model is reached for this highly flexible biomolecule.     

Diffusion and segmental dynamics of rodlike molecules by fluorescence correlation spectroscopy

The dynamics of weakly bending polymers is analyzed on the basis of a Gaussian semiflexible chain model and the fluorescence correlation spectroscopy (FCS) correlation function is determined. Particular attention is paid to the influence of the rotational motion on the decay of the FCS correlation function. An analytical expression for the correlation function is derived, from which the averaged segmental mean square displacement can be determined independent of any specific model for the polymer dynamics. The theoretical analysis exhibits a strong dependence of the correlation function on the rotational motion for semiflexible polymers with typical lengths and persistence lengths of actin filaments or fd viruses. Hence, FCS allows for a measurement of the rotational motion of such semiflexible polymers. The theoretical results agree well with experimental measurements on actin filaments and confirm the importance of large relaxation times.     

Networks of coupled rotators: relationship between structures and internal dynamics in metal-binding proteins. Applications to apo- and holo-calbindin

This article presents an analysis of the internal dynamics of the Ca2+-binding protein calbindin, based on the Networks of Coupled Rotators (NCRs) introduced recently. Several fundamental and practical issues raised by this approach are investigated. The roles of various parameters of the model are examined. The NCR model is shown to account for the modifications of the internal dynamics upon Ca2+ binding by calbindin. Two alternative strategies to estimate local internal effective correlation times of the protein are proposed, which offer good agreement between predictions and experiment.     

Diffusion Spectrum of Polymer Melt Measured by Varying Magnetic Field Gradient Pulse Width in PGSE NMR

The translational motion of polymers is a complex process and has a big impact on polymer structure and chemical reactivity. The process can be described by the segment velocity autocorrelation function or its diffusion spectrum, which exhibit several characteristic features depending on the observational time scale—from the Brownian delta function on a large time scale, to complex details in a very short range. Several stepwise, more-complex models of translational dynamics thus exist—from the Rouse regime over reptation motion to a combination of reptation and tube-Rouse motion. Accordingly, different methods of measurement are applicable, from neutron scattering for very short times to optical methods for very long times. In the intermediate regime, nuclear magnetic resonance (NMR) is applicable—for microseconds, relaxometry, and for milliseconds, diffusometry. We used a variation of the established diffusometric method of pulsed gradient spin-echo NMR to measure the diffusion spectrum of a linear polyethylene melt by varying the gradient pulse width. We were able to determine the characteristic relaxation time of the first mode of the tube-Rouse motion. This result is a deviation from a Rouse model of polymer chain displacement at the crossover from a square-root to linear time dependence, indicating a new long-term diffusion regime in which the dynamics of the tube are also described by the Rouse model.     

A structural mode-coupling approach to 15N NMR relaxation in proteins.

The two-body Slowly Relaxing Local Structure (SRLS) model was applied to (15)N NMR spin relaxation in proteins and compared with the commonly used original and extended model-free (MF) approaches. In MF, the dynamic modes are assumed to be decoupled, local ordering at the N-H sites is represented by generalized order parameters, and internal motions are described by effective correlation times. SRLS accounts for dynamical coupling between the global diffusion of the protein and the internal motion of the N-H bond vector. The local ordering associated with the coupling potential and the internal N-H diffusion are tensors with orientations that may be tilted relative to the global diffusion and magnetic frames. SRLS generates spectral density functions that differ from the MF formulas. The MF spectral densities can be regarded as limiting cases of the SRLS spectral density. SRLS-based model-fitting and model-selection schemes similar to the currently used MF-based ones were devised, and a correspondence between analogous SRLS and model-free parameters was established. It was found that experimental NMR data are sensitive to the presence of mixed modes. Our results showed that MF can significantly overestimate order parameters and underestimate local motion correlation times in proteins. The extent of these digressions in the derived microdynamic parameters is estimated in the various parameter ranges, and correlated with the time scale separation between local and global motions. The SRLS-based analysis was tested extensively on (15)N relaxation data from several isotropically tumbling proteins. The results of SRLS-based fitting are illustrated with RNase H from E. coli, a protein extensively studied previously with MF.     

Sensitivity of 2H NMR spectroscopy to motional models: proteins and highly viscous liquids as examples

In order to study to what extent mechanisms of molecular motion can be unambiguously revealed by (2)H NMR spectroscopy, (2)H spectra for proteins (chicken villin protein headpiece HP36, selectively methyl-deuterated at leucine-69, C(δ) D(3)) and binary systems of high viscosity (benzene-d(6) in tricresyl phosphate) have been carefully analyzed as illustrative examples (the spectra are taken from the literature). In the first case, a model of restricted diffusion mediated by jumps between rotameric orientations has been tested against jump- and free diffusion models which describe rotational motion combined with jump dynamics. It has been found that the set of (2)H spectra of methyl-deuterated at leucine-69 chicken villin protein headpiece HP36 can be consistently explained by different motional models as well as by a gaussian distribution of correlation times assuming isotropic rotation (simple brownian diffusion model). The last finding shows that when the possible distribution of correlation times is not very broad one might not be able to distinguish between heterogeneous and homogenous (but more complex) dynamics by analyzing (2)H lineshapes. For benzene-d(6) in tricresyl phosphate, the dynamics is heterogeneous and it has been demonstrated that a gaussian distribution of correlation times reproduces well the experimental lineshapes, while for a Cole-Davidson distribution the agreement is somewhat worse. For inquires into the sensitivity of quadrupolar NMR spectral analysis (by "quadrupolar NMR spectroscopy we understand NMR spectroscopy of nuclei possessing quadrupole moment), the recently presented theoretical approach [Kruk et al., J. Chem. Phys. 135, 224511 (2011)] has been used as it allows simulating quadrupolar spectra for arbitrary motional conditions by employing the stochastic Liouville equation.     

Dynamics and conformations of PEO chains chemically bonded on silica: comparison between 1H and 2H NMR

1H NMR was used to study the motion of monomer units in a layer of poly(ethylene oxide) chains grafted on silica. First, the dependence of the relaxation times on the grafting ratios is discussed qualitatively from a phenomenological point of view. Next, the NMR line narrowing effect by high-speed rotation is observed in the same samples with different grafting ratios. The magic angle spinning technique permits determination of two correlation times for each grafting ratio: tau(c) characteristic of an environment with a fast motion and tau(l) characteristic of an environment with a slow motion. In addition, the dynamics of these grafted chains are investigated by deuterium NMR (2H NMR), which is sensitive to the anisotropy of molecular motion. The evolution has been studied for two extreme grafting ratios and each time as a function of temperature. The anisotropy is more marked at low temperatures and for a low grafting ratio. The results are consistent with the 1H NMR relaxation times measured as a function of temperature.     

Conformational flexibility of the group B meningococcal polysaccharide in solution

To elucidate the role of secondary structure in the immune response against alpha(2-->8)-linked polysialic acid, the capsular polysaccharide of Group B meningococci, we have investigated its solution dynamics by using specific models of molecular motion and hydrodynamic modeling to interpret experimental NMR data. (13)C-[(1)H] NMR relaxation times and steady-state NOE enhancements were measured for two aqueous solutions of alpha(2-->8)-linked sialic acid polysaccharides. Each contained a unique distribution of polysaccharide chain lengths, with average lengths estimated at 40 or 400 residues. Models for rigid molecule tumbling, including two based on helical conformations proposed for the polysaccharide,(31) could not explain the NMR measurements. In general for these helices, the correlation times for their overall tumbling that best account for the NMR data correspond to polysaccharide chains between 9 and 18 residues in length, far short of the average lengths estimated for either solution. The effects of internal motions incorporated into these helices was modeled with an effective correlation time representing helix tumbling as well as internal motion. This modeling demonstrated that even with extreme amounts of internal motion, "flexible helices" of 25 residues or more still could not produce the NMR measurements. All data are consistent with internal and segmental motions dominating the nuclear magnetic relaxation of the polysaccharide and not molecular tumbling. Statistical distributions of correlation times have been found specifically for the pyranose rings, linkage groups, and methoxy groups that can account for the measured relaxation times and NOE enhancements. The distributions suggest that considerable flexibility attends the polysaccharide in solution, and the ranges of motional frequencies for the linkage groups and pyranose rings are comparable. We conclude that the Group B meningococcal polysaccharide is a random coil chain in solution, and therefore, does not have antigenic epitopes dependent upon a rigid, ordered conformation.     

The internal dynamic modes of charged self-assembled peptide fibrils

Photon correlation spectroscopy is used to study the internal dynamics of self-assembled charged peptide fibrils. Short neutral and charged polymeric aggregates have diffusive modes due to whole macromolecular motion. For long semiflexible fibrils the logarithm of the intermediate scattering function follows a q(2)t(3/4) scaling at long times consistent with a Kratky-Porod free energy and preaveraged Oseen hydrodynamics. Persistence lengths on the order of micrometers are calculated for the peptide fibrils consistent with estimates from the liquid-crystalline phase behavior. Fibril diameters (5-35 nm) calculated from the initial decay of the correlation functions are in agreement with transmission electron microscopy measurements.     

Fractional protein dynamics seen by nuclear magnetic resonance spectroscopy: Relating molecular dynamics simulation and experiment

We propose a fractional Brownian dynamics model for time correlation functions characterizing the internal dynamics of proteins probed by NMR relaxation spectroscopy. The time correlation functions are represented by a broad distribution of exponential functions which are characterized by two parameters. We show that the model describes well the restricted rotational motion of N-H vectors in the amide groups of lysozyme obtained from molecular dynamics simulation and that reliable predictions of experimental relaxation rates can be obtained on that basis.     

Further studies of multiple nuclear spin relaxation and local motions is dissolved 1,1-dichloro-2,2-bis(4-hydroxyphenyl)ethylene polyformal

Deuterium, carbon-13 and proton spin-lattice relaxation times at two fields are reported for dilute solutions of 1,1-dichloro-2,2,-bis(4-hydroxyphenyl)ethylene polyformal. The carbon-13 and proton relaxation measurements were made at a concentration of 10% (w/w) in deuterated s-tetrachloroethane and as a function of temperature. A partially deuterated analog with deuterated methylene groups was used in order to remove cross-relaxation effects from the phenylene proton relaxation. In addition, deuterium relaxation measurements were made on this sample at a concentration of 10% (w/w) in tetrachloroethane as a function of temperature. The data are interpreted in terms of segmental motion arising within the bisphenol units and anisotropic internal rotations of the other structural components. Motions of the phenylene groups in the backbone are described by the Hall-Helfand segmental correlation function plus the Woessner anisotropic internal-rotation correlation function. Motions of the formal linkage are described by the same segmental correlation function plus an internal correlation function based on restricted double rotation about the two carbon/oxygen bonds. The local motion of the formal group is discussed in terms of confomational transitions that are likely in a polyformal in view of the conformational energy surface. A Helfand Type II motion of the formal group corresponding to a transition from gg′ to tg is identified as the most plausible rearrangement of this unit.     

2H NMR studies of glycerol dynamics in protein matrices

We use (2)H NMR spectroscopy to investigate the rotational motion of glycerol molecules in matrices provided by the connective tissue proteins elastin and collagen. Analyzing spin-lattice relaxation, line-shape properties, and stimulated-echo decays, we determine the rates and geometries of the motion as a function of temperature and composition. It is found that embedding glycerol in an elastin matrix leads to a mild slowdown of glycerol reorientation at low temperatures and glycerol concentrations, while the effect vanishes at ambient temperatures or high solvent content. Furthermore, it is observed that the nonexponential character of the rotational correlation functions is much more prominent in the elastin matrix than in the bulk liquid. Results from spin-lattice relaxation and line shape measurements indicate that, in the mixed systems, the strong nonexponentiality is in large part due to the existence of distributions of correlation times, which are broader on the long-time flank and, hence, more symmetric than in the neat system. Stimulated-echo analysis of slow glycerol dynamics reveals that, when elastin is added, the mechanism for the reorientation crosses over from small-angle jump dynamics to large-angle jump dynamics and the geometry of the motion changes from isotropic to anisotropic. The results are discussed against the background of present and previous findings for glycerol and water dynamics in various protein matrices and compared with observations for other dynamically highly asymmetric mixtures so as to ascertain in which way the viscous freezing of a fast component in the matrix of a slow component differs from the glassy slowdown in neat supercooled liquids.     

Frequency dependent study of the correlation functions in EPR spectroscopy--the Cole-Davidson approach 1. Perdeuterated 2,2,6,6-tetramethyl-4-piperidone N-oxide in toluene

EPR linewidth measurements of PD-Tempone in toluene at 1 (L-Band), 4 (S-Band), 9 (X-Band) and 34 GHz (Q-Band) microwave frequencies indicate the presence of a distribution of relaxation times. The empirical response parameter introduced by Cole-Davidson for the analysis of dielectric relaxation in liquids has been used for the analysis of EPR relaxation data in the low frequency region. The Cole-Davidson parameter can assume values in the range 0 < beta < or = 1. When beta = 1, one obtains the Debye-type spectral density. The calculated linewidth data at 1 GHz agrees with a Cole-Davidson distribution function with a width parameter 0.83 +/- 0.04 for a spherical solute. Beta < 1 at L-band suggests the presence of an asymmetrical distribution of relaxation times associated with different modes of relaxation mechanisms or internal molecular motions. This study shows EPR experiments at low microwave frequencies are more sensitive to the shape of the correlation function.     

Use of a cooperative-motion domain model to analyze brillouin light scattering measurements of the dynamic modulus in triblock copolymers

The use of the relaxation function is widespread in the study of polymer dynamics. Since the popular empirical KWW relaxation function consistently underestimates dielectric loss at high frequency, several models dealing explicitly with intermolecular cooperativity have been proposed as alternatives. In this article, the domain model proposed by Matsuoka, previously used only to analyze dielectric relaxation results, is used to analyze Brillouin light scattering results from polystyrene–polybutadiene–polystyrene triblock copolymers. A single relaxation time analysis and the KWW model are both compared to the domain model. Neither of these models fits the Brillouin data well. The single relaxation time analysis gives a physically unrealistic results; the KWW analysis fits the data at low frequency, but fails in the high-frequency region by underestimating the attenuation. The domain model fits the Brillouin data well over the entire temperature/frequency range. The results show that in order to understand the full range of dynamics in these materials and in polymeric materials in general, the KWW model is insufficient due to its underestimation of attenuation at high frequency. A model including cooperative motion is crucial to fully understand polymer dynamics. © 2000 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 38: 2170–2178, 2000     

Orientational dynamics in the isotropic phase of a calamitic liquid-crystal model

We report a molecular dynamics simulation study on the isotropic phase of an idealized calamitic liquid crystal model with a length-to-width ratio of approximately 5-6. The study focuses on the characterization of single-particle and collective orientational dynamics on approaching the phase transition to the nematic phase. Recent experimental and simulation works have suggested that a power law behavior exists at relatively short times in the decay of the time derivative of the orientational correlation functions. Qualitatively, our simulation data are consistent with these findings. Both single-particle and collective time correlation function derivatives possess, in their respective log-log plots, a linear region at very short times, whose slope is essentially independent from the thermodynamic state. Nevertheless, the single-particle orientational correlation functions are better described by a function which is the sum of a fast exponential, an intermediate stretched-exponential and a slow exponential, while the collective orientational correlation functions are satisfactorily described by a sum of two exponentials, at higher density, or by just one exponential, at lower density.     

The trehalose coating effect on the internal protein dynamics

(15)N and (13)C NMR experiments were applied to conduct a comparative study of a cold shock protein (Csp) in two states-lyophilized powder and a protein embedded in a glassy trehalose matrix. Both samples were studied at various levels of rehydration. The experiments used (measuring relaxation rates R(1) and R(1ρ), motionally averaged dipolar couplings and solid state exchange method detecting reorientation of the chemical shift anisotropy tensor) allow obtaining abundant information on the protein structural features and internal motions in a range of correlation times from nanoseconds to seconds. The main results are: (a) the trehalose coating makes the protein structure more native in comparison with the dehydrated lyophilized powder, however, trehalose still cannot remove all non-native hydrogen bonds which are present in a dehydrated protein; (b) trehalose has an appreciable effect on the internal dynamics: the motion of the backbone N-H groups in the nanosecond and microsecond time scales becomes slower while the motional amplitude remains constant; (c) upon adding water to the Csp-trehalose mixture, water molecules accumulate around proteins forming a layer between the protein surface and the trehalose matrix. The protein dynamics become faster, however, not as fast as in the fully hydrated state; (d) the hydration response of dynamics of the NH and CH(CH(2)) groups in a protein is qualitatively different: upon increasing protein hydration, the correlation times of the N-H motions become shorter and the amplitude remains stable, and for CH(CH(2)) groups the motional amplitude increases and the correlation times do not change. This can be explained by a different ability of the NH and CH(CH(2)) groups to form hydrogen bonds.