Predicting NMR Relaxation Rates in Anisotropically Tumbling Proteins through Networks of Coupled Rotators

We show that the prediction of ¹⁵N relaxation rates in proteins can be extended to systems with anisotropic global rotational diffusion by using a network of coupled rotators (NCR), starting from a three‐dimensional structure. The relaxation rates predicted by this method are confronted in several examples with experiments performed by other groups. The NCR spectral density functions are compared with the results obtained from reduced spectral density mapping. The consequence of the timescales of internal motions on the predicted relaxation rates and the effects of the predicted local anisotropy—present only in the case of anisotropic overall tumbling—on dynamic parameters, are discussed.     

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.     

Study of internal ring rotation and molecular reorientation in the liquid crystal 5O.7 by deuterium N.M.R. spectroscopy

The spectral densities of motion were determined by deuterium N.M.R. relaxation measurements in the nematic, smectic A and smectic C phases of 4-n-pentyloxybenzylidene-d₁-4'-heptylaniline and 4-n-pentyloxybenzylidene-4'-heptylaniline-2,3,5,6-d₄. By examining two atomic sites on a 5O.7 molecule, we were able to gain information on the reorientation motion and internal rotation of the aniline ring. It was also found that director fluctuations make some contribution to the spectral density J₁ (ω). We use the superimposed rotations model to account for the internal ring motion and the small step rotational diffusion model for the molecular reorientation. The derived rotational diffusion constants for the spinning and tumbling motions appear to give physically plausible activation energies in the mesophases of 5O.7.     

Molecular dynamics of a ferroelectric smectogen in its smectic phases by means of 2H NMR spectroscopy

In this work the dynamic behaviour of the ferroelectric liquid crystal (-)-(S )-[4-(2-methylbutyloxycarbonyl)phenyl] 4-n-heptylbiphenylcarboxylate (MBHB) in its smectic A (SmA), unwound chiral smectic C (uSmC*) and chiral smectic C (SmC*) phases has been studied by means of ²H NMR spectroscopy. Zeeman (T _1Z) and quadrupolar (T _1Q) spin-lattice relaxation times have been analysed to extract dynamic parameters (diffusion coefficients and activation energies). The small step rotation diffusion model in the uniaxial approximation has been used to describe overall spinning and tumbling motions, and the strong collision model to describe the internal reorientations of the aromatic fragment. Relaxation data in the SmC* phase have been analysed by using a theoretical approach. The dynamic features obtained in the smectic phases of this mesogen are here presented and discussed in comparison with the results obtained in other ferroelectric liquid crystals, focusing on the fast regime of motions.     

Study of spin–lattice relaxation and local motion in dissolved poly[2,2-propanediylbis(3,5-dimethyl-4, hydroxyphenyl)carbonate]

Proton and carbon-13 spin–lattice relaxation times are reported for 10-wt % solutions of tetramethyl bisphenol-A polycarbonate. The relaxation times for both nuclei were measured at two Larmor frequencies and as a function of temperature. These relaxation times are interpreted in terms of three motions: segmental motion, restricted rotational diffusion, and backbone methyl-group rotation. The Hall–Helfand correlation function is used to describe the segmental motion. Internal rotation is described by the usual Woessner approach and restricted anisotropic rotational diffusion by the Gronski approach. As demonstrated by its higher activation energy, correlated segmental motion appears to be slower than the unsubstituted polycarbonate of BPA. In addition, the single-transition processes seem to be still less important than correlated backbone transitions. Phenylene-group rotation is described in terms of restricted rotational diffusion instead of complete anisotropic rotation. The time scale for backbone methyl-group rotation is comparable to that in BPA, a fact indicative of weaker cooperativity between this motion and the other motions. Rotation of the methyl group attached to the phenylene ring is too fast to significantly contribute to relaxation except by partially averaging the dipole–dipole interactions. The higher activation energies for segmental motion observed in solution for this methyl-substituted polycarbonate relative to the unsubstituted polycarbonate parallel a significant increase in the glass transition temperature observed for the substituted material. The restricted pheylene-group rotation in solution is also parallelled by a large upward shift of the low-temperature loss peak in the glassy polymer.     

A model of interdomain mobility in a multidomain protein

Domain mobility plays an essential role in the biological function of multidomain systems. The characteristic times of domain motions fall into the interval from nano- to milliseconds, amenable to NMR studies. Proper analysis of NMR relaxation data for these systems in solution has to account for interdomain motions, in addition to the overall tumbling and local intradomain dynamics. Here we propose a model of interdomain mobility in a multidomain protein, which considers domain reorientations as exchange/interconversion between two distinct conformational states of the molecule, combined with fully anisotropic overall tumbling. Analysis of 15N-relaxation data for Lys48-linked diubiquitin at pH 4.5 and 6.8 showed that this model adequately fits the experimental data and allows characterization of both structural and motional properties of diubiquitin, thus providing information about the relative orientation of ubiquitin domains in both interconverting states. The analysis revealed that the two domains reorient on a time scale of 9-30 ns, with the amplitudes sufficient for allowing a protein ligand access to the binding sites sequestered at the interface in the closed conformation. The analysis of a possible mechanism controlling the equilibrium between the interconverting states in diubiquitin points toward protonation of His68, which results in three different charged states of the molecule, with zero, +e, and +2e net charge. Only two of the three states are noticeably populated at pH 4.5 or 6.8, which assures applicability of the two-state model to diubiquitin at these conditions. We also compare our model with the "extended model-free" approach and discuss possible future developments of the model.     

Evaluating rotational diffusion from protein MD simulations

It is now feasible to carry out molecular dynamics simulations of proteins in water that are long compared to the overall tumbling of the molecule. Here, we examine rotational diffusion in four small, globular proteins (ubiquitin, binase, lysozyme, and fragment B3 of protein G) with the TIP3P, TIP4P/EW, and SPC/E water models, in simulations that are 6 to 60 times as long as the mean rotational tumbling time. We describe a method for extracting diffusion tensors from such simulations and compare the results to experimental values extracted from NMR relaxation measurements. The simulation results accurately follow a diffusion equation, even for spherical harmonic correlation functions with l as large as 8. However, the best-fit tensors are significantly different from experiment, especially for the commonly used TIP3P water model. Simulations that are 20 to 100 times longer than the rotational tumbling times are needed for good statistics. A number of residues exhibit internal motions on the nanosecond time scale, but in all cases examined here, a product of internal and overall time-correlation functions matches the total time-correlation function well.     

Study of internal ring rotation and molecular reorientation in the liquid crystal 5O.7 by deuterium N.M.R. spectroscopy

Abstract

The spectral densities of motion were determined by deuterium N.M.R. relaxation measurements in the nematic, smectic A and smectic C phases of 4-n-pentyloxybenzylidene-d ₁-4′-heptylaniline and 4-n-pentyloxybenzylidene-4′-heptylaniline-2,3,5,6-d ₄. By examining two atomic sites on a 5O.7 molecule, we were able to gain information on the reorientation motion and internal rotation of the aniline ring. It was also found that director fluctuations make some contribution to the spectral density J ₁ (ω). We use the superimposed rotations model to account for the internal ring motion and the small step rotational diffusion model for the molecular reorientation. The derived rotational diffusion constants for the spinning and tumbling motions appear to give physically plausible activation energies in the mesophases of 5O.7.     

Dynamics of large elongated RNA by NMR carbon relaxation

We present an NMR strategy for characterizing picosecond-to-nanosecond internal motions in uniformly 13C/15N-labeled RNAs that combines measurements of R1, R1rho, and heteronuclear 13C{1H} NOEs for protonated base (C2, C5, C6, and C8) and sugar (C1') carbons with a domain elongation strategy for decoupling internal from overall motions and residual dipolar coupling (RDC) measurements for determining the average RNA global conformation and orientation of the principal axis of the axially symmetric rotational diffusion. TROSY-detected pulse sequences are presented for the accurate measurement of nucleobase carbon R1 and R1rho rates in large RNAs. The relaxation data is analyzed using a model free formalism which takes into account the very high anisotropy of overall rotational diffusion (Dratio approximately 4.7), asymmetry of the nucleobase CSAs and noncollinearity of C-C, C-H dipolar and CSA interactions under the assumption that all interaction tensors for a given carbon experience identical isotropic internal motions. The approach is demonstrated and validated on an elongated HIV-1 TAR RNA (taum approximately 18 ns) both in free form and bound to the ligand argininamide (ARG). Results show that, while ARG binding reduces the amplitude of collective helix motions and local mobility at the binding pocket, it leads to a drastic increase in the local mobility of "spacer" bulge residues linking the two helices which undergo virtually unrestricted internal motions (S2 approximately 0.2) in the ARG bound state. Our results establish the ability to quantitatively study the dynamics of RNAs which are significantly larger and more anisotropic than customarily studied by NMR carbon relaxation.     

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.     

Experimental and theoretical analysis of the reorientational dynamics of fullerene C70 in various aromatic solvents

A previous study of C70 in deuterated chlorobenzene generated evidence suggesting C70 was experiencing unique reorientational behavior at given temperatures. The present study explores the possibility that this behavior is present across other solvents. The 13C spin-lattice relaxation rates for four carbon resonances in C70 were analyzed in benzene-d6, chlorobenzene-d5, and o-dichlorobenzene-d4, and as a function of temperature, to probe the reorientational dynamics of this fullerene. Anisotropic behavior was observed at the lowest (283 K) and highest temperatures (323 K), isotropic diffusion was seen between 293 and 303 K, and quasi-isotropic at 313 K. When anisotropic motion was present, diffusion about the figure axis was seen to be higher than diffusion of the figure axis. Experimentally obtained diffusion coefficients generated reorientational correlation times that were in excellent agreement with experimental values. Theoretical predictions generated by a modified Gierer-Wirtz model provided acceptable predictions of the diffusion constants; with DX usually being more closely reproduced and DZ values generally being underestimated. Overall, the results indicate that the factors affecting rotational behavior are complex and that multiple solvent factors are necessary to characterize the overall motion of C70 in these solvents. Although a solvent's viscosity is normally sufficient to characterize the tumbling motion, the spinning motion is less sensitive to solvent viscosity but more responsive to solvent structure. The balance and collective influence of these factors ultimately determines the overall rotational behavior.     

How to detect internal motion by homonuclear NMR

NOESY and ROESY cross-peak intensities depend on internuclear distances and internal motion. Internal motion is usually ignored, and NOESY cross-peak intensities are interpreted in terms of internuclear distances only. Off-resonance ROESY experiments measure a weighted average of NOE and ROE. The weight can be described and experimentally set by an angle theta;. For large enough molecules, NOE and ROE have opposite signs. Therefore, each cross-peak intensity becomes zero for an angle theta;(0). For any sample, the maximum angle theta;(0) is determined by the overall motion of the molecule. Smaller theta;(0) values reflect the angular component of internal motions. Because individual cross-peaks are analyzed, the method evaluates internal motions of individual H,H vectors. The reduction of theta;(0) is largest for internal motions on a time scale of 100-300 ps. The sensitivity of theta;(0) for internal motions decreases with increasing molecular weight. We estimate that detecting internal motions will be practicable for molecules up to about 15 kDa. We describe a protocol to measure theta;(0) from a series of off-resonance ROESY spectra. For such a series, we describe the choice of experimental parameters, a procedure to extract theta;(0) from the raw data, and the interpretation of theta;(0) in terms of internal motions. In the small protein BPTI, we analyzed 75 cross-peaks. The precision of theta;(0) was 0.25 degrees, as compared to typical reductions of theta;(0) of 3 degrees. We found a well-defined maximum theta;(0) for cross-peaks in rigid parts of the molecule, which reflects the overall motion of the molecule. For BPTI, also many structurally important long-range cross-peaks appear rigid. The lower theta;(0) values of long-range contacts involving methyl groups are consistent with methyl rotation on the 25-ps time scale. The lower theta;(0) values of the flexible C-terminus and of flexible side chains translate into upper limits for the angular order parameter of 0.4 and 0.5-0.8, respectively. Off-resonance ROESY can monitor internal motions of H,H contacts that are used in a structure calculation. Because no isotope labeling is needed, off-resonance ROESY can be used to detect internal motions in a wide range of natural products.     

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.     

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.     

Internal dynamics features in the free jet rotational spectrum of the acetaldehyde-Kr molecular complex

The structure and the dynamics of internal motions in the complex formed between acetaldehyde and Kr are studied by free jet absorption microwave spectroscopy performed in the range 60-78 GHz. The fourfold structure of each rotational line is evidence of the vibration-rotation coupling between the overall rotation of the complex, a tunneling motion of the Kr atom between two equivalent positions and the internal rotation of the methyl group in the acetaldehyde moiety. The four sets of transitions could be fitted with a coupled Hamiltonian which allows for the Coriolis interaction obtaining the energy separation between the vibrational energy levels related to the tunneling motion, while the observed splittings due to the methyl group internal rotation were analyzed independently with an appropriate model. The potential energy barriers for the tunneling motion and the internal rotation of the methyl group have been calculated and the interaction of the rare gas atom with the acetaldehyde moiety is reflected in the change of the V(3) barrier to internal rotation in going from the molecule to the weakly bound complex.     

NMR line-width, self-diffusion and relaxation studies of neopentanol in liquid and solid phases

The translational and rotational dynamics of the liquid and disordered (solid I) phases of neopentanol are investigated using high-field multinuclear NMR. The extensive line-narrowing of the ¹H resonances for solid I is ascribed to the onset of translational diffusion whereas the line-narrowing of the deuteron and carbon-13 signals is dominated by molecular reorientations. The activation energy of the molecular self-diffusion is 34 and 71 kJ mol⁻¹ for the liquid and solid I phases, respectively. The self-diffusion coefficient of solid I is 3.2 × 10⁻¹³ m² s⁻¹ at the melting point.

A thorough analysis of the multinuclear T₁ data is presented. The activation energy of the overall tumbling motion in the liquid and solid I phases, obtained from the hydroxyl deuteron T₁ data, is 36 and 52 kJ mol⁻¹, respectively. The internal reorientations have a profound effect on the spin-lattice relaxation times of the methyl and methylene groups by reducing the effective correlation time by an order of magnitude relative to the overall tumbling motion in solid I. The long correlation time (22 and 58 ps of liquid and solid neopentanol at the melting point) and high activation energy suggest that the overall tumbling motion in the liquid and disordered phases involves hydrogen-bonded aggregates.     

Dynamics of attractive vesicles in shear flow

A nonequilibrium molecular dynamics (NEMD) method is employed to study the dynamics of two identical vesicles with attractive interactions immersed in shear flow. The dynamics behaviors of attractive vesicles depend on the attractive interactions and the shear rates simultaneously. There are four motion types for attractive vesicles in shear flow: a coupled-tumbling (CTB) motion, a coupled-trembling (CTR) motion, a collision/rotation mixture (CRM) motion and a separated-tank-treading (STT) motion, which are determined by the competition between the shear flow and the attractive interactions. Furthermore, the dynamics behavior of an individual vesicle shows three main motion types such as tumbling, trembling and tank-treading motions, and relies mainly on the shear rates. Meanwhile, comparisons with rigid vesicles for the dynamics behaviors are made, and the collision/rotation mixture (M) motion isn’t observed for rigid vesicles.     

Furanose dynamics in the HhaI methyltransferase target DNA studied by solution and solid-state NMR relaxation

Both solid-state and solution NMR relaxation measurements are routinely used to quantify the internal dynamics of biomolecules, but in very few cases have these two techniques been applied to the same system, and even fewer attempts have been made so far to describe the results obtained through these two methods through a common theoretical framework. We have previously collected both solution 13C and solid-state 2H relaxation measurements for multiple nuclei within the furanose rings of several nucleotides of the DNA sequence recognized by HhaI methyltransferase. The data demonstrated that the furanose rings within the GCGC recognition sequence are very flexible, with the furanose rings of the cytidine, which is the methylation target, experiencing the most extensive motions. To interpret these experimental results quantitatively, we have developed a dynamic model of furanose rings based on the analysis of solid-state 2H line shapes. The motions are modeled by treating bond reorientations as Brownian excursions within a restoring potential. By applying this model, we are able to reproduce the rates of 2H spin-lattice relaxation in the solid and 13C spin-lattice relaxation in solution using comparable restoring force constants and internal diffusion coefficients. As expected, the 13C relaxation rates in solution are less sensitive to motions that are slower than overall molecular tumbling than to the details of global molecular reorientation, but are somewhat more sensitive to motions in the immediate region of the Larmor frequency. Thus, we conclude that the local internal motions of this DNA oligomer in solution and in the hydrated solid state are virtually the same, and we validate an approach to the conjoint analysis of solution and solid-state NMR relaxation and line shapes data, with wide applicability to many biophysical problems.     

Reorientational dynamics of the dodecahydro-closo-dodecaborate anion in Cs2B12H12

Rapid reorientational motions of the B(12)H(12)(2-) icosahedral anion, a key intermediate in borohydride dehydrogenation, are revealed by quasielastic neutron scattering (QENS) measurements of Cs(2)B(12)H(12) between 430 and 530 K. At 430 K, over the range of momentum transfers collected, the elastic incoherent structure factor (EISF) is consistent with a model for reorientational jumps about a single molecular axis. At temperatures of 480 K and higher, however, the EISF suggests the emergence of multiaxis reorientation by dynamically similar, independent jumps about two axes, on average, preserving crystallographic order. Alternatively, if one assumes that the anions are undergoing temperature-dependent rotational trapping, then the EISF is also consistent with a jump model involving a temperature-dependent mobile fraction of anions statistically tumbling between discrete crystallographic sites. Although neutron vibrational spectra demonstrate that the anion torsional modes soften dramatically with increasing temperature, the QENS-derived activation energy of 333 meV for reorientation clearly shows that the anions are not undergoing isotropic rotational diffusion.