Internuclear Vector Research Articles

Coherent Raman spectroscopy of nuclear quadrupole resonance of La aroundPr3+inLaF3

Coherent Raman spectroscopy was applied for an optical-rf double-resonance study of a ${\mathrm{LaF}}_{3}$ crystal doped with ${\mathrm{Pr}}^{3+}.$ The coherent nuclear spin Raman scattering was detected as a function of the applied rf frequency. Exciting the resonance condition of the ${}^{3}{\stackrel{\ensuremath{\rightarrow}}{{H}_{4}}}^{3}{P}_{0}$ transition of ${\mathrm{Pr}}^{3+}$ (20 925 ${\mathrm{cm}}^{\mathrm{\ensuremath{-}}1}$), only the La nuclei surrounding the ${\mathrm{Pr}}^{3+}$ ion were observed through their nuclear quadrupole resonance (NQR). The double resonance between the optical transition of the ion and the NQR transition of its neighboring nucleus is theoretically described by analyzing the magnetic dipolar interaction that is affected by the optical excitation. Under a certain restriction, which the present system fulfills, the intensity of the Raman heterodyne signal can be described by an analytical function of the internuclear vector and the orientation of the electric-field gradient (EFG) at the Pr and La nuclei. Five different neighboring La nuclei were observed. They are different from the bulk La in both magnitude and orientation of the EFG. In addition, it was found that the laser frequency jitter affects the relative signal intensity of the different La NQR transitions through the optical pumping of the La spin levels.

Multipole Analysis of the Electron Density in Topaz Using X-ray Diffraction Data

The static deformation electron density, Laplacian of the electron density and critical points in the electron density were analyzed in topaz Al2[SiO4]F2, using high-precision X-ray diffraction data. The electron deformation density, positive values of the Laplacian at (3,−1) bond critical points and the net atomic charges indicate a closed-shell-type interaction in the polyhedra. Anion valence-shell charge depletions are revealed. Maxima in the Laplacian of the electron density are displaced towards the close-packed plane owing to the mutual repulsion of the anion valence shells. The relationship between the topological features of the electron density and the close-packing concept is discussed. Shifts of the critical points from the internuclear vectors reflect the strain in the structure.

Trans content in atactic polystyrene estimated by double-quantum solid-state NMR

The conformational statistics of atactic polystyrene have been investigated by solid-state NMR, on a sample labeled with 25% 13 CH 2 groups. Double-quantum pulse sequences are used to select statistically formed 13 C– 13 C spin pairs. An experiment correlating the 13 C– 13 C internuclear vector with the 13 C chemical-shift anisotropy determines the conformational statistics of a single backbone torsion angle. Double-quantum experiments correlating the 13 CH 2 chemical-shift anisotropies of two adjacent segments provide information on two consecutive torsion angles. The analysis of the spectra yields a trans content of 68% (±10%). This represents the first direct experimental estimate of the trans/ gauche ratio in atactic polystyrene. The trans content found here agrees well with rotational-isomeric-state models but is inconsistent with results from more elaborate atomistic models that take packing effects into account.

Modulation of the Alignment Tensor of Macromolecules Dissolved in a Dilute Liquid Crystalline Medium

Dipolar couplings measured for pairs of nuclei constrain the possible orientations of internuclear vectors relative to the molecule’s alignment tensor.1-3 For macromolecules such as proteins and nucleic acids, the required small degree of alignment with the magnetic field can result from the molecules’ own magnetic susceptibility anisotropy2-4 or can be induced by the use of a liquid crystalline medium.1 Provided the alignment is sufficiently weak, only directly bonded pairs of atoms give rise to a measurable dipolar coupling, which manifests itself as a change in the corresponding scalar interaction. The spectral simplicity of the high-resolution NMR spectrum is therefore retained, and measurement of dipolar couplings is straightforward. The most commonly used liquid crystalline medium for inducing macromolecular alignment consists of large, disk-shaped phospholipid particles, often referred to as bicelles.5,6 The volume fraction of phospholipid particles can be adjusted to obtain the desired degree of alignment.7 For the case where the diagonalized alignment tensor is axially symmetric, and the internuclear distance is accurately known, each dipolar coupling restrains the position of the corresponding internuclear vector to a cone about the unique axis of the alignment tensor. For the general case, where the alignment tensor is asymmetric, the cone is distorted and “taco-shaped”.8 Because the direction of a second rank tensor interaction cannot be distinguished from its inverse, the dipolar coupling actually defines two cones of possible bond vector orientations, in opposing directions. Although such information is clearly very useful in determining macromolecular structures more accurately,4,6 the continuum of possible orientations makes it difficult to determine local geometry in the absence of other information. Here we demonstrate that the degeneracy can be lifted by recording a second set of dipolar couplings under conditions where the orientation and/or rhombicity of the alignment tensor is altered. Three different approaches are shown to be successful: (1) addition of an unstructured, so-called His-tag peptide at the C-terminus of the protein, (2) changing the sample pH, and (3) changing the net charge of the bicelles. For recombinant proteins, it is frequently possible to add a short unstructured peptide at either the Nor C-terminus. Although such an additional peptide fragment does not alter the structure of the globular domain, it adds a flexible “tail” to the protein which affects the molecule’s alignment tensor. For example, a so-called His-tag sequence frequently is added to a heterologously expressed protein to facilitate its purification. Histidine residues have a pK of ∼7, and their net charge can be altered by lowering the pH from 7.5 to 5. If there is a small amount of net charge on the surface of the bicelles, this will change the very weak electrostatic attraction/repulsion between the protein and the bicelle, thereby altering the alignment tensor. Finally, we demonstrate that deliberately adding a net charge to the bicelles, recently shown to increase the stability of the liquid crystalline phase at low volume fractions,9 also can be used to alter the alignment tensor, in either the presence or absence of the Histag sequence. Experiments are carried out for samples of uniformly 13C/15Nenriched ubiquitin, either with an additional SHHHHHH peptide at its C-terminus (Martek Biosciences, Columbia, MD) or without (VLI Research, Southeastern, PA). Samples were ∼0.7 mM, in 93% H2O, 7% D2O, containing 4-5% (w/v) bicelles (consisting of a 3:1 molar ratio of DMPC to DHPC).5,6 As a second test, measurements were conducted for a sample of uniformly 15Nlabeled aprotinin (often referred to as BPTI) (0.2 mM in 4-5% (w/v) bicelles). The dipolar coupling between two nuclei, A and B, in a solute macromolecule of fixed shape is related to the traceless alignment tensor according to

Determining the Magnitude of the Fully Asymmetric Diffusion Tensor from Heteronuclear Relaxation Data in the Absence of Structural Information

Determining the Magnitude of the Fully Asymmetric Diffusion Tensor from Heteronuclear Relaxation Data in the Absence of Structural Information

Relationships between bridging oxygen O17 quadrupolar coupling parameters and structure in alkali silicates

We have performed ab initio calculations on the model cluster (OH)3Si–O–Si(OH)3 with lithium or sodium cations coordinated to the central oxygen in order to refine the relationships between O17 quadrupolar coupling parameters and the local structure around the bridging oxygen in the case of alkali silicates. We have also used a point charge model to derive approximate expressions to describe the dependence of the O17 electric field gradient tensor on the orientation of the alkali cation-bridging oxygen internuclear vector(s). From these calculations we predict that the previously established trend in O17 quadrupolar coupling constant, Cq, with Si–O–Si angle is systematically shifted to lower magnitudes with increasing number and field strength of coordinating alkali cations and that Cq will be relatively insensitive to variations in the alkali cation-bridging oxygen internuclear vector orientation. The previously established trend in O17 quadrupolar coupling asymmetry parameter, ηq, with Si–O–Si angle is systematically shifted to higher values by the presence of one coordinating alkali cation, and only slightly shifted to higher values by the presence of two coordinating alkali cations. As with the quadrupolar coupling constant, the magnitude of the shift in asymmetry parameter increases with increasing field strength of the coordinating alkali cation(s). In contrast to Cq, ηq is additionally dependent on variations in the alkali cation-bridging oxygen internuclear vector orientation. With two or more coordinating alkali cations, however, the magnitude of these variations are reduced, and for certain alkali cation configurations are eliminated. Finally, the parametric dependence of Cq on ηq via the Si–O–Si angle can be used to distinguish between bridging oxygen environments with differing number of coordinating alkali cations.

Radiofrequency-Driven and Slow-Magic-Angle-Sample-Spinning Polarization-Transfer Techniques: A Comparative Study

The rate constant of radiofrequency-driven (RF-driven) polarization transfer and that of polarization transfer under slow-magic-angle sample spinning (S-MAS) are compared using a model system, polycrystalline α-α′-13C2-phthalic acid. While the rate constant under RF irradiation in static samples strongly depends on the orientation of the internuclear vector, the rate constant under S-MAS is hardly sensitive to that orientation and, thus, depends almost exclusively on the internuclear distance. Consequently, polarization-transfer rate constants obtained under S-MAS can be interpreted more simply when used to study local order in polycrystalline or amorphous solids.

Rotational Correlation Times of Internuclear Vectors in a DNA Duplex with G–A Mismatch Determined in Aqueous Solution by Complete Relaxation Matrix Analysis of Off-Resonance ROESY (O-ROESY) Spectra

Complete relaxation matrix analysis of the off-resonance ROESY (O-ROESY) spectra is presented and demonstrated for a synthetic DNA duplex with two G–A mismatches,d(GCTGTCGAAAGC)2, in solution. The internuclear distance and the rotational correlation time of the internuclear vector could be garnered simultaneously using complete relaxation matrix analysis of O-ROESY, by which spin diffusion effects could be accommodated. Correlation times in the terminal and the mismatched regions were significantly reduced compared to those in other regions, indicating the conformational flexibility of the mismatched pair. The average structure obtained by restrained molecular dynamics simulation with inclusion of variations of the rotational correlation times also indicated a general tendency of the mismatched and contiguous bases to flip to the outside of the double strand. Off-resonance ROESY combined with the complete relaxation matrix analysis method may offer an alternative way to investigate the structures and dynamics of biological macromolecules.

Use of dipolar 1H-15N and 1H-13C couplings in the structure determination of magnetically oriented macromolecules in solution.

Anisotropy of the molecular magnetic susceptibility gives rise to a small degree of alignment. The resulting residual dipolar couplings, which can now be measured with the advent of higher magnetic fields in NMR, contain information on the orientation of the internuclear vectors relative to the molecular magnetic susceptibility tensor, thereby providing information on long range order that is not accessible by any of the solution NMR parameters currently used in structure determination. Thus, the dipolar couplings constitute unique and powerful restraints in determining the structures of magnetically oriented macromolecules in solution. The method is demonstrated on a complex of the DNA-binding domain of the transcription factor GATA-1 with a 16 base pair oligodeoxyribonucleotide.

Calculation of nuclear magnetic resonance order parameters in proteins by normal mode analysis

An analytic formula is developed for calculating the generalized NMR order parameters in a protein from the normal mode analysis (NMA). The generalized order parameter, S̃2, is given as thermal ensemble average of a Taylor series in powers of Δ, the displacement of internuclear vector from its mean. Henry and Szabo developed a method to calculate the ensemble average based on the NMA carried out in the Cartesian coordinate space (CCS). However, atomic motions in each individual CCS normal modes are linear in the three-dimensional space, which may cause interatomic distances even between covalently bonded atoms to change significantly. In this situation Henry and Szabo proposed to use a special formula for S̃2 which includes a trick to compensate such changes. We showed that by carrying out the NMA in the dihedral angle space (DAS) and by interpreting each DAS normal mode into curved atomic motions, S̃2 can be calculated reliably for spin pairs separated up to about 10 intervening covalent bonds by a natural formula without any trick.

Analytic Solutions for the Time Evolution of Dipolar-Dephasing NMR Signals

The time evolution of dipolar-dephasing NMR signals reveals the strength of dipolar couplings in solids. In most instances where microcrystalline samples are investigated, numerical simulations of dipolar-dephasing intensities provide fits to experimental data in order to determine internuclear distances. In this paper, analytic solutions are derived for the time evolution of signals in heteronuclear SEDOR, REDOR, and TEDOR experiments. These solutions, in terms of infinite series of Bessel functions of the first kind, are exact and replace a numerical integration over the possible orientations of the internuclear vectors in a powdered or amorphous sample. Results from the analytic solutions converge rapidly and are compared here with results from numerical integration. A method for efficient least-squares fitting of dipolar-dephasing data is also demonstrated, including fits to experimental SEDOR and TEDOR dephasing curves.

Off-Resonance Irradiation Effects on the Relaxation Process of the Protein and the Nucleic Acid System. (I).

Off-resonace irradiation effects were investigated using Bloch formalism, spin temperature and density matrix formalism. 1st model is suitable for the 1D off-resonance rotating frame experiments. 2nd model is important if dipolar order contributes to the relaxation process. 3rd model is used in the off-resonance ROESY (O-ROESY) experiment for the assessment of the internal motions of specific internuclear vectors.

Off-Resonance Rotating-Frame Nuclear Overhauser Effect Spectroscopy

The incorporation of off-resonance spin-lock effects into the ROESY experiment enables the evaluation of nuclear motion and internuclear distances in macromolecules by employing the functional dependence of the cross-peak intensity on off-resonance spin-lock frequency. The salient difference between the off-resonance ROESY and the ROESY experiments is that the on-resonance spin lock was replaced by an off-resonance spin lock applied at frequency νoff and at field strength B2 delivered in the form of a shifted laminar pulse. Variation of νoff at constant B2 creates a suite of two-dimensional spectra with frequency-dependent dispersion behavior in the cross-peak intensity established by the interacting spin-12 nuclei. Simulations are presented describing the functional dependence of cross-peak intensity dispersion behavior on macromolecular parameters. This experiment, designated O-ROESY, was implemented in compensated form and yields an amplitude-corrected cross-peak intensity dispersion curve with a smoothly increasing NOESY contribution, at the expense of ROESY, as the off-resonance spin-lock frequency is increased. Application of the O-ROESY experiment was made to lysozyme, demonstrating the feasibility of obtaining the correlation time of specific internuclear vectors, the internuclear vector order parameter, the correlation time of overall reorientational motion, and the relative internuclear distance, from an analysis of O-ROESY cross-peak intensity dispersion curve behavior for several Cα proton pairs in the stable β-strand region of the protein.

A Test of the Model-Free Formulas. Effects of Anisotropic Rotational Diffusion and Dimerization

The effects of anisotropy of rotational diffusion and extent of dimerization on the performance of the simple and extended model-free formulas are investigated. Numerically exact 15N NMR R 1, R 2, and NOE data are simulated for cylindrically symmetric species and also for mixtures of spherical monomers and dimers with the same internal dynamics. The relevant internuclear vectors are assumed to move in isotropic deflection potentials fixed at different orientations in the molecule and to exhibit a single relaxation time in their internal correlation functions. The simple model-free formula is fitted to these simulated data in order to obtain the best-fit order parameter and internal relaxation time for each nucleus. Fitting is accomplished by the standard data-analysis protocol, in which a single common global correlation time is adjusted, and also by an alternative protocol, in which the global correlation time is adjusted separately for each nucleus. The extended model-free formula is likewise fitted to these same data. With noise-free data, the simple model-free formula and standard protocol yield remarkably good best-fit internal motion parameters up to moderate anisotropies ( r = D ||/ D ⊥ = 2.0), but some or many of the NMR relaxation data are not fitted well even for quite modest anisotropies ( r = 1.3). The extended model-free formula yields an improved fit to the NMR data, but predicts substantial amplitudes of nonexistent slow internal motions (τ ≳ 0.2 ns) for many of the nuclei for all r ≥ 1.3. The simple model-free formula with the alternate protocol yields even better internal motion parameters than the standard protocol and also an excellent fit to the NMR relaxation data. The best-fit global correlation time for each nucleus corresponds very closely to the theoretical correlation time defined herein. Knowledge of these times would allow one not only to estimate the anisotropy of diffusion but also in favorable cases to infer the existence of slow internal motions. Inclusion of typical statistical errors in R 1, R 2, and NOE, or modest exchange contributions to R 2, seriously degrades the performance of the simple model-free formula with either protocol, especially in regard to the internal relaxation times, which can exhibit very large deviations from their input value even for spherical diffusers. When the fraction of monomers existing as dimers lies in the range 0.1 ≤ ƒ d ≤ 0.8, none of the three model-free approaches tested yields reliable internal motion parameters.

NMR relaxation and molecular reorientation over unequal potential barriers

Modulation of the dipolar Hamiltonian owing to complex motion of the internuclear vector which undergoes an isotropic overall motion associated with jumps between two or three potential minima of various shapes has been considered and general equations describing nuclear magnetic relaxation have been derived. The calculations take into account the change in the internuclear vector length during the internal motion. If changes in the internuclear vector take place internal motion of a frequency lower than the frequency of the overall motion can affect the relaxation rate significantly. There is evidence of splitting of the minimum of the function In T 1 versus the inverse of the temperature due to two modes of motion between three inequivalent minima.

The amplitude of local angular motion of purines in DNA in solution.

Nuclear magnetic resonance and optical experiments are combined to determine the rms amplitude of local angular motion of purines in DNA in solution. A 12 base-pair duplex DNA with the sequence d(CGCGAATTCGCG)2 is deuterated at the H8 positions of adenine and guanine by exchange with solvent at 55 degrees C. The deuterium nmr spectrum of this DNA is measured at 30 mg/mL at 30 degrees C in an 11.76 Tesla magnet (76.75 MHz). The time-resolved fluorescence polarization anisotropies (FPA) of this same sample and also a greatly diluted sample (0.215 mg/mL) were measured after addition of ethidium. FPA measurements of the dilute sample yield the hydrodynamic radius, RH = 9.94 +/- 0.2 A, while those at the nmr concentration are employed to characterize the collective motions in terms of either an enhanced viscosity or dimer formation. The rms amplitude of local angular motion was determined by analyzing the 2H-nmr spectrum, in particular the line width, using recently developed theory for the transverse relaxation rate (RQ2) together with essential information about the collective motions from these and other optical studies. When the principal-axis frame of the electric field gradient tensor is assumed to undergo overdamped libration around each of its three body-fixed axes in an isotropic deflection potential, then the rms amplitude of local angular motion around any single axis is found to lie in the range 10 degrees-11 degrees, provided the high DNA concentration acts to enhance the viscosity, and is about 9 degrees-11 degrees, if it acts to produce end-to-end dimers. The proton nmr relaxation data of Eimer et al. are reanalyzed and shown to yield an rms amplitude of angular motion of the cytosine H5-H6 internuclear vector of 9 degrees-10 degrees, depending upon its orientation with respect to the helix axis. In all of these analyses, full account is taken of the collective twisting and bending deformations, which have a small but significant effect on the results. It is shown that the rms amplitudes of local angular motion do not depend strongly on the model (potential), provided that isotropic rotation around the same number of axes is allowed and that one compares rms angles of the same dimensionality. The rms amplitudes of local angular motion in solution are comparable to those observed for the same sequence at low levels of hydration in the solid state.

Renormalized multiple-scattering theory of photoelectron diffraction.

The current multiple-scattering cluster techniques for the calculation of x-ray photoelectron and Auger-electron diffraction patterns consume much computer time in the intermediate-energy range (200--1000 eV); in fact, because of the large value of the electron mean free path and of the large forward-scattering amplitude at such energies, the electron samples a relatively large portion of the crystal, so that the number of paths to be considered becomes dramatically high. An alternative method is developed in the present paper: instead of calculating the individual contribution from each single path, the scattering matrix of each plane parallel to the surface is calculated with a renormalization process that calculates every scattering event in the plane up to infinite order. Similarly the scattering between two planes is calculated up to infinite order, and the double-plane scattering matrix is introduced. The process may then be applied to the calculation of a larger set of atomic layers. The advantage of the method is that a relatively small number of internuclear vectors have been used to obtain convergence in the calculation.

Ethanol Molecular Dynamics Measured by Coupled Spin Relaxation Exhibiting Cross Correlation between Dipole-Dipole and Chemical-Shift Anisotropy

The motional dynamics of ethanol in chloroform are studied by carbon- 13 multiplet spin-lattice relaxation over a wide range of temperatures. The coupled methylene ( 13CH 2) spin system is perturbed from thermal equilibrium using a variety of nonselective and selective pulse techniques. Simultaneous fitting of the resulting partially relaxed transition intensities with a model derived from Redfield theory allows the extraction of both auto-and cross-correlated dipolar spectral densities, from which the three principal elements of a diffusion tensor may be determined. The cross correlation between the chemical-shift anisotropy, CSA, and the dipolar, D, interactions is found to be appreciable. This cross term couples together magnetization modes with different spin-inversion symmetries. Inclusion of the D-CSA interference terms in the model improves the precision of the measurements of dipolar spectral densities, and the determination of the D-CSA interference term itself provides an estimate of the anisotropy of shielding tensors. It is found that the diffusion of ethanol can be described by an axially symmetric diffusion tensor with its unique axis about 5° from the C-O internuclear vector. Calculation of the D-CSA interference terms for an asymmetric shielding tensor under anisotropic diffusion is reported. The experiments were conducted using CD 3CH 2OD so that interference from intramolecular interactions with the adjacent protons is minimized. When other isotopomers of ethanol, CD 3CH 2OH, CH 3CH 2OD, and CH 3CH 2OH, were used, measurably different relaxation was observed. However, the experimental results in all cases can be analyzed by grouping these other interactions into a random-field interaction term. The determined dipolar spectral densities and D-CSA interference terms for each isotopomer are within experimental error of the values obtained for CD 3CH 2OD. As the adjacent protons only minimally affect the evaluation of anisotropic diffusion, this work supports the extension of this technique to systems where more complex isotopic enrichment schemes are not practical.

Computationally efficient gradients for relaxation matrix-based structure refinement including the accommodation of internal motions.

A general method for deriving analytical gradients based on NOESY cross-peak intensities is presented. This method allows for very rapid calculation of exact gradients of cross-peak intensities with respect to parameters directly related to the pairwise dipole-dipole interaction giving rise to the cross peak. In the simplest case, gradients with respect to internuclear separation can be calculated, which allows for the rational modification of distance constraints used in structural refinement. In the general case, any arbitrary level of knowledge of the internal dynamics of the molecule can be introduced, thereby providing a pathway for the experimental determination of motional parameters. The motional characteristics of the internuclear vectors defining dipole-dipole interactions are cast in model-free terms. The form of the gradient circumvents many of the limitations of gradients expressed in terms of Cartesian or dihedral variables. The gradients presented are simple, direct, and exact and require little computational effort to calculate.

A simple theoretical treatment of quadrupolar effects on magic-angle-spinning solid-state NMR spectra of [formula omitted] nuclei in the limit of large quadrupole coupling constants

A simple approach is discussed for studying the effect of quadrupolar nuclei on the magic-angle-spinning solid-state NMR lines of spin- 1 2 nuclei in the limit of large quadrupole coupling constants. Equations are derived both for the isotropic shifts and the Pake-like powder patterns for any quadrupolar spin and for arbitrary orientations of the internuclear vector with respect to the unique axis of an axially symmetric quadrupole tensor. First-order effects due to a small Zeeman perturbation on these lines are explored, as well as deviations from axial symmetry in the electric field gradient when S = 3 2 quadrupolar nuclei are involved. Spectral parameters likely to be observed in the case of coupling between 31P and 201Hg are also discussed.