PISEMA powder patterns and PISA wheels

Resonance patterns observed in 2D PISEMA (polarization inversion spin exchange at magic angle) spectra from a transmembrane alpha-helix have been demonstrated to yield structural details of the protein. This paper presents a mathematical discussion of the PISEMA powder spectrum as the image in the frequency plane of a quadratic function from the sphere of unit vectors. The simplicity of this function allows easy calculation of the powder spectrum. Based on this analysis of powder patterns, four degeneracies are discussed which arise in determining possible orientations associated with PISA spectra. This paper also gives parametric equations for PISA wheels, which are specific patterns observed in PISEMA spectra of oriented peptides. These wheels are useful both in assigning the resonances and in determining the orientation of the helix with respect to the magnetic field. The union of these PISA wheels gives the entire powder spectrum.     

Structural fitting of PISEMA spectra of aligned proteins

An algorithm for fitting protein structures to PISEMA spectra is described, and its application to helical proteins in aligned samples is demonstrated using both simulated and experimental results. The formulation of the algorithm in terms of rotation operators yields compact recursion relations that provide a fast and effective way of obtaining peptide plane orientations from chemical and torsion angle constraints. The algorithm in combination with experimental solid-state NMR data results in a method for determining the backbone structures of proteins, since it yields the orientation of a helix as a whole, including its tilt and twist angles, and describes kinks and curves with atomic resolution. Although the algorithm can be applied in an "assignment-free" manner to spectra of uniformly labeled proteins, the precision of the structural fitting is improved by the addition of assignment information, for example the identification of resonances by residue type from spectra of selectively labeled proteins.     

14N Polarization Inversion Spin Exchange at Magic Angle (PISEMA)

Polarization Inversion Spin Exchange at Magic Angle (PISEMA) is a powerful experiment for determining peptide orientation in uniformly aligned samples such as planar membranes. In this paper, we present (14)N-PISEMA experiment which correlates (14)N quadrupolar coupling and (14)N-(1)H dipolar coupling. (14)N-PISEMA enables the use of (14)N quadrupolar coupling tensor as an ultra sensitive probe for peptide orientation and can be carried out without the need of isotope enrichment. The experiment is based on selective spin-exchange between a proton and a single-quantum transition of (14)N spins. The spin-exchange dynamics is described and the experiment is demonstrated with a natural abundant N-acetyl valine crystal sample.     

Magnitudes and orientations of the 15N chemical shift tensor of [1-15N]-2'-deoxyguanosine determined on a polycrystalline sample by two-dimensional solid-state NMR spectroscopy.

The magnitudes and orientations of the 15N chemical shift tensor of [1-15N]-2'-deoxyguanosine were determined from a polycrystalline sample using the two-dimensional PISEMA experiment. The magnitudes of the principal values of the 15N chemical shift tensor of the N1 nitrogen of [1-15N]-2'-deoxyguanosine were found to be sigma11 = 54 ppm, sigma22 = 148 ppm, and sigma33 = 201 ppm with respect to (15NH4)2SO4 in aqueous solution. Comparisons of experimental and simulated two-dimensional powder pattern spectra show that sigma33N is approximately collinear with the N-H bond. The tensor orientation of sigma33N for N1 of [1-15N]-2'-deoxyguanosine is similar to the values obtained for the side chain residues of 15Nepsilon1-tryptophan and 15Npi-histidine even though the magnitudes differ significantly.     

Determining a helical protein structure using peptide pixels

The residual dipolar coupling-periodicity planarity correlation makes it possible to determine peptide plane orientations in regular periodic protein secondary structure elements. Each peptide plane orientation represents a "pixel" of protein structure, and is expressed in terms of three angles referred to as tilt, phase, and pitch angles. In this report, we present the novel "3P" (periodicity, planarity, and pixels) method that allows one to determine secondary and tertiary structure of alpha-helical proteins. We demonstrate the 3P method by determining the structure of domain 1 of the receptor-associated protein (RAP) to a backbone accuracy of 1.0 Angstrom using RDCs measured in a single alignment medium, together with a minimal number of NOE distance restraints, using a new Xplor-NIH module.     

NMR in rapidly rotating samples of nematic liquid crystals

By rapidly rotating a bulk nematic sample in a magnetic field it is possible to obtain a stable configuration where the directors all lie in the plane of rotation. A random distribution of orientations in the plane of rotation gives rise to a unique NMR “powder pattern”. The use of this technique in identifying spin—lattice relaxation mechanisms as well as its possible use in observing biaxial molecular order is given.     

A rapid algorithm for simulating motionally averaged powder patterns

An efficient algorithm is presented for calculating motionally averaged powder patterns for two- or three-site exchange. It reduces the required CPU time by constructing a matrix which describes the change in chemical shift with exchange for all orientations to the field. The powder pattern is then calculated from this matrix via Bloch's equations including exchange. This approach allows the mean chemical shift of the species undergoing exchange to be eliminated from the calculation, which is then only dependent on the chemical-shift difference between the species, and the exchange rate. The application of the algorithm to motional averaging by molecular jump reorientations is described in detail.     

2₄-SEMA as a sensitive and offset compensated SLF sequence

Separated Local Field (SLF) spectroscopy is a powerful tool for the determination of structure and dynamics of oriented systems such as membrane proteins oriented in lipid bilayers and liquid crystals. Of many SLF techniques available, Polarization Inversion Spin Exchange at Magic Angle (PISEMA) has found wide application due to its many favorable characteristics. However the pulse sequence suffers from its sensitivity to proton resonance frequency offset. Recently we have proposed a new sequence named 2(4)-SEMA (J. Chem. Phys. 132 (2010) 134301) that overcomes this problem of PISEMA. The present work demonstrates the advantage of 2(4)-SEMA as a highly sensitive SLF technique even for very large proton offset. 2(4)-SEMA has been designed for obtaining reliable dipolar couplings by switching the magic-angle spin-lock for protons over four quadrants as against the use of only two quadrants in PISEMA. It is observed that for on-resonance condition, 2(4)-SEMA gives rise to signal intensity comparable to or slightly higher than that from PISEMA. But under off-resonance conditions, intensities from 2(4)-SEMA are several fold higher than those from PISEMA. Comparison with another offset compensated pulse sequence, SAMPI4, also indicates a better intensity profile for 2(4)-SEMA. Experiments carried out on a single crystal of (15)N labeled N-acetyl-dl-valine and simulations have been used to study the relative performance of the pulse sequences considered.     

Spin dynamics of polarization inversion spin exchange at the magic angle in multiple spin systems

Polarization inversion spin exchange at the magic angle (PISEMA) [J. Magn. Reson. A 109, 270 (1994)] is an important experiment in NMR structural characterization of membrane proteins in oriented lipid bilayers. This paper presents a theoretical and experimental study of the spin dynamics in PISEMA to investigate the line-narrowing mechanism. The study focuses on the effect of neighboring protons on the spin exchange of a strongly coupled spin pair. The spin exchange is solved analytically for simple spin systems and is numerically simulated for many-spin systems. The results show that the dipolar couplings from the neighboring protons of a strongly coupled spin pair perturb the spin exchange only in the second order, therefore it has little contribution to the linewidth of PISEMA spectra in comparison to the separated-local-field spectra. The effects from proton resonance offset and the mismatch of the Hartmann-Hahn condition are also discussed along with experimental results using model single-crystal samples.     

A distributed-mass approach for dynamic analysis of Timoshenko plane frames

The dynamic stiffness method is the exact method for the dynamic analysis of plane frames using the continuous-coordinate system to consider the effect of mass distribution in beam elements. The dynamic stiffness method may create some null modes where the joints of beam element have null deformation. Unlike the Bernoulli–Euler frames, adding an interior node at the middle of the beam elements cannot normalize all the null modes of flexural vibration in the Timoshenko frames. The floating interior-node scheme is proposed to eliminate the null modes of flexural vibration in the Timoshenko frames. Orthogonal properties of vibration modes in Timoshenko plane frames are theoretically derived, through which the equations of motion in beam elements can be transformed into the decoupled equations of motion in terms of mode amplitudes.     

Periodicity, planarity, residual dipolar coupling, and structures

The periodic behavior of residual dipolar couplings (RDCs) arising from nucleic acid and protein secondary structures is shown to be more complex and information-rich than previously believed. We have developed a theoretical framework which allows the bond vector orientation of nucleic acids and the peptide plane orientations of protein secondary structures to be extracted from their Dipolar waves. In this article, we focus on utilizing "Dipolar waves" of peptides to extract structure information, and describe in more detail the fundamental principles of the relationship between the periodicities in structure and RDCs, the practical procedure to extract peptide plane orientation information from RDC data, and assessment of errors using Monte-Carlo simulations. We demonstrate the utility of our method for two model alpha-helices, one kinked and one curved, and as well as an irregular beta-strand.     

Three-dimensional solid-state NMR spectroscopy is essential for resolution of resonances from in-plane residues in uniformly (15)N-labeled helical membrane proteins in oriented lipid bilayers

Uniformly (15)N-labeled samples of membrane proteins with helices aligned parallel to the membrane surface give two-dimensional PISEMA spectra that are highly overlapped due to limited dispersions of (1)H-(15)N dipolar coupling and (15)N chemical shift frequencies. However, resolution is greatly improved in three-dimensional (1)H chemical shift/(1)H-(15)N dipolar coupling/(15)N chemical shift correlation spectra. The 23-residue antibiotic peptide magainin and a 54-residue polypeptide corresponding to the cytoplasmic domain of the HIV-1 accessory protein Vpu are used as examples. Both polypeptides consist almost entirely of alpha-helices, with their axes aligned parallel to the membrane surface. The measurement of three orientationally dependent frequencies for Val17 of magainin enabled the three-dimensional orientation of this helical peptide to be determined in the lipid bilayer.     

On the powder diffraction pattern of crystals with stacking faults

The so-called direct solution of the powder diffraction pattern for a faulted layered crystal is discussed. It is shown how, in the general case, peak profiles can be split into a symmetric and an antisymmetric component. The relationships between peak profile parameters and the underlying faulting structure, as given by the probability correlation function, are evidenced. The formalism reduces to known equations when applied to particular faulting models. Warren's equations for peak profile of fcc materials with {111} planar faulting are derived within the framework of a general theory. Possible strategies for incorporating the proposed formalism into a general powder pattern refinement procedure are also discussed.     

Dimers Belonging to Three Orientations on Plane Honeycomb Lattices

It is well known that there are three types of dimers belonging to the three different orientations in a honeycomb lattice, and in each type all dimers are mutually parallel. Based on a previous result, we can compute the partition function of the dimer problem of the plane (free boundary) honeycomb lattices with three different activities by using the number of its pure dimer coverings (perfect matchings). The explicit expression of the partition function and free energy per dimer for many types of plane honeycomb lattices with fixed shape of boundaries is obtained in this way (for a shape of plane honeycomb lattices, the procedure that the size goes to infinite, corresponds to a way that the honeycomb lattice goes to infinite). From these results, an interesting phenomena is observed. In the case of the regions of the plane honeycomb lattice has zero entropy per dimer—when its size goes to infinite—though in the thermodynamic limit, there is no freedom in placing a dimer at all, but if we distinguish three types of dimers with nonzero activities, then its free energy per dimer is nonzero. Furthermore, a sufficient condition for the plane honeycomb lattice with zero entropy per dimer (when the three activities are equal to 1) is obtained. Finally, the difference between the plane honeycomb lattices and the plane quadratic lattices is discussed and a related problem is proposed.     

An Alternative Algorithm for the Symmetry Classification of Ordinary Differential Equations

This is the first paper on symmetry classification for ordinary differential equations (ODEs) based on Wu’s method. We carry out symmetry classification of two ODEs, named the generalizations of the Kummer-Schwarz equations which involving arbitrary function. First, Lie algorithm is used to give the determining equations of symmetry for the given equations, which involving arbitrary functions. Next, differential form Wu’s method is used to decompose determining equations into a union of a series of zero sets of differential characteristic sets, which are easy to be solved relatively. Each branch of the decomposition yields a class of symmetries and associated parameters. The algorithm makes the classification become direct and systematic. Yuri Dimitrov Bozhkov, and Pammela Ramos da Conceição have used the Lie algorithm to give the symmetry classifications of the equations talked in this paper in 2020. From this paper, we can find that the differential form Wu’s method for symmetry classification of ODEs with arbitrary function (parameter) is effective, and is an alternative method.     

Using low-E resonators to reduce RF heating in biological samples for static solid-state NMR up to 900 MHz

RF heating of solid-state biological samples is known to be a destabilizing factor in high-field NMR experiments that shortens the sample lifetime by continuous dehydration during the high-power cross-polarization and decoupling pulses. In this work, we describe specially designed, large volume, low-E 15N-1H solid-state NMR probes developed for 600 and 900 MHz PISEMA studies of dilute membrane proteins oriented in hydrated and dielectrically lossy lipid bilayers. The probes use an orthogonal coil design in which separate resonators pursue their own aims at the respective frequencies, resulting in a simplified and more efficient matching network. Sample heating at the 1H frequency is minimized by a loop-gap resonator which produces a homogeneous magnetic field B1 with low electric field E. Within the loop-gap resonator, a multi-turn solenoid closely matching the shape of the sample serves as an efficient observe coil. We compare power dissipation in a typical lossy bilayer sample in the new low-E probe and in a previously reported 15N-1H probe which uses a double-tuned 4-turn solenoid. RF loss in the sample is measured in each probe by observing changes in the 1H 360 degrees pulse lengths. For the same values of 1H B1 field, sample heating in the new probe was found to be smaller by an order of magnitude. Applications of the low-E design to the PISEMA study of membrane proteins in their native hydrated bilayer environment are demonstrated at 600 and 900 MHz.     

Imaging the cone state of the spin reorientation transition

The magnetization behavior and the domain pattern in remanence are studied in Co/Pt multilayers. The reorientation of magnetization from perpendicular to in plane is found to happen via the state of canted magnetization. In the transition from an easy axis to an easy plane a stable domain pattern in the in-plane magnetization components is found for Co/Pt multilayers. The analysis of the domain pattern reveals that the magnetization canting is such that all in-plane orientations of magnetization are equally occupied. The found structure is appointed to the cone state.     

Rotation axis calibration of a turntable using constrained global optimization

For determining the rotation axis as the baseline to transform between camera and turntable coordinate frames, rotation axis calibration of a turntable using constrained global optimization is proposed. At the beginning, camera parameters are calibrated using multiple views of only one pattern positioned differently and 3D corner points are retrieved with the pattern placed onto the rotation turntable in a 360° view. Then the rotation axis direction vector is decided under the constraint of the distance between each closed circle trajectory plane and each circle with its center is determined by fitting the circle with global least squares method, respectively. Experimental results demonstrate that the proposed method is effective for estimating the parameters of rotation axis.