Analytical solution to the coupled evolution of multidimensional NMR data

A substantial time savings in the collection of multidimensional NMR data can be achieved by coupling the evolution of nuclei in the indirect dimensions. In order to save time, the sampling of the indirect dimensions is inherently incomplete. Therefore, many algorithms and samplings schemes have been developed aimed at separating the coevolved frequencies into analyzable data with limited artifacts. This paper extends the use of circulant matrices to describe coupled evolution with convolutions. By understanding the data in terms of convolutions, there is an exact solution to the inversion problem of extracting the orthogonal vectors from the coupled dimensions. Previously, this inversion problem has been solved using peak coordinates extracted from spectra. In contrast, the method described here uses spectra directly. This solution suggests a simple sampling scheme of collecting N orthogonal spectra, and N + 1 projections at specific projection angles, however, the theory developed can be extended generally to arbitrary projection angles. The circulant matrix methodology is demonstrated for simulated and real data. Further, an algorithm for separating overlapped signals in the detected dimension is presented. The algorithm involves the forward calculation of the coupled spectra from the orthogonal spectra, followed by back calculation of the orthogonal spectra from the coupled spectra, thus permitting rigorous cross-validation. This algorithm is shown to be robust in that erroneous solutions give rise to large artifacts. Electronic supplementary material  The online version of this article (doi:) contains supplementary material, which is available to authorized users.     

Application of iterative soft thresholding for fast reconstruction of NMR data non-uniformly sampled with multidimensional Poisson Gap scheduling

The fast Fourier transformation has been the gold standard for transforming data from time to frequency domain in many spectroscopic methods, including NMR. While reliable, it has as a drawback that it requires a grid of uniformly sampled data points. This needs very long measuring times for sampling in multidimensional experiments in all indirect dimensions uniformly and even does not allow reaching optimal evolution times that would match the resolution power of modern high-field instruments. Thus, many alternative sampling and transformation schemes have been proposed. Their common challenges are the suppression of the artifacts due to the non-uniformity of the sampling schedules, the preservation of the relative signal amplitudes, and the computing time needed for spectra reconstruction. Here we present a fast implementation of the Iterative Soft Thresholding approach (istHMS) that can reconstruct high-resolution non-uniformly sampled NMR data up to four dimensions within a few hours and make routine reconstruction of high-resolution NUS 3D and 4D spectra convenient. We include a graphical user interface for generating sampling schedules with the Poisson-Gap method and an estimation of optimal evolution times based on molecular properties. The performance of the approach is demonstrated with the reconstruction of non-uniformly sampled medium and high-resolution 3D and 4D protein spectra acquired with sampling densities as low as 0.8%. The method presented here facilitates acquisition, reconstruction and use of multidimensional NMR spectra at otherwise unreachable spectral resolution in indirect dimensions.     

High resolution 4-D spectroscopy with sparse concentric shell sampling and FFT-CLEAN

Recent efforts to reduce the measurement time for multidimensional NMR experiments have fostered the development of a variety of new procedures for sampling and data processing. We recently described concentric ring sampling for 3-D NMR experiments, which is superior to radial sampling as input for processing by a multidimensional discrete Fourier transform. Here, we report the extension of this approach to 4-D spectroscopy as Randomized Concentric Shell Sampling (RCSS), where sampling points for the indirect dimensions are positioned on concentric shells, and where random rotations in the angular space are used to avoid coherent artifacts. With simulations, we show that RCSS produces a very low level of artifacts, even with a very limited number of sampling points. The RCSS sampling patterns can be adapted to fine rectangular grids to permit use of the Fast Fourier Transform in data processing, without an apparent increase in the artifact level. These artifacts can be further reduced to the noise level using the iterative CLEAN algorithm developed in radioastronomy. We demonstrate these methods on the high resolution 4-D HCCH-TOCSY spectrum of protein G's B1 domain, using only 1.2% of the sampling that would be needed conventionally for this resolution. The use of a multidimensional FFT instead of the slow DFT for initial data processing and for subsequent CLEAN significantly reduces the calculation time, yielding an artifact level that is on par with the level of the true spectral noise.     

FM reconstruction of non-uniformly sampled protein NMR data at higher dimensions and optimization by distillation

Non-uniform sampling (NUS) enables recording of multidimensional NMR data at resolutions matching the resolving power of modern instruments without using excessive measuring time. However, in order to obtain satisfying results, efficient reconstruction methods are needed. Here we describe an optimized version of the Forward Maximum entropy (FM) reconstruction method, which can reconstruct up to three indirect dimensions. For complex datasets, such as NOESY spectra, the performance of the procedure is enhanced by a distillation procedure that reduces artifacts stemming from intense peaks.     

NMR experiments for the rapid identification of P=O···H–X type hydrogen bonds in nucleic acids

Implementation of a new algorithm, SMILE, is described for reconstruction of non-uniformly sampled two-, three- and four-dimensional NMR data, which takes advantage of the known phases of the NMR spectrum and the exponential decay of underlying time domain signals. The method is very robust with respect to the chosen sampling protocol and, in its default mode, also extends the truncated time domain signals by a modest amount of non-sampled zeros. SMILE can likewise be used to extend conventional uniformly sampled data, as an effective multidimensional alternative to linear prediction. The program is provided as a plug-in to the widely used NMRPipe software suite, and can be used with default parameters for mainstream application, or with user control over the iterative process to possibly further improve reconstruction quality and to lower the demand on computational resources. For large data sets, the method is robust and demonstrated for sparsities down to ca 1%, and final all-real spectral sizes as large as 300 Gb. Comparison between fully sampled, conventionally processed spectra and randomly selected NUS subsets of this data shows that the reconstruction quality approaches the theoretical limit in terms of peak position fidelity and intensity. SMILE essentially removes the noise-like appearance associated with the point-spread function of signals that are a default of five-fold above the noise level, but impacts the actual thermal noise in the NMR spectra only minimally. Therefore, the appearance and interpretation of SMILE-reconstructed spectra is very similar to that of fully sampled spectra generated by Fourier transformation.     

Application of nonlinear sampling schemes to COSY-type spectra

Summary Nonlinear sampling along the t₁ dimension is applied to COSY-type spectra. The sine dependence of the time domain signals for the cross peaks is matched by a nonlinear sampling scheme that samples most densely around the maximum of the sine function. Data are processed by maximum entropy reconstruction, using a modified implementation of the Cambridge algorithm of Skilling and Bryan. The procedure is demonstrated for P.E.COSY spectra recorded on a cyclic hexapeptide and on a 126-residue domain of the protein villin. The number of t₁ values in the nonlinearly sampled experiments was reduced by a factor of four compared to linear sampling. The sensitivity and resolution of the resulting spectra are comparable to those achieved by conventional methods. The method described can thus significantly reduce the measuring time for COSY-type spectra.Abbreviations COSY two-dimensional correlation spectroscopy - E.COSY exclusive COSY - P.E.COSY primitive E.COSY - TPPI time proportional phase incrementation - FID free induction decay To whom correspondence should be addressed.     

NMR spectroscopy reveals unexpected structural variation at the protein–protein interface in MHC class I molecules

It is well established that non-uniform sampling (NUS) allows acquisition of multi-dimensional NMR spectra at a resolution that cannot be obtained with traditional uniform acquisition through the indirect dimensions. However, the impact of NUS on the signal-to-noise ratio (SNR) and sensitivity are less well documented. SNR and sensitivity are essential aspects of NMR experiments as they define the quality and extent of data that can be obtained. This is particularly important for spectroscopy with low concentration samples of biological macromolecules. There are different ways of defining the SNR depending on how to measure the noise, and the distinction between SNR and sensitivity is often not clear. While there are defined procedures for measuring sensitivity with high concentration NMR standards, such as sucrose, there is no clear or generally accepted definition of sensitivity when comparing different acquisition and processing methods for spectra of biological macromolecules with many weak signals close to the level of noise. Here we propose tools for estimating the SNR and sensitivity of NUS spectra with respect to sampling schedule and reconstruction method. We compare uniformly acquired spectra with NUS spectra obtained in the same total measuring time. The time saving obtained when only 1/k of the Nyquist grid points are sampled is used to measure k-fold more scans per increment. We show that judiciously chosen NUS schedules together with suitable reconstruction methods can yield a significant increase of the SNR within the same total measurement time. Furthermore, we propose to define the sensitivity as the probability to detect weak peaks and show that time-equivalent NUS can considerably increase this detection sensitivity. The sensitivity gain increases with the number of NUS indirect dimensions. Thus, well-chosen NUS schedules and reconstruction methods can significantly increase the information content of multidimensional NMR spectra of challenging biological macromolecules.     

CONNJUR Workflow Builder: a software integration environment for spectral reconstruction

CONNJUR Workflow Builder (WB) is an open-source software integration environment that leverages existing spectral reconstruction tools to create a synergistic, coherent platform for converting biomolecular NMR data from the time domain to the frequency domain. WB provides data integration of primary data and metadata using a relational database, and includes a library of pre-built workflows for processing time domain data. WB simplifies maximum entropy reconstruction, facilitating the processing of non-uniformly sampled time domain data. As will be shown in the paper, the unique features of WB provide it with novel abilities to enhance the quality, accuracy, and fidelity of the spectral reconstruction process. WB also provides features which promote collaboration, education, parameterization, and non-uniform data sets along with processing integrated with the Rowland NMR Toolkit (RNMRTK) and NMRPipe software packages. WB is available free of charge in perpetuity, dual-licensed under the MIT and GPL open source licenses.     

Fast acquisition of NMR spectra using Fourier transform of non-equispaced data

Rapid acquisition of high-resolution 2D and 3D NMR spectra is essential for studying biological macromolecules. In order to minimize the experimental time, a non-linear sampling scheme is proposed for the indirect dimensions of multidimensional experiments. These data can be processed using the algorithm proposed by Dutt and Rokhlin (Appl. Comp. Harm. Anal. 1995, 2, 85–100) for fast Fourier transforms of non equispaced data. Examples of ¹H−¹⁵N HSQC spectra are shown, where crowded correlation peaks can be resolved using non-linear acquisition. Simulated data have been used to analyze the artefacts produced by the Lagrange interpolation. As compared to non-linear processing methods, this algorithm is simple and highly robust since no parameters need to be adjusted by the user.     

Al NMR: a novel NMR data processing program optimized for sparse sampling

Sparse sampling in biomolecular multidimensional NMR offers increased acquisition speed and resolution and, if appropriate conditions are met, an increase in sensitivity. Sparse sampling of indirectly detected time domains combined with the direct truly multidimensional Fourier transform has elicited particular attention because of the ability to generate a final spectrum amenable to traditional analysis techniques. A number of sparse sampling schemes have been described including radial sampling, random sampling, concentric sampling and variations thereof. A fundamental feature of these sampling schemes is that the resulting time domain data array is not amenable to traditional Fourier transform based processing and phasing correction techniques. In addition, radial sampling approaches offer a number of advantages and capabilities that are also not accessible using standard NMR processing techniques. These include sensitivity enhancement, sub-matrix processing and determination of minimal sets of sampling angles. Here we describe a new software package (Al NMR) that enables these capabilities in the context of a general NMR data processing environment.     

Improved resolution in triple-resonance spectra by nonlinear sampling in the constant-time domain

Summary Nonlinear sampling along the constant-time dimension is applied to the constant-time HNCO spectrum of the dimerization domain of Ga14. Nonlinear sampling was used for the nitrogen dimension, while the carbon and proton dimensions were sampled linearly. A conventional ct-HNCO spectrum is compared with a nonlinearly sampled spectrum, where the gain in experiment time obtained from nonlinear sampling is used to increase the resolution in the carbonyl dimension. Nonlinearly sampled data are processed by maximum entropy reconstruction. It is shown that the nonlinearly sampled spectrum has a higher resolution, although it was recorded in less time. The constant intensity of the signal in the constant-time dimension allows for a variety of sampling schedules. A schedule of randomly distributed sampling points yields the best results. This general method can be used to significantly increase the quality of heteronuclear constant-time spectra.     

Rapid analysis of protein backbone resonance assignments using cryogenic probes,a distributed Linux-based computing architecture,and an integrated set of spectral analysis tools

Rapid data collection, spectral referencing, processing by time domain deconvolution, peak picking and editing, and assignment of NMR spectra are necessary components of any efficient integrated system for protein NMR structure analysis. We have developed a set of software tools designated AutoProc, AutoPeak, and AutoAssign, which function together with the data processing and peak-picking programs NMRPipe and Sparky, to provide an integrated software system for rapid analysis of protein backbone resonance assignments. In this paper we demonstrate that these tools, together with high-sensitivity triple resonance NMR cryoprobes for data collection and a Linux-based computer cluster architecture, can be combined to provide nearly complete backbone resonance assignments and secondary structures (based on chemical shift data) for a 59-residue protein in less than 30 hours of data collection and processing time. In this optimum case of a small protein providing excellent spectra, extensive backbone resonance assignments could also be obtained using less than 6 hours of data collection and processing time. These results demonstrate the feasibility of high throughput triple resonance NMR for determining resonance assignments and secondary structures of small proteins, and the potential for applying NMR in large scale structural proteomics projects.Abbreviations: BPTI – bovine pancreatic trypsin inhibitor; LP – linear prediction; FT – Fourier transform; S/N – signal-to-noise ratio; FID – free induction decay     

Interpolating and extrapolating with hmsIST: seeking a t<Subscript>max</Subscript> for optimal sensitivity,resolution and frequency accuracy

Non-Uniform Sampling has the potential to exploit the optimal resolution of high-field NMR instruments. This is not possible in 3D and 4D NMR experiments when using traditional uniform sampling due to the long overall measurement time. Nominally, uniformly sampled time domain data acquired to a maximum evolution time t_max can be extended to high resolution via a virtual maximum evolution time t*_max while extrapolating with linear prediction or iterative soft thresholding (IST). At the high resolution obtainable with extrapolation of US data, however, the accuracy of peak positions is compromised as observed when comparing inter- and intra-residue peaks in a 3D HNCA experiment. However, the accuracy of peak positions is largely improved by spreading the same number of acquired time domain data points non-uniformly over a larger evolution time to an optimal t_max followed by extrapolation to a total t*_max and processing the data with an appropriate reconstruction method, such as hmsIST. To explore the optimum value of experimentally measured t_max to be reached non-uniformly with a given number of sampling points we have created test situations of time-equivalent experiments and evaluate sensitivity and accuracy of peak positions. Here we use signal-to-maximum-noise ratio as the decisive measure of sensitivity. We find that both sensitivity and resolution are optimal when PoissonGap sampling to a t_max of about ½*T₂ ^*. Digital resolution is further enhanced by extrapolating the range of acquired time domain data to 2*T₂ ^* but without measuring experimental points beyond ½*T₂ ^*.     

PR-CALC: A Program for the Reconstruction of NMR Spectra from Projections

Projection-reconstruction NMR (PR-NMR) has attracted growing attention as a method for collecting multidimensional NMR data rapidly. The PR-NMR procedure involves measuring lower-dimensional projections of a higher-dimensional spectrum, which are then used for the mathematical reconstruction of the full spectrum. We describe here the program PR-CALC, for the reconstruction of NMR spectra from projection data. This program implements a number of reconstruction algorithms, highly optimized to achieve maximal performance, and manages the reconstruction process automatically, producing either full spectra or subsets, such as regions or slices, as requested. The ability to obtain subsets allows large spectra to be analyzed by reconstructing and examining only those subsets containing peaks, offering considerable savings in processing time and storage space. PR-CALC is straightforward to use, and integrates directly into the conventional pipeline for data processing and analysis. It was written in standard C+ + and should run on any platform. The organization is flexible, and permits easy extension of capabilities, as well as reuse in new software. PR-CALC should facilitate the widespread utilization of PR-NMR in biomedical research. Electronic supplementary material Electronic supplementary material is available for this article at and accessible for authorised users.     

Pitfalls in compressed sensing reconstruction and how to avoid them

Multidimensional NMR can provide unmatched spectral resolution, which is crucial when dealing with samples of biological macromolecules. The resolution, however, comes at the high price of long experimental time. Non-uniform sampling (NUS) of the evolution time domain allows to suppress this limitation by sampling only a small fraction of the data, but requires sophisticated algorithms to reconstruct omitted data points. A significant group of such algorithms known as compressed sensing (CS) is based on the assumption of sparsity of a reconstructed spectrum. Several papers on the application of CS in multidimensional NMR have been published in the last years, and the developed methods have been implemented in most spectral processing software. However, the publications rarely show the cases when NUS reconstruction does not work perfectly or explain how to solve the problem. On the other hand, every-day users of NUS develop their rules-of-thumb, which help to set up the processing in an optimal way, but often without a deeper insight. In this paper, we discuss several sources of problems faced in CS reconstructions: low sampling level, missassumption of spectral sparsity, wrong stopping criterion and attempts to extrapolate the signal too much. As an appendix, we provide MATLAB codes of several CS algorithms used in NMR. We hope that this work will explain the mechanism of NUS reconstructions and help readers to set up acquisition and processing parameters. Also, we believe that it might be helpful for algorithm developers.     

大窑遗址二道沟地点石制品研究的抽样方法设计

结合简单随机抽样、整群抽样和连续重复抽样的特点,基于中心极限定理,本文设计了适用于大批量石制品抽样调查的具体方案,通过边抽样、边评价的方式,利用有限的样本量推断总体特征,并将其实际应用到大窑遗址二道沟地点石制品的研究中,取得了良好的效果。此外,该抽样方案对于石制品的物源追溯、遗址区域调查等也有一定的实用价值。     

Reduction of Metal Artifact in Single Photon-Counting Computed Tomography by Spectral-Driven Iterative Reconstruction Technique

PurposeThe exciting prospect of Spectral CT (SCT) using photon-counting detectors (PCD) will lead to new techniques in computed tomography (CT) that take advantage of the additional spectral information provided. We introduce a method to reduce metal artifact in X-ray tomography by incorporating knowledge obtained from SCT into a statistical iterative reconstruction scheme. We call our method Spectral-driven Iterative Reconstruction (SPIR).MethodThe proposed algorithm consists of two main components: material decomposition and penalized maximum likelihood iterative reconstruction. In this study, the spectral data acquisitions with an energy-resolving PCD were simulated using a Monte-Carlo simulator based on EGSnrc C++ class library. A jaw phantom with a dental implant made of gold was used as an object in this study. A total of three dental implant shapes were simulated separately to test the influence of prior knowledge on the overall performance of the algorithm. The generated projection data was first decomposed into three basis functions: photoelectric absorption, Compton scattering and attenuation of gold. A pseudo-monochromatic sinogram was calculated and used as input in the reconstruction, while the spatial information of the gold implant was used as a prior. The results from the algorithm were assessed and benchmarked with state-of-the-art reconstruction methods.ResultsDecomposition results illustrate that gold implant of any shape can be distinguished from other components of the phantom. Additionally, the result from the penalized maximum likelihood iterative reconstruction shows that artifacts are significantly reduced in SPIR reconstructed slices in comparison to other known techniques, while at the same time details around the implant are preserved. Quantitatively, the SPIR algorithm best reflects the true attenuation value in comparison to other algorithms.ConclusionIt is demonstrated that the combination of the additional information from Spectral CT and statistical reconstruction can significantly improve image quality, especially streaking artifacts caused by the presence of materials with high atomic numbers.     

Tilted view reconstruction in optical microscopy. Three-dimensional reconstruction of Drosophila melanogaster embryo nuclei.

The resolution along the optical axis (z) is much less than the in-plane resolution in any current optical microscope, conventional or otherwise. We have used mutually tilted, through-focal section views of the same object to provide a solution to this problem. A tilting specimen stage was constructed for an optical microscope, which with the use of a coverslip-free water immersion lens, allowed the collection of data sets from intact Drosophila melanogaster embryos at viewing directions up to 90 degrees apart. We have devised an image processing scheme to determine the relative tilt, translation, and sampling parameters of the different data sets. This involves the use of a modified phase cross-correlation function, which produces a very sharp maximum. Finally the data sets are merged using figure-of-merit and local area scaling techniques borrowed from x-ray protein crystallography. We demonstrate the application of this technique to data sets from a metaphase plate in an embryo of Drosophila melanogaster. As expected, the merged reconstruction combined the highest resolution available in the individual data sets. As estimated from the Fourier transform, the final resolution is 0.25 microns in x and y and 0.4 microns in z. In the final reconstruction all ten chromosome arms can be easily delineated; this was not possible in the individual data sets. Within many of the arms the two individual chromatids can be seen. In some cases the chromatids are wrapped around each other helically, in others they lie alongside each other in a parallel arrangement.     

MR Image Reconstruction Using Block Matching and Adaptive Kernel Methods

An approach to Magnetic Resonance (MR) image reconstruction from undersampled data is proposed. Undersampling artifacts are removed using an iterative thresholding algorithm applied to nonlinearly transformed image block arrays. Each block array is transformed using kernel principal component analysis where the contribution of each image block to the transform depends in a nonlinear fashion on the distance to other image blocks. Elimination of undersampling artifacts is achieved by conventional principal component analysis in the nonlinear transform domain, projection onto the main components and back-mapping into the image domain. Iterative image reconstruction is performed by interleaving the proposed undersampling artifact removal step and gradient updates enforcing consistency with acquired k-space data. The algorithm is evaluated using retrospectively undersampled MR cardiac cine data and compared to k-t SPARSE-SENSE, block matching with spatial Fourier filtering and k-t ℓ₁-SPIRiT reconstruction. Evaluation of image quality and root-mean-squared-error (RMSE) reveal improved image reconstruction for up to 8-fold undersampled data with the proposed approach relative to k-t SPARSE-SENSE, block matching with spatial Fourier filtering and k-t ℓ₁-SPIRiT. In conclusion, block matching and kernel methods can be used for effective removal of undersampling artifacts in MR image reconstruction and outperform methods using standard compressed sensing and ℓ₁-regularized parallel imaging methods.     

3D Mapping of the SPRY2 domain of ryanodine receptor 1 by single-particle cryo-EM

The type 1 skeletal muscle ryanodine receptor (RyR1) is principally responsible for Ca(2+) release from the sarcoplasmic reticulum and for the subsequent muscle contraction. The RyR1 contains three SPRY domains. SPRY domains are generally known to mediate protein-protein interactions, however the location of the three SPRY domains in the 3D structure of the RyR1 is not known. Combining immunolabeling and single-particle cryo-electron microscopy we have mapped the SPRY2 domain (S1085-V1208) in the 3D structure of RyR1 using three different antibodies against the SPRY2 domain. Two obstacles for the image processing procedure; limited amount of data and signal dilution introduced by the multiple orientations of the antibody bound in the tetrameric RyR1, were overcome by modifying the 3D reconstruction scheme. This approach enabled us to ascertain that the three antibodies bind to the same region, to obtain a 3D reconstruction of RyR1 with the antibody bound, and to map SPRY2 to the periphery of the cytoplasmic domain of RyR1. We report here the first 3D localization of a SPRY2 domain in any known RyR isoform.