Ultra-high resolution 3D NMR spectra from limited-size data sets

The advantage of the filter diagonalization method (FDM) for analysis of triple-resonance NMR experiments is demonstrated by application to a 3D constant time (CT) HNCO experiment. With a 15N-,13C-labeled human ubiquitin sample (1.0 mM), high spectral resolution was obtained at 500 MHz in 25 min with only 6-8 increments in each of the CT dimensions. This data set size is about a factor of 50-100 smaller than typically required, yet FDM analysis results in a fully resolved spectrum with a sharp peak for each HNCO resonance. Unlike Fourier transform (FT) processing, in which spectral resolution in each dimension is inversely proportional to the acquisition time in this dimension, FDM is a true multi-dimensional method; the resolution in all dimensions is determined by the total information content of the entire signal. As the CT dimensions of the 3D HNCO signal have approximate time-reversal symmetry, they can each be doubled by combining the usual four hyper-complex data sets. This apparent quadrupling of the data is important to the success of the method. Thus, whenever raw sensitivity is not limiting, well-resolved n-dimensional spectra can now be obtained in a small fraction of the usual time. Alternatively, to maximize sensitivity, evolution periods of faster relaxing nuclei may be radically shortened, the total required resolution being obtained through chemical shift encoding of other, more slowly relaxing, spins. Improvements similar to those illustrated with a 3D HNCO spectrum are expected for other triple-resonance spectra, where CT evolution in the indirect dimensions is implemented.     

Phase-sensitive spectral estimation by the hybrid filter diagonalization method

A more robust way to obtain a high-resolution multidimensional NMR spectrum from limited data sets is described. The Filter Diagonalization Method (FDM) is used to analyze phase-modulated data and cast the spectrum in terms of phase-sensitive Lorentzian "phase-twist" peaks. These spectra are then used to obtain absorption-mode phase-sensitive spectra. In contrast to earlier implementations of multidimensional FDM, the absolute phase of the data need not be known beforehand, and linear phase corrections in each frequency dimension are possible, if they are required. Regularization is employed to improve the conditioning of the linear algebra problems that must be solved to obtain the spectral estimate. While regularization smoothes away noise and small peaks, a hybrid method allows the true noise floor to be correctly represented in the final result. Line shape transformation to a Gaussian-like shape improves the clarity of the spectra, and is achieved by a conventional Lorentzian-to-Gaussian transformation in the time-domain, after inverse Fourier transformation of the FDM spectra. The results obtained highlight the danger of not using proper phase-sensitive line shapes in the spectral estimate. The advantages of the new method for the spectral estimate are the following: (i) the spectrum can be phased by conventional means after it is obtained; (ii) there is a true and accurate noise floor; and (iii) there is some indication of the quality of fit in each local region of the spectrum. The method is illustrated with 2D NMR data for the first time, but is applicable to n-dimensional data without any restriction on the number of time/frequency dimensions.     

Rapid high-resolution four-dimensional NMR spectroscopy using the filter diagonalization method and its advantages for detailed structural elucidation of oligosaccharides

Four-dimensional nuclear magnetic resonance spectroscopy with high resolution of signals in the indirect dimensions is reported as an implementation of the filter diagonalization method (FDM). Using an oligosaccharide derivatized with 13C-labeled acetyl isotags, a four-dimensional constant-time pulse sequence was tailored for conjoint use with the FDM. Results demonstrate that high resolution in all dimensions can be achieved using a relatively short experimental time period (19 h), even though the spectrum is highly congested in the direct and all three indirect dimensions. The combined use of isotags, constant-time pulse sequences, and FDM permits rapid isolation of sugar ring proton spin systems in multiple dimensions and enables all endocyclic J-couplings to be simply measured, the key goal to assigning sugar stereochemistry and anomeric configuration. A general method for rapid, unambiguous elucidation of spin systems in oligosaccharides has been a long-sought goal of carbohydrate NMR, and isotags combined with the FDM now enable this to be easily performed. Additional general advantages of the FDM program for generating high-resolution 2D slices in any dimension from a 4D spectrum are emphasized.     

The multidimensional filter diagonalization method

The theory and numerical aspects of the recently developed multidimensional version of the filter diagonalization method (FDM) are described in detail. FDM can construct various "ersatz" or "hybrid" spectra from multidimensional time signals. Spectral resolution is not limited by the time-frequency uncertainty principle in each separate frequency dimension, but rather by the total joint information content of the signal, i.e., N(total) = N(1) x N(2) x vertical ellipsis x N(D), where some of the interferometric dimensions do not have to be represented by more than a few (e.g., two) time increments. It is shown that FDM can be used to compute various reduced-dimensionality projections of a high-dimensional spectrum directly, i.e., avoiding construction of the latter. A subsequent paper (J. Magn. Reson. 144, 357-366 (2000)) is concerned with applications of the method to 2D, 3D, and 4D NMR experiments.     

Rapid 3D NMR using the filter diagonalization method: application to oligosaccharides derivatized with 13C-labeled acetyl groups

Rapid 3D NMR spectroscopy of oligosaccharides having isotopically labeled acetyl "isotags" was made possible with high resolution in the indirect dimensions using the filter diagonalization method (FDM). A pulse sequence was designed for the optimal correlation of acetyl methyl protons, methyl carbons, and carbonyl carbons. The multi-dimensional nature of the FDM, coupled with the advantages of constant-time evolution periods, resulted in marked improvements over Fourier transform (FT) and mirror-image linear prediction (MI-LP) processing methods. The three methods were directly compared using identical data sets. A highly resolved 3D spectrum was achieved with the FDM using a very short experimental time (28 min).     

The Multidimensional Filter Diagonalization Method: II. Application to 2D Projections of 2D, 3D, and 4D NMR Experiments

The theory of the multidimensional filter diagonalization method (FDM) described in the previous paper (V. A. Mandelshtam, 2000, J. Magn. Reson. 144, 343–356 (2000)) is applied to NMR time signals with up to four independent time variables. Direct projections of the multidimensional time signals produce new kinds of 2D spectra. The resolution obtained by FDM can be far superior to that obtained by conventional phase-sensitive FT processing, and correlation peaks in heteronuclear and homonuclear experiments can be condensed to sharp singlets, removing all spin–spin couplings. Examples of singlet-HSQC and singlet-TOCSY spectra show big gains in resolution. It is not necessary to have a finely digitized spectrum, in which the individual multiplet components are resolved, for the methods to work. Examples of FDM spectra, ranging from simple organic molecules and steroids to metalloproteins, are shown.     

The Multidimensional Filter Diagonalization Method: I. Theory and Numerical Implementation

The theory and numerical aspects of the recently developed multidimensional version of the filter diagonalization method (FDM) are described in detail. FDM can construct various “ersatz” or “hybrid” spectra from multidimensional time signals. Spectral resolution is not limited by the time-frequency uncertainty principle in each separate frequency dimension, but rather by the total joint information content of the signal, i.e., N_total = N₁ × N₂ × × N_D, where some of the interferometric dimensions do not have to be represented by more than a few (e.g., two) time increments. It is shown that FDM can be used to compute various reduced-dimensionality projections of a high-dimensional spectrum directly, i.e., avoiding construction of the latter. A subsequent paper (J. Magn. Reson. 144, 357–366 (2000)) is concerned with applications of the method to 2D, 3D, and 4D NMR experiments.     

Indirect covariance NMR spectroscopy of through-bond homo-nuclear correlations for quadrupolar nuclei in solids under high-resolution

Indirect covariance NMR spectroscopy is demonstrated in solids, and we show that it can be used to obtain through-bond 2D homo-nuclear correlation spectra for quadrupolar nuclei under high-resolution. These spectra, generated with indirect covariance from a hetero-nuclear correlation spectrum, are equivalent to those recorded with the through-bond homo-nuclear hetero-nuclear single-quantum correlation (H-HSQC) method very recently proposed. However, the indirect covariance method can save a lot of experiment time, compared to the H-HSQC experiments, which allows introducing a high-resolution quadrupolar filter, thus providing a much better resolution, even on medium-field spectrometers. The covariance concept can be used to generate many different "indirectly-detected" high-resolution homo-nuclear correlation spectra with through-space or through-bond correlations for spin 1/2 or quadrupolar nuclei. We also propose a simple method that decreases the noise in all (direct or indirect) covariance methods.     

Regularization of the two-dimensional filter diagonalization method: FDM2K

We outline an important advance in the problem of obtaining a two-dimensional (2D) line list of the most prominent features in a 2D high-resolution NMR spectrum in the presence of noise, when using the Filter Diagonalization Method (FDM) to sidestep limitations of conventional FFT processing. Although respectable absorption-mode spectra have been obtained previously by the artifice of “averaging” several FDM calculations, no 2D line list could be directly obtained from the averaged spectrum, and each calculation produced numerical artifacts that were demonstrably inconsistent with the measured data, but which could not be removed a posteriori. By regularizing the intrinsically ill-defined generalized eigenvalue problem that FDM poses, in a particular quite plausible way, features that are weak or stem from numerical problems are attenuated, allowing better characterization of the dominant spectral features. We call the new algorithm FDM2K.     

Covariance NMR spectroscopy by singular value decomposition

Covariance NMR is demonstrated for homonuclear 2D NMR data collected using the hypercomplex and TPPI methods. Absorption mode 2D spectra are obtained by application of the square-root operation to the covariance matrices. The resulting spectra closely resemble the 2D Fourier transformation spectra, except that they are fully symmetric with the spectral resolution along both dimensions determined by the favorable resolution achievable along omega2. An efficient method is introduced for the calculation of the square root of the covariance spectrum by applying a singular value decomposition (SVD) directly to the mixed time-frequency domain data matrix. Applications are shown for 2D NOESY and 2QF-COSY data sets and computational benchmarks are given for data matrix dimensions typically encountered in practice. The SVD implementation makes covariance NMR amenable to routine applications.     

基于近似l0范数最小化的NMR波谱稀疏重建算法

在核磁共振（NMR）波谱中,过长的数据采集时间会使很多化学以及分子生物学领域的高分辨率多维谱应用难以实现. 传统的解决办法是使用随机非均匀采样代替奈奎斯特采样,但这样会使谱图质量受损. 压缩传感的出现为此提供了更好的解决办法,合适的压缩传感重建算法可以通过很少的随机非均匀采样将谱图高质量的重建出来. 该文先介绍了一种可用于谱图重建的压缩传感重建算法,名为“平滑l₀范数最小化法”,然后针对该算法对采样噪声鲁棒性较差的缺点进行了改进. 通过将改进后的算法与原算法在一维实数域信号以及NMR波谱信号重建实验中进行对比后表明,改进后的算法对噪声的鲁棒性明显提高,并能获得更好的重建性能.     

Improvement of spectral resolution by spectroscopic imaging

We propose to use three-dimensional spectroscopic imaging (SI) to increase the spectral resolution for biological samples for which strong susceptibility effects (or poor magnetic homogeneity) cause significant line broadening. Due to susceptibility effects (or poor field homogeneity) the SI voxel spectra even from a uniform sample are shifted with respect to each other and much less broadened than the total sample spectrum. Realignment of the spectra from individual voxels prior to their coaddition produces a total-volume spectrum with significantly narrower lines.     

Application of the Filter Diagonalization Method to One- and Two-Dimensional NMR Spectra

A new non-Fourier data processing algorithm, the filter diagonalization method (FDM), is presented and applied to phase-sensitive 1D and 2D NMR spectra. FDM extracts parameters (peak positions, linewidths, amplitudes, and phases) directly from the time-domain data by fitting the data to a sum of damped complex sinusoids. Grounded in a quantum-mechanical formalism, FDM shares some of the features of linear prediction and other linear algebraic approaches, but is numerically more efficient, scaling like the fast Fourier transform algorithm with respect to data size, and has the ability to correctly handle spectra with thousands or even millions of lines where the competing methods break down. Results obtained on complex spectra are promising.     

The single basis filter diagonalization method: a rapid multidimensional data processing scheme

A new way to apply the filter diagonalization method (FDM) that results in a large increase in the speed of calculation of multidimensional NMR spectra is presented. The speed increase is accompanied by slight differences in spectral lineshapes, although frequency estimates remain essentially identical. For contoured spectra, the method does not result in appreciable differences from the full FDM calculation. Optimal parameter sets for an FDM calculation can be estimated far more rapidly, which makes the FDM more straightforward to employ in practice. The performance of the method versus the full FDM was investigated for both model and experimental signals. The effect of noise on the method was also studied.     

Singular spectrum analysis for an automated solvent artifact removal and baseline correction of 1D NMR spectra

NMR spectroscopy in biology and medicine is generally performed in aqueous solutions, thus in (1)H NMR spectroscopy, the dominant signal often stems from the partly suppressed solvent and can be many orders of magnitude larger than the resonances of interest. Strong solvent signals lead to a disappearance of weak resonances of interest close to the solvent artifact and to base plane variations all over the spectrum. The AUREMOL-SSA/ALS approach for automated solvent artifact removal and baseline correction has been originally developed for multi-dimensional NMR spectroscopy. Here, we describe the necessary adaptations for an automated application to one-dimensional NMR spectra. Its core algorithm is still based on singular spectrum analysis (SSA) applied on time domain signals (FIDs) and it is still combined with an automated baseline correction (ALS) in the frequency domain. However, both steps (SSA and ALS) have been modified in order to achieve optimal results when dealing with one-dimensional spectra. The performance of the method has been tested on one-dimensional synthetic and experimental spectra including the back-calculated spectrum of HPr protein and an experimental spectrum of a human urine sample. The latter has been recorded with the typically used NOESY-type 1D pulse sequence including water pre-saturation. Furthermore, the fully automated AUREMOL-SSA/ALS procedure includes the managing of oversampled, digitally filtered and zero-filled data and the correction of the frequency domain phase shift caused by the group delay time shift from the digital finite response filtering.     

Restricted linear least square treatment processing of heteronuclear spectra of biomolecules using the ANAFOR strategy

We explore the use of a processing procedure based on restricted least square minimization as a tool for reducing the time versus resolution dilemma often encountered for biomolecular multidimensional spectra. Using a 2D spectrum as a reference, we obtain the necessary input of frequency components and linewidths. Combined even with a limited time evolution in the indirect dimension, the amplitudes of the correlation peaks in all planes of the 3D spectra can be extracted, and can be used to reconstruct the interferograms in the third dimension. Parameters such as number of lines, threshold choice, resolution, lineshape, number of experimental data points and finally signal to noise ratio of the spectrum are examined starting from a triple-resonance HNCA spectrum of ubiquitin.     

基于核磁共振的统计全相关谱在大鼠肾脏组织中的应用

生物组织是基于NMR的代谢组学研究的主要对象之一，广泛应用于分子病理学、毒理学、生物医学等众多领域. 但是，为了保证测定的准确，组织的NMR实验往往需要在较低的温度下和较短时间内完成，以防止由于组织内酶的降解和扩散而导致的某些代谢物质的分析信息被破坏. 统计全相关谱(Statistical Total Correlation Spectroscopy, STOCSY)是依靠一维谱来实现二维谱的一些功能的方法，不需要额外的实验时间，已经被广泛应用于代谢组学研究中. 本文采用STOCSY方法，通过对一系列¹H高分辨魔角旋转谱的统计分析和计算，得到了肾脏组织的准二维相关谱，其中共振峰之间的相关较为准确的反应了物质之间的耦合信息，为物质的归属提供了帮助.     

Use of spatial encoding in NMR photography

NMR photography has gained significant attention as a method of storing and retrieving information using NMR spectroscopy. Among the commonly practiced methods the most important is the frequency encoding by use of a multi-frequency pulse on a liquid crystal molecule. We propose and demonstrate another robust method which relies on spatial encoding. Spatial information is mapped onto the spectrum, if excited and recorded in the presence of a gradient. The encoding is performed by applying a multi-frequency pulse in the presence of a gradient. The subsequent acquisition, under a gradient, helps storing this spatial information on a one-dimensional spectrum. Series of such spectra can also store two-dimensional patterns. This procedure is described and exemplified in this paper.     

Triple-quantum MAS-NMR of quadrupolar nuclei

From two-dimensional multiquantum NMR spectra of quadrupolar nuclei, it is now possible to obtain much greater resolution than in a classical single-quantum magic-angle spinning or variable-angle spinning spectrum. We describe here a very simple pulse scheme which efficiently excites the desired multiquantum NMR coherence and a new acquisition procedure which yields to pure-absorption mode 2D spectra. Experimental spectra for ⁸⁷Rb in polycrystalline rubidium nitrate illustrate the method.     

Anomalous Rotational Resonance Spectra in Magic-Angle Spinning NMR

Magic-angle spinning NMR spectra of samples containing dilute spin-1/2 pairs display broadenings or splittings when a rotational resonance condition is satisfied, meaning that a small integer multiple of the spinning frequency matches the difference in the two isotropic shift frequencies. We show experimental rotational resonance NMR spectra of a 13C2-labeled retinal which are in qualitative disagreement with existing theory. We propose an explanation of these anomalous rotational spectra involving residual heteronuclear couplings between the 13C nuclei and the neighboring 1H nuclei. These couplings strongly influence the rotational resonance 13C spectrum, despite the presence of a strong radiofrequency decoupling field at the 1H Larmor frequency. We model the residual heteronuclear couplings by differential transverse relaxation of the 13C single-quantum coherences. We present a superoperator theory of the phenomenon and describe a numerical algorithm for rapid Liouville space simulations in periodic systems. Good agreement with experimental results is obtained by using a biexponential transverse relaxation model for each spin site.