Lineshapes and artifacts in Multidimensional Fourier Transform of arbitrary sampled NMR data sets

Revisiting aliasing noise to build more robust sparsity in nonuniform sampling 2D-NMR

A continuing priority is to better understand and resolve the barriers to using nonuniform sampling (NUS) in challenging small molecule 2D NMR with subsampling of the Nyquist grid (a.k.a. coverage) below 50%. Possible causes for artifacts, often termed sampling noise, in 1D-NUS of 2D-NMR are revisited here, where weak aliasing artifacts are a growing concern as NUS becomes sparser. As NUS schedules become sparser, repeat sequences are shown to occur in the dense sampling regions early in the sampling schedule, causing aliasing artifacts in resulting spectra. An intuitive screening approach that detects patterns in sampling schedules based on a convolutional filter was implemented. Sampling schedules that have low proportions of repeat sequences show significantly reduced artifacts. Another route to remediate early repeat sequences is a short period of uniform sampling at the beginning of the schedule, which also leads to a significant suppression of unwanted sampling noise. Combining the repeat sequence filter with a survey of HSQC and LR-HSQMBC experiments, it is shown that very short initial uniform regions of about 2%-4% of the sampling space can ameliorate repeat sequences in sparser NUS and lead to robust spectral reconstructions by iterative soft thresholding (IST), even when the point spread function is unchanged. Using the principles developed here, a suite of 'one-click' schedules was developed for broader use.

13C Direct Detected NMR for Challenging Systems

Thanks to recent improvements in NMR spectrometer hardware and pulse sequence design, modern 13C NMR has become a useful tool for biomolecular applications. The complete assignment of a protein can be accomplished by using 13C detected multinuclear experiments and it can provide unique information relevant for the study of a variety of different biomolecules including paramagnetic proteins and intrinsically disordered proteins. A wide range of NMR observables can be measured, concurring to the structural and dynamic characterization of a protein in isolation, as part of a larger complex, or even inside a living cell. We present the different properties of 13C with respect to 1H, which provide the rationale for the experiments developed and their application, the technical aspects that need to be faced, and the many experimental variants designed to address different cases. Application areas where these experiments successfully complement proton NMR are also described.

Clustered sparsity and Poisson-gap sampling

Non-uniform sampling (NUS) is a popular way of reducing the amount of time taken by multidimensional NMR experiments. Among the various non-uniform sampling schemes that exist, the Poisson-gap (PG) schedules are particularly popular, especially when combined with compressed-sensing (CS) reconstruction of missing data points. However, the use of PG is based mainly on practical experience and has not, as yet, been explained in terms of CS theory. Moreover, an apparent contradiction exists between the reported effectiveness of PG and CS theory, which states that a "flat" pseudo-random generator is the best way to generate sampling schedules in order to reconstruct sparse spectra. In this paper we explain how, and in what situations, PG reveals its superior features in NMR spectroscopy. We support our theoretical considerations with simulations and analyses of experimental data from the Biological Magnetic Resonance Bank (BMRB). Our analyses reveal a previously unnoticed feature of many NMR spectra that explains the success of "blue-noise" schedules, such as PG. We call this feature "clustered sparsity". This refers to the fact that the peaks in NMR spectra are not just sparse but often form clusters in the indirect dimension, and PG is particularly suited to deal with such situations. Additionally, we discuss why denser sampling in the initial and final parts of the clustered signal may be useful.

Order from disorder in the sarcomere: FATZ forms a fuzzy but tight complex and phase-separated condensates with α-actinin

In sarcomeres, α-actinin cross-links actin filaments and anchors them to the Z-disk. FATZ (filamin-, α-actinin-, and telethonin-binding protein of the Z-disk) proteins interact with α-actinin and other core Z-disk proteins, contributing to myofibril assembly and maintenance. Here, we report the first structure and its cellular validation of α-actinin-2 in complex with a Z-disk partner, FATZ-1, which is best described as a conformational ensemble. We show that FATZ-1 forms a tight fuzzy complex with α-actinin-2 and propose an interaction mechanism via main molecular recognition elements and secondary binding sites. The obtained integrative model reveals a polar architecture of the complex which, in combination with FATZ-1 multivalent scaffold function, might organize interaction partners and stabilize α-actinin-2 preferential orientation in Z-disk. Last, we uncover FATZ-1 ability to phase-separate and form biomolecular condensates with α-actinin-2, raising the question whether FATZ proteins can create an interaction hub for Z-disk proteins through membraneless compartmentalization during myofibrillogenesis.

Temperature as an Extra Dimension in Multidimensional Protein NMR Spectroscopy

NMR spectroscopy is a particularly informative method for studying protein structures and dynamics in solution; however, it is also one of the most time-consuming. Modern approaches to biomolecular NMR spectroscopy are based on lengthy multidimensional experiments, the duration of which grows exponentially with the number of dimensions. The experimental time may even be several days in the case of 3D and 4D spectra. Moreover, the experiment often has to be repeated under several different conditions, for example, to measure the temperature-dependent effects in a spectrum (temperature coefficients (TCs)). Herein, a new approach that involves joint sampling of indirect evolution times and temperature is proposed. This allows TCs to be measured through 3D spectra in even less time than that needed to acquire a single spectrum by using the conventional approach. Two signal processing methods that are complementary, in terms of sensitivity and resolution, 1) dividing data into overlapping subsets followed by compressed sensing reconstruction, and 2) treating the complete data set with a variant of the Radon transform, are proposed. The temperature-swept 3D HNCO spectra of two intrinsically disordered proteins, osteopontin and CD44 cytoplasmic tail, show that this new approach makes it possible to determine TCs and their non-linearities effectively. Non-linearities, which indicate the presence of a compact state, are particularly interesting. The complete package of data acquisition and processing software for this new approach are provided.

Enhancing Compression Level for More Efficient Compressed Sensing and Other Lessons from NMR Spectroscopy

Modern nuclear magnetic resonance spectroscopy (NMR) is based on two- and higher-dimensional experiments that allow the solving of molecular structures, i.e., determine the relative positions of single atoms very precisely. However, rich chemical information comes at the price of long data acquisition times (up to several days). This problem can be alleviated by compressed sensing (CS)—a method that revolutionized many fields of technology. It is known that CS performs the most efficiently when measured objects feature a high level of compressibility, which in the case of NMR signal means that its frequency domain representation (spectrum) has a low number of significant points. However, many NMR spectroscopists are not aware of the fact that various well-known signal acquisition procedures enhance compressibility and thus should be used prior to CS reconstruction. In this study, we discuss such procedures and show to what extent they are complementary to CS approaches. We believe that the survey will be useful not only for NMR spectroscopists but also to inspire the broader signal processing community.

Structural analysis of 25-hydroxycholesterol stereoisomers differing in configuration in position 17 and 20, by three-dimensional NMR spectra

The application of 3D NMR experiments and DFT calculations enabled the structure investigation of C-17 epimer of 3-(25-hydroxycholest-5-enyl) acetate is presented. The H-17 and H-20 protons features the same values of ¹H chemical shift, what causes that the structure elucidation require additional resolution enabled by 3D NMR experiments. The NMR experiments and theoretical calculations allowed for: the resonance assignment (3D COSY-HMBC and 3D TOCSY-HSQC techniques), the prediction of spatial structure (3D NOESY-HSQC and 3D ROESY-HSQC experiments), and the precise measurement of heteronuclear coupling constants (3D HSQC-TOCSY spectra with E.COSY-type multiplets).

Nonuniform sampling by quantiles

A flexible strategy for choosing samples nonuniformly from a Nyquist grid using the concept of statistical quantiles is presented for broad classes of NMR experimentation. Quantile-directed scheduling is intuitive and flexible for any weighting function, promotes reproducibility and seed independence, and is generalizable to multiple dimensions. In brief, weighting functions are divided into regions of equal probability, which define the samples to be acquired. Quantile scheduling therefore achieves close adherence to a probability distribution function, thereby minimizing gaps for any given degree of subsampling of the Nyquist grid. A characteristic of quantile scheduling is that one-dimensional, weighted NUS schedules are deterministic, however higher dimensional schedules are similar within a user-specified jittering parameter. To develop unweighted sampling, we investigated the minimum jitter needed to disrupt subharmonic tracts, and show that this criterion can be met in many cases by jittering within 25-50% of the subharmonic gap. For nD-NUS, three supplemental components to choosing samples by quantiles are proposed in this work: (i) forcing the corner samples to ensure sampling to specified maximum values in indirect evolution times, (ii) providing an option to triangular backfill sampling schedules to promote dense/uniform tracts at the beginning of signal evolution periods, and (iii) providing an option to force the edges of nD-NUS schedules to be identical to the 1D quantiles. Quantile-directed scheduling meets the diverse needs of current NUS experimentation, but can also be used for future NUS implementations such as off-grid NUS and more. A computer program implementing these principles (a.k.a. QSched) in 1D- and 2D-NUS is available under the general public license.

Triple resonance ¹⁵Ν NMR relaxation experiments for studies of intrinsically disordered proteins

Description of protein dynamics is known to be essential in understanding their function. Studies based on a well established [Formula: see text] NMR relaxation methodology have been applied to a large number of systems. However, the low dispersion of [Formula: see text] chemical shifts very often observed within intrinsically disordered proteins complicates utilization of standard 2D HN correlated spectra because a limited number of amino acids can be characterized. Here we present a suite of triple resonance HNCO-type NMR experiments for measurements of five [Formula: see text] relaxation parameters ([Formula: see text], [Formula: see text], NOE, cross-correlated relaxation rates [Formula: see text] and [Formula: see text]) in doubly [Formula: see text],[Formula: see text]-labeled proteins. We show that the third spectral dimension combined with non-uniform sampling provides relaxation rates for almost all residues of a protein with extremely poor chemical shift dispersion, the C terminal domain of [Formula: see text]-subunit of RNA polymerase from Bacillus subtilis. Comparison with data obtained using a sample labeled by [Formula: see text] only showed that the presence of [Formula: see text] has a negligible effect on [Formula: see text], [Formula: see text], and on the cross-relaxation rate (calculated from NOE and [Formula: see text]), and that these relaxation rates can be used to calculate accurate spectral density values. Partially [Formula: see text]-labeled sample was used to test if the observed increase of [Formula: see text] [Formula: see text] in the presence of [Formula: see text] corresponds to the [Formula: see text] dipole-dipole interactions in the [Formula: see text],[Formula: see text]-labeled sample.

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.

Subrandom methods for multidimensional nonuniform sampling

Methods of nonuniform sampling that utilize pseudorandom number sequences to select points from a weighted Nyquist grid are commonplace in biomolecular NMR studies, due to the beneficial incoherence introduced by pseudorandom sampling. However, these methods require the specification of a non-arbitrary seed number in order to initialize a pseudorandom number generator. Because the performance of pseudorandom sampling schedules can substantially vary based on seed number, this can complicate the task of routine data collection. Approaches such as jittered sampling and stochastic gap sampling are effective at reducing random seed dependence of nonuniform sampling schedules, but still require the specification of a seed number. This work formalizes the use of subrandom number sequences in nonuniform sampling as a means of seed-independent sampling, and compares the performance of three subrandom methods to their pseudorandom counterparts using commonly applied schedule performance metrics. Reconstruction results using experimental datasets are also provided to validate claims made using these performance metrics.

Deterministic multidimensional nonuniform gap sampling

Born from empirical observations in nonuniformly sampled multidimensional NMR data relating to gaps between sampled points, the Poisson-gap sampling method has enjoyed widespread use in biomolecular NMR. While the majority of nonuniform sampling schemes are fully randomly drawn from probability densities that vary over a Nyquist grid, the Poisson-gap scheme employs constrained random deviates to minimize the gaps between sampled grid points. We describe a deterministic gap sampling method, based on the average behavior of Poisson-gap sampling, which performs comparably to its random counterpart with the additional benefit of completely deterministic behavior. We also introduce a general algorithm for multidimensional nonuniform sampling based on a gap equation, and apply it to yield a deterministic sampling scheme that combines burst-mode sampling features with those of Poisson-gap schemes. Finally, we derive a relationship between stochastic gap equations and the expectation value of their sampling probability densities.

Applications of high dimensionality experiments to biomolecular NMR

High dimensionality NMR experiments facilitate resonance assignment and precise determination of spectral parameters such as coupling constants. Sparse non-uniform sampling enables acquisition of experiments of high dimensionality with high resolution in acceptable time. In this review we present and compare some significant applications of NMR experiments of dimensionality higher than three in the field of biomolecular studies in solution.

NMR assignment of intrinsically disordered self-processing module of the FrpC protein of Neisseria meningitidis

The self-processing module (SPM) is an internal segment of the FrpC protein (P415-F591) secreted by the pathogenic Gram-negative bacterium Neisseria meningitidis during meningococcal infection of human upper respiratory tract. SPM mediates 'protein trans-splicing', a unique natural mechanism for editing of proteins, which involves a calcium-dependent autocatalytic cleavage of the peptide bond between D414 and P415 and covalent linkage of the cleaved fragment through its carboxy-terminal group of D414 to [Formula: see text]-amino group of lysine residue within a neighboring polypeptide chain. We present an NMR resonance assignment of the calcium-free SPM, which displays characteristic features of intrinsically disordered proteins. Non-uniformly sampled 5D HN(CA)CONH, 4D HCBCACON, and HCBCANCO spectra were recorded to resolve poorly dispersed resonance frequencies of the disordered protein and 91 % of SPM residues were unambiguously assigned. Analysis of the chemical shifts revealed that two regions of the intrinsically disordered SPM (A95-S101 and R120-I127) have a tendency to form a helical structure, whereas the residues P1-D7 and G36-A40 have the propensity to adopt a [Formula: see text]-structure.

Non-Uniform Sampling and J-UNIO Automation for Efficient Protein NMR Structure Determination

High-resolution structure determination of small proteins in solution is one of the big assets of NMR spectroscopy in structural biology. Improvements in the efficiency of NMR structure determination by advances in NMR experiments and automation of data handling therefore attracts continued interest. Here, non-uniform sampling (NUS) of 3D heteronuclear-resolved [(1)H,(1)H]-NOESY data yielded two- to three-fold savings of instrument time for structure determinations of soluble proteins. With the 152-residue protein NP_372339.1 from Staphylococcus aureus and the 71-residue protein NP_346341.1 from Streptococcus pneumonia we show that high-quality structures can be obtained with NUS NMR data, which are equally well amenable to robust automated analysis as the corresponding uniformly sampled data.

(13)C-detected NMR experiments for automatic resonance assignment of IDPs and multiple-fixing SMFT processing

Intrinsically disordered proteins (IDPs) have recently attracted much interest, due to their role in many biological processes, including signaling and regulation mechanisms. High-dimensional (13)C direct-detected NMR experiments have proven exceptionally useful in case of IDPs, providing spectra with superior peak dispersion. Here, two such novel experiments recorded with non-uniform sampling are introduced, these are 5D HabCabCO(CA)NCO and 5D HNCO(CA)NCO. Together with the 4D (HACA)CON(CA)NCO, an extension of the previously published 3D experiments (Pantoja-Uceda and Santoro in J Biomol NMR 59:43-50, 2014. doi: 10.1007/s10858-014-9827-1), they form a set allowing for complete and reliable resonance assignment of difficult IDPs. The processing is performed with sparse multidimensional Fourier transform based on the concept of restricting (fixing) some of spectral dimensions to a priori known resonance frequencies. In our study, a multiple-fixing method was developed, that allows easy access to spectral data. The experiments were tested on a resolution-demanding alpha-synuclein sample. Due to superior peak dispersion in high-dimensional spectrum and availability of the sequential connectivities between four consecutive residues, the overwhelming majority of resonances could be assigned automatically using the TSAR program.

Nonuniform sampling of hypercomplex multidimensional NMR experiments: Dimensionality, quadrature phase and randomization

Nonuniform sampling (NUS) in multidimensional NMR permits the exploration of higher dimensional experiments and longer evolution times than the Nyquist Theorem practically allows for uniformly sampled experiments. However, the spectra of NUS data include sampling-induced artifacts and may be subject to distortions imposed by sparse data reconstruction techniques, issues not encountered with the discrete Fourier transform (DFT) applied to uniformly sampled data. The characterization of these NUS-induced artifacts allows for more informed sample schedule design and improved spectral quality. The DFT-Convolution Theorem, via the point-spread function (PSF) for a given sampling scheme, provides a useful framework for exploring the nature of NUS sampling artifacts. In this work, we analyze the PSFs for a set of specially constructed NUS schemes to quantify the interplay between randomization and dimensionality for reducing artifacts relative to uniformly undersampled controls. In particular, we find a synergistic relationship between the indirect time dimensions and the "quadrature phase dimension" (i.e. the hypercomplex components collected for quadrature detection). The quadrature phase dimension provides additional degrees of freedom that enable partial-component NUS (collecting a subset of quadrature components) to further reduce sampling-induced aliases relative to traditional full-component NUS (collecting all quadrature components). The efficacy of artifact reduction is exponentially related to the dimensionality of the sample space. Our results quantify the utility of partial-component NUS as an additional means for introducing decoherence into sampling schemes and reducing sampling artifacts in high dimensional experiments.

NMR Methods for the Study of Instrinsically Disordered Proteins Structure, Dynamics, and Interactions: General Overview and Practical Guidelines

Thanks to recent improvements in NMR instrumentation, pulse sequence design, and sample preparation, a panoply of new NMR tools has become available for atomic resolution characterization of intrinsically disordered proteins (IDPs) that are optimized for the particular chemical and spectroscopic properties of these molecules. A wide range of NMR observables can now be measured on increasingly complex IDPs that report on their structural and dynamic properties in isolation, as part of a larger complex, or even inside an entire living cell. Herein we present basic NMR concepts, as well as optimised tools available for the study of IDPs in solution. In particular, the following sections are discussed hereafter: a short introduction to NMR spectroscopy and instrumentation (Sect. 3.1), the effect of order and disorder on NMR observables (Sect. 3.2), particular challenges and bottlenecks for NMR studies of IDPs (Sect. 3.3), 2D HN and CON NMR experiments: the fingerprint of an IDP (Sect. 3.4), tools for overcoming major bottlenecks of IDP NMR studies (Sect. 3.5), 13C detected experiments (Sect. 3.6), from 2D to 3D: from simple snapshots to site-resolved characterization of IDPs (Sect. 3.7), sequential NMR assignment: 3D experiments (Sect. 3.8), high-dimensional NMR experiments (nD, with n>3) (Sect. 3.9) and conclusions and perspectives (Sect. 3.10).

Modified OMP algorithm for exponentially decaying signals

A group of signal reconstruction methods, referred to as compressed sensing (CS), has recently found a variety of applications in numerous branches of science and technology. However, the condition of the applicability of standard CS algorithms (e.g., orthogonal matching pursuit, OMP), i.e., the existence of the strictly sparse representation of a signal, is rarely met. Thus, dedicated algorithms for solving particular problems have to be developed. In this paper, we introduce a modification of OMP motivated by nuclear magnetic resonance (NMR) application of CS. The algorithm is based on the fact that the NMR spectrum consists of Lorentzian peaks and matches a single Lorentzian peak in each of its iterations. Thus, we propose the name Lorentzian peak matching pursuit (LPMP). We also consider certain modification of the algorithm by introducing the allowed positions of the Lorentzian peaks' centers. Our results show that the LPMP algorithm outperforms other CS algorithms when applied to exponentially decaying signals.

Accurate scoring of non-uniform sampling schemes for quantitative NMR

Non-uniform sampling (NUS) in NMR spectroscopy is a recognized and powerful tool to minimize acquisition time. Recent advances in reconstruction methodologies are paving the way for the use of NUS in quantitative applications, where accurate measurement of peak intensities is crucial. The presence or absence of NUS artifacts in reconstructed spectra ultimately determines the success of NUS in quantitative NMR. The quality of reconstructed spectra from NUS acquired data is dependent upon the quality of the sampling scheme. Here we demonstrate that the best performing sampling schemes make up a very small percentage of the total randomly generated schemes. A scoring method is found to accurately predict the quantitative similarity between reconstructed NUS spectra and those of fully sampled spectra. We present an easy-to-use protocol to batch generate and rank optimal Poisson-gap NUS schedules for use with 2D NMR with minimized noise and accurate signal reproduction, without the need for the creation of synthetic spectra.

Quantitative 2D liquid-state NMR

Two-dimensional (2D) liquid-state NMR has a very high potential to simultaneously determine the absolute concentration of small molecules in complex mixtures, thanks to its capacity to separate overlapping resonances. However, it suffers from two main drawbacks that probably explain its relatively late development. First, the 2D NMR signal is strongly molecule-dependent and site-dependent; second, the long duration of 2D NMR experiments prevents its general use for high-throughput quantitative applications and affects its quantitative performance. Fortunately, the last 10 years has witnessed an increasing number of contributions where quantitative approaches based on 2D NMR were developed and applied to solve real analytical issues. This review aims at presenting these recent efforts to reach a high trueness and precision in quantitative measurements by 2D NMR. After highlighting the interest of 2D NMR for quantitative analysis, the different strategies to determine the absolute concentrations from 2D NMR spectra are described and illustrated by recent applications. The last part of the manuscript concerns the recent development of fast quantitative 2D NMR approaches, aiming at reducing the experiment duration while preserving - or even increasing - the analytical performance. We hope that this comprehensive review will help readers to apprehend the current landscape of quantitative 2D NMR, as well as the perspectives that may arise from it.

Novel methods based on (13)C detection to study intrinsically disordered proteins

Intrinsically disordered proteins (IDPs) are characterized by highly flexible solvent exposed backbones and can sample many different conformations. These properties confer them functional advantages, complementary to those of folded proteins, which need to be characterized to expand our view of how protein structural and dynamic features affect function beyond the static picture of a single well defined 3D structure that has influenced so much our way of thinking. NMR spectroscopy provides a unique tool for the atomic resolution characterization of highly flexible macromolecules in general and of IDPs in particular. The peculiar properties of IDPs however have profound effects on spectroscopic parameters. It is thus worth thinking about these aspects to make the best use of the great potential of NMR spectroscopy to contribute to this fascinating field of research. In particular, after many years of dealing with exclusively heteronuclear NMR experiments based on (13)C direct detection, we would like here to address their relevance when studying IDPs.

Toward optimal-resolution NMR of intrinsically disordered proteins

Proteins, which, in their native conditions, sample a multitude of distinct conformational states characterized by high spatiotemporal heterogeneity, most often termed as intrinsically disordered proteins (IDPs), have become a target of broad interest over the past 15years. With the growing evidence of their important roles in fundamental cellular processes, there is an urgent need to characterize the conformational behavior of IDPs at the highest possible level. The unique feature of NMR spectroscopy in the context of IDPs is its ability to supply details of their structural and temporal alterations at atomic-level resolution. Here, we briefly review recently proposed NMR-based strategies to characterize transient states populated by IDPs and summarize the latest achievements and future prospects in methodological development. Because low chemical shift dispersion represents the major obstacle encountered when studying IDPs by nuclear magnetic resonance, particular attention is paid to techniques allowing one to approach the physical limits of attainable resolution.

Ultrafast 2D NMR: an emerging tool in analytical spectroscopy

Two-dimensional nuclear magnetic resonance (2D NMR) spectroscopy is widely used in chemical and biochemical analyses. Multidimensional NMR is also witnessing increased use in quantitative and metabolic screening applications. Conventional 2D NMR experiments, however, are affected by inherently long acquisition durations, arising from their need to sample the frequencies involved along their indirect domains in an incremented, scan-by-scan nature. A decade ago, a so-called ultrafast (UF) approach was proposed, capable of delivering arbitrary 2D NMR spectra involving any kind of homo- or heteronuclear correlation, in a single scan. During the intervening years, the performance of this subsecond 2D NMR methodology has been greatly improved, and UF 2D NMR is rapidly becoming a powerful analytical tool experiencing an expanded scope of applications. This review summarizes the principles and main developments that have contributed to the success of this approach and focuses on applications that have been recently demonstrated in various areas of analytical chemistry--from the real-time monitoring of chemical and biochemical processes, to extensions in hyphenated techniques and in quantitative applications.

NASR: an effective approach for simultaneous noise and artifact suppression in NMR spectroscopy

As a powerful tool for biological analysis, especially protein structure and dynamic studies, nuclear magnetic resonance (NMR) spectroscopy suffers from intrinsic low signal to nose ratio (SNR) and long acquisition time required for multidimensional (nD) experiments. Nonuniform sampling (NUS) can effectively speed up the experiment but often introduces artifacts into the spectrum. In addition to the development of highly sensitive hardware and NMR pulse sequences, data postprocessing is a relative simple and cost-effective method to improve the SNR and suppress the artifacts. In this work, we propose an effective approach for simultaneously suppressing noise and artifacts based on the resampling principle. The method is named NASR for short and tested using one-, two-, and three-dimensional (1D, 2D, and 3D) NMR spectra that were acquired using ether conventional or NUS (spiral and random, for 3D) approaches. The results reveal that the NASR is fast and applicable for improving the quality of 1D to nD NMR spectra with all kinds of sampling schemes.

Exploring signal-to-noise ratio and sensitivity in non-uniformly sampled multi-dimensional NMR spectra

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.

Rapid protein global fold determination using ultrasparse sampling, high-dynamic range artifact suppression, and time-shared NOESY

In structural studies of large proteins by NMR, global fold determination plays an increasingly important role in providing a first look at a target's topology and reducing assignment ambiguity in NOESY spectra of fully protonated samples. In this work, we demonstrate the use of ultrasparse sampling, a new data processing algorithm, and a 4-D time-shared NOESY experiment (1) to collect all NOEs in (2)H/(13)C/(15)N-labeled protein samples with selectively protonated amide and ILV methyl groups at high resolution in only four days, and (2) to calculate global folds from this data using fully automated resonance assignment. The new algorithm, SCRUB, incorporates the CLEAN method for iterative artifact removal but applies an additional level of iteration, permitting real signals to be distinguished from noise and allowing nearly all artifacts generated by real signals to be eliminated. In simulations with 1.2% of the data required by Nyquist sampling, SCRUB achieves a dynamic range over 10000:1 (250× better artifact suppression than CLEAN) and completely quantitative reproduction of signal intensities, volumes, and line shapes. Applied to 4-D time-shared NOESY data, SCRUB processing dramatically reduces aliasing noise from strong diagonal signals, enabling the identification of weak NOE crosspeaks with intensities 100× less than those of diagonal signals. Nearly all of the expected peaks for interproton distances under 5 Å were observed. The practical benefit of this method is demonstrated with structure calculations for 23 kDa and 29 kDa test proteins using the automated assignment protocol of CYANA, in which unassigned 4-D time-shared NOESY peak lists produce accurate and well-converged global fold ensembles, whereas 3-D peak lists either fail to converge or produce significantly less accurate folds. The approach presented here succeeds with an order of magnitude less sampling than required by alternative methods for processing sparse 4-D data.

A comparison of convex and non-convex compressed sensing applied to multidimensional NMR

The resolution of multidimensional NMR spectra can be severely limited when regular sampling based on the Nyquist-Shannon theorem is used. The theorem binds the sampling rate with a bandwidth of a sampled signal and thus implicitly creates a dependence between the line width and the time of experiment, often making the latter one very long. Recently, Candès et al. (2006) [25] formulated a non-linear sampling theorem that determines the required number of sampling points to be dependent mostly on the number of peaks in a spectrum and only slightly on the number of spectral points. The result was pivotal for rapid development and broad use of signal processing method called compressed sensing. In our previous work, we have introduced compressed sensing to multidimensional NMR and have shown examples of reconstruction of two-dimensional spectra. In the present paper we discuss in detail the accuracy and robustness of two compressed sensing algorithms: convex (iterative soft thresholding) and non-convex (iteratively re-weighted least squares with local ℓ(0)-norm) in application to two- and three-dimensional datasets. We show that the latter method is in many terms more effective, which is in line with recent works on the theory of compressed sensing. We also present the comparison of both approaches with multidimensional decomposition which is one of the established methods for processing of non-linearly sampled data.

Speeding up sequence specific assignment of IDPs

The characterization of intrinsically disordered proteins (IDPs) by NMR spectroscopy is made difficult by the extensive spectral overlaps. To overcome the intrinsic low-resolution of the spectra the introduction of high-dimensionality experiments is essential. We present here a set of high-resolution experiments based on direct (13)C-detection which proved useful in the assignment of α-synuclein, a paradigmatic IDP. In particular, we describe the implementation of 4D HCBCACON, HCCCON, HCBCANCO, 4/5D HNCACON and HNCANCO and 3/4D HCANCACO experiments, specifically tailored for spin system identification and backbone resonances sequential assignment. The use of non-uniform-sampling in the indirect dimension and of the H-flip approach to achieve longitudinal relaxation enhancement rendered the experiments very practical.

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.

Covariance spectroscopy with a non-uniform and consecutive acquisition scheme for signal enhancement of the NMR experiments

Two-dimensional covariance (COV2D) spectroscopy with non-uniform and consecutive acquisition (NUCA) scheme is introduced. This NUCA-COV2D method allows the number of t(1) points to be reduced by a factor of 1.5-3 without any broadening of the linewidth. Furthermore, the signal-to-noise ratio (S/N) can be increased up to 50%, which can further save experimental time by another factor of 2. This method has been demonstrated with model samples and the microcrystalline proteins. In all cases, the total experimental time can be reduced by a factor of 3-6, without any loss of resolution and S/N, with respect to what is necessary with the FT2D NMR.

Measurement and applications of long-range heteronuclear scalar couplings: recent experimental and theoretical developments

The use of long-range heteronuclear couplings, in association with (1)H-(1)H scalar couplings and NOE restraints, has acquired growing importance for the determination of the relative stereochemistry, and structural and conformational information of organic and biological molecules. However, the routine use of such couplings is hindered by the inherent difficulties in their measurement. Prior to the advancement in experimental techniques, both long-range homo- and heteronuclear scalar couplings were not easily accessible, especially for very large molecules. The development of a large number of multidimensional NMR experimental methodologies has alleviated the complications associated with the measurement of couplings of smaller strengths. Subsequent application of these methods and the utilization of determined J-couplings for structure calculations have revolutionized this area of research. Problems in organic, inorganic and biophysical chemistry have also been solved by utilizing the short- and long-range heteronuclear couplings. In this minireview, we discuss the advantages and limitations of a number of experimental techniques reported in recent times for the measurement of long-range heteronuclear couplings and a few selected applications of such couplings. This includes the study of medium- to larger-sized molecules in a variety of applications, especially in the study of hydrogen bonding in biological systems. The utilization of these couplings in conjunction with theoretical calculations to arrive at conclusions on the hyperconjugation, configurational analysis and the effect of the electronegativity of the substituents is also discussed.

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.

Suppression of sampling artefacts in high-resolution four-dimensional NMR spectra using signal separation algorithm

The development of non-uniform sampling (NUS) strategies permits to obtain high-dimensional spectra with increased resolution in significantly reduced experimental time. We extended a previously proposed signal separation algorithm (SSA) to process sparse four-dimensional NMR data. It is employed for two experiments carried out for a partially unstructured 114-residue construct of chicken Engrailed 2 protein, namely 4D HCCH-TOCSY and 4D C,N-edited NOESY. The SSA allowed us to obtain high-quality spectra using only as little as 0.16% of the available samples, with low sampling artefacts approaching the thermal noise level in most spectral regions. It is demonstrated that NUS 4D HCCH-TOCSY is dominated by sampling noise and requires efficient artefact suppression. On the other hand, 4D C,N-edited NOESY is a particularly attractive experiment for NUS, as the absence of diagonal peaks renders the problem of artefacts less critical. We also present a transverse-relaxation optimized sequence for HMQC that is especially designed for longer evolution periods in the indirectly detected proton dimension in high-dimensional pulse sequences. In conjunction with novel sampling strategies and efficient processing methods, this improvement enabled us to obtain unique structural information about aliphatic-amide contacts.

Deterministic schedules for robust and reproducible non-uniform sampling in multidimensional NMR

We show that a simple, general, and easily reproducible method for generating non-uniform sampling (NUS) schedules preserves the benefits of random sampling, including inherently reduced sampling artifacts, while removing the pitfalls associated with choosing an arbitrary seed. Sampling schedules are generated from a discrete cumulative distribution function (CDF) that closely fits the continuous CDF of the desired probability density function. We compare random and deterministic sampling using a Gaussian probability density function applied to 2D HSQC spectra. Data are processed using the previously published method of Spectroscopy by Integration of Frequency and Time domain data (SIFT). NUS spectra from deterministic sampling schedules were found to be at least as good as those from random schedules at the SIFT critical sampling density, and significantly better at half that sampling density. The method can be applied to any probability density function and generalized to greater than two dimensions.

Parallel receivers and sparse sampling in multidimensional NMR

The recent introduction of NMR spectrometers with multiple receivers permits spectra from several different nuclear species to be recorded in parallel, and several standard pulse sequences to be combined into a single entity. It is shown how these improvements in the flow and quality of spectral information can be significantly augmented by compressive sensing techniques--controlled aliasing, Hadamard spectroscopy, single-point evaluation of evolution space (SPEED), random sampling, projection-reconstruction, and hyperdimensional NMR. Future developments of these techniques are confidently expected to mitigate one of the most serious limitations in multidimensional NMR--the excessive duration of the measurements.

Boosting protein dynamics studies using quantitative nonuniform sampling NMR spectroscopy

NMR spectroscopy is uniquely suited to study protein dynamics over a wide range of time scales at atomic resolution. However, existing NMR relaxation methods require highly serial, lengthy data collection, ultimately limiting their application to short-lived samples, such as proteins in living cells. In recent years, the utility of nonuniform sampling (NUS) NMR methodologies has been increasingly recognized, but their application has been rare in relaxation measurements where highly accurate spectral quantification is demanded. Recently, Matsuki et al. developed a new NUS-processing method, SIFT (Spectroscopy by Integration of Frequency and Time domain information), which is highly robust and faithful in reproducing signals. In this work, we demonstrate the gains that are possible with more aggressive use of frequency domain information than was employed previously. This improvement is crucial for SIFT to be used in accelerating relaxation measurements while preserving full analytical accuracy. By taking the KIX domain of mouse CREB-binding protein (CBP) as an example, we demonstrate that this quantitative NUS processing method enables total 10-fold expedition of the R(2) relaxation dispersion measurements. The advanced SIFT processing should be equally useful for other NMR relaxation measurements.

Random phase detection in multidimensional NMR

Despite advances in resolution accompanying the development of high-field superconducting magnets, biomolecular applications of NMR require multiple dimensions in order to resolve individual resonances, and the achievable resolution is typically limited by practical constraints on measuring time. In addition to the need for measuring long evolution times to obtain high resolution, the need to distinguish the sign of the frequency constrains the ability to shorten measuring times. Sign discrimination is typically accomplished by sampling the signal with two different receiver phases or by selecting a reference frequency outside the range of frequencies spanned by the signal and then sampling at a higher rate. In the parametrically sampled (indirect) time dimensions of multidimensional NMR experiments, either method imposes an additional factor of 2 sampling burden for each dimension. We demonstrate that by using a single detector phase at each time sample point, but randomly altering the phase for different points, the sign ambiguity that attends fixed single-phase detection is resolved. Random phase detection enables a reduction in experiment time by a factor of 2 for each indirect dimension, amounting to a factor of 8 for a four-dimensional experiment, albeit at the cost of introducing sampling artifacts. Alternatively, for fixed measuring time, random phase detection can be used to double resolution in each indirect dimension. Random phase detection is complementary to nonuniform sampling methods, and their combination offers the potential for additional benefits. In addition to applications in biomolecular NMR, random phase detection could be useful in magnetic resonance imaging and other signal processing contexts.

5D 13C-detected experiments for backbone assignment of unstructured proteins with a very low signal dispersion

Two novel 5D NMR experiments (CACONCACO, NCOCANCO) for backbone assignment of disordered proteins are presented. The pulse sequences exploit relaxation properties of the unstructured proteins and combine the advantages of (13)C-direct detection, non-uniform sampling, and longitudinal relaxation optimization to maximize the achievable resolution and minimize the experimental time. The pulse sequences were successfully tested on the sample of partially disordered delta subunit from RNA polymerase from Bacillus subtilis. The unstructured part of this 20 kDa protein consists of 81 amino acids with frequent sequential repeats. A collection of 0.0003% of the data needed for a conventional experiment with linear sampling was sufficient to perform an unambiguous assignment of the disordered part of the protein from a single 5D spectrum.

Recording 2-D Nutation NQR Spectra by Random Sampling Method

The method of random sampling was introduced for the first time in the nutation nuclear quadrupole resonance (NQR) spectroscopy where the nutation spectra show characteristic singularities in the form of shoulders. The analytic formulae for complex two-dimensional (2-D) nutation NQR spectra (I = 3/2) were obtained and the condition for resolving the spectral singularities for small values of an asymmetry parameter η was determined. Our results show that the method of random sampling of a nutation interferogram allows significant reduction of time required to perform a 2-D nutation experiment and does not worsen the spectral resolution.

Non-uniform frequency domain for optimal exploitation of non-uniform sampling

Random sampling of NMR signal, not limited by Nyquist Theorem, yields up to thousands-fold gain in the experiment time required to obtain desired spectral resolution. Discrete Fourier transform (DFT), that can be used for processing of randomly sampled datasets, provides rarely exploited possibility to introduce irregular frequency domain. Here we demonstrate how this feature opens an avenue to NMR techniques of ultra-high resolution and dimensionality. We present the application of high resolution 5D experiments for protein backbone assignment and measurements of coupling constants from the 4D E.COSY multiplets. Spectral data acquired with the use of proposed techniques allow easy assignment of protein backbone resonances and precise determination of coupling constants.

Iterative algorithm of discrete Fourier transform for processing randomly sampled NMR data sets

Spectra obtained by application of multidimensional Fourier Transformation (MFT) to sparsely sampled nD NMR signals are usually corrupted due to missing data. In the present paper this phenomenon is investigated on simulations and experiments. An effective iterative algorithm for artifact suppression for sparse on-grid NMR data sets is discussed in detail. It includes automated peak recognition based on statistical methods. The results enable one to study NMR spectra of high dynamic range of peak intensities preserving benefits of random sampling, namely the superior resolution in indirectly measured dimensions. Experimental examples include 3D (15)N- and (13)C-edited NOESY-HSQC spectra of human ubiquitin.

Gridding and fast Fourier transformation on non-uniformly sparse sampled multidimensional NMR data

For multidimensional NMR method, indirect dimensional non-uniform sparse sampling can dramatically shorten acquisition time of the experiments. However, the non-uniformly sampled NMR data cannot be processed directly using fast Fourier transform (FFT). We show that the non-uniformly sampled NMR data can be reconstructed to Cartesian grid with the gridding method that has been wide applied in MRI, and sequentially be processed using FFT. The proposed gridding-FFT (GFFT) method increases the processing speed sharply compared with the previously proposed non-uniform Fourier Transform, and may speed up application of the non-uniform sparse sampling approaches.

Fast acquisition of high resolution 4-D amide-amide NOESY with diagonal suppression, sparse sampling and FFT-CLEAN

Amide-amide NOESY provides important distance constraints for calculating global folds of large proteins, especially integral membrane proteins with beta-barrel folds. Here, we describe a diagonal-suppressed 4-D NH-NH TROSY-NOESY-TROSY (ds-TNT) experiment for NMR studies of large proteins. The ds-TNT experiment employs a spin state selective transfer scheme that suppresses diagonal signals while providing TROSY optimization in all four dimensions. Active suppression of the strong diagonal peaks greatly reduces the dynamic range of observable signals, making this experiment particularly suitable for use with sparse sampling techniques. To demonstrate the utility of this method, we collected a high resolution 4-D ds-TNT spectrum of a 23kDa protein using randomized concentric shell sampling (RCSS), and we used FFT-CLEAN processing for further reduction of aliasing artifacts - the first application of these techniques to a NOESY experiment. A comparison of peak parameters in the high resolution 4-D dataset with those from a conventionally-sampled 3-D control spectrum shows an accurate reproduction of NOE crosspeaks in addition to a significant reduction in resonance overlap, which largely eliminates assignment ambiguity. Likewise, a comparison of 4-D peak intensities and volumes before and after application of the CLEAN procedure demonstrates that the reduction of aliasing artifacts by CLEAN does not systematically distort NMR signals.