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

Deciphering the Path of S-nitrosation of Human Thioredoxin: Evidence of an Internal NO Transfer and Implication for the Cellular Responses to NO

Nitric oxide (NO) is a free radical with a signaling capacity. Its cellular functions are achieved mainly through S-nitrosation where thioredoxin (hTrx) is pivotal in the S-transnitrosation to specific cellular targets. In this study, we use NMR spectroscopy and mass spectrometry to follow the mechanism of S-(trans)nitrosation of hTrx. We describe a site-specific path for S-nitrosation by measuring the reactivity of each of the 5 cysteines of hTrx using cysteine mutants. We showed the interdependence of the three cysteines in the nitrosative site. C73 is the most reactive and is responsible for all S-transnitrosation to other cellular targets. We observed NO internal transfers leading to C62 S-nitrosation, which serves as a storage site for NO. C69-SNO only forms under nitrosative stress, leading to hTrx nuclear translocation.

Phase separation driven by interchangeable properties in the intrinsically disordered regions of protein paralogs

Paralogs, arising from gene duplications, increase the functional diversity of proteins. Protein functions in paralog families have been extensively studied, but little is known about the roles that intrinsically disordered regions (IDRs) play in their paralogs. Without a folded structure to restrain them, IDRs mutate more diversely along with evolution. However, how the diversity of IDRs in a paralog family affects their functions is unexplored. Using the RNA-binding protein Musashi family as an example, we applied multiple structural techniques and phylogenetic analysis to show how members in a paralog family have evolved their IDRs to different physicochemical properties but converge to the same function. In this example, the lower prion-like tendency of Musashi-1's IDRs, rather than Musashi-2's, is compensated by its higher α-helical propensity to assist their assembly. Our work suggests that, no matter how diverse they become, IDRs could evolve different traits to a converged function, such as liquid-liquid phase separation.

Reducing the measurement time of exact NOEs by non-uniform sampling

We have previously reported on the measurement of exact NOEs (eNOEs), which yield a wealth of additional information in comparison to conventional NOEs. We have used these eNOEs in a variety of applications, including calculating high-resolution structures of proteins and RNA molecules. The collection of eNOEs is challenging, however, due to the need to measure a NOESY buildup series consisting of typically four NOESY spectra with varying mixing times in a single measurement session. While the 2D version can be completed in a few days, a fully sampled 3D-NOESY buildup series can take 10 days or more to acquire. This can be both expensive as well as problematic in the case of samples that are not stable over such a long period of time. One potential method to significantly decrease the required measurement time of eNOEs is to use non-uniform sampling (NUS) to decrease the number of points measured in the indirect dimensions. The effect of NUS on the extremely tight distance restraints extracted from eNOEs may be very pronounced. Therefore, we investigated the fidelity of eNOEs measured from three test cases at decreasing NUS densities: the 18.4 kDa protein human Pin1, the 4.1 kDa WW domain of Pin1 (both in 3D), and a 4.6 kDa 14mer RNA UUCG tetraloop (2D). Our results show that NUS imparted negligible error on the eNOE distances derived from good quality data down to 10% sampling for all three cases, but there is a noticeable decrease in the eNOE yield that is dependent upon the underlying sparsity, and thus complexity, of the sample. For Pin1, this transition occurred at roughly 40% while for the WW domain and the UUCG tetraloop it occurred at lower NUS densities of 20% and 10%, respectively. We rationalized these numbers through reconstruction simulations under various conditions. The extent of this loss depends upon the number of scans taken as well as the number of peaks to be reconstructed. Based on these findings, we have created guidelines for choosing an optimal NUS density depending on the number of peaks needed to be reconstructed in the densest region of a 2D or 3D NOESY spectrum.

Monitoring Hydrogenation Reactions using Benchtop 2D NMR with Extraordinary Sensitivity and Spectral Resolution

Low-field benchtop nuclear magnetic resonance (BT-NMR) spectrometers with Halbach magnets are being increasingly used in science and industry as cost-efficient tools for the monitoring of chemical reactions, including hydrogenation. However, their use of low-field magnets limits both resolution and sensitivity. In this paper, we show that it is possible to alleviate these two problems through the combination of parahydrogen-induced polarization (PHIP) and fast correlation spectroscopy with time-resolved non-uniform sampling (TR-NUS). PHIP can enhance NMR signals so that substrates are easily detectable on BT-NMR spectrometers. The interleaved acquisition of one- and two-dimensional spectra with TR-NUS provides unique insight into the consecutive moments of hydrogenation reactions, with a spectral resolution unachievable in a standard approach. We illustrate the potential of the technique with two examples: the hydrogenation of ethylphenyl propiolate and the hydrogenation of a mixture of two substrates - ethylphenyl propiolate and ethyl 2-butynoate.

TReNDS-Software for reaction monitoring with time-resolved non-uniform sampling

NMR spectroscopy, used routinely for structure elucidation, has also become a widely applied tool for process and reaction monitoring. However, the most informative of NMR methods-correlation experiments-are often useless in this kind of applications. The traditional sampling of a multidimensional FID is usually time-consuming, and thus, the reaction-monitoring toolbox was practically limited to 1D experiments (with rare exceptions, e.g., single-scan or fast-sampling experiments). Recently, the technique of time-resolved non-uniform sampling (TR-NUS) has been proposed, which allows to use standard multidimensional pulse sequences preserving the temporal resolution close to that achievable in 1D experiments. However, the method existed only as a prototype and did not allow on-the-fly processing during the reaction. In this paper, we introduce TReNDS: free, user-friendly software kit for acquisition and processing of TR-NUS data. The program works on Bruker, Agilent, and Magritek spectrometers, allowing to carry out up to four experiments with interleaved TR-NUS. The performance of the program is demonstrated on the example of enzymatic hydrolysis of sucrose.

A CON-based NMR assignment strategy for pro-rich intrinsically disordered proteins with low signal dispersion: the C-terminal domain of histone H1.0 as a case study

The C-terminal domain of histone H1.0 (C-H1.0) is involved in DNA binding and is a main determinant of the chromatin condensing properties of histone H1.0. Phosphorylation at the (S/T)-P-X-(K/R) motifs affects DNA binding and is crucial for regulation of C-H1.0 function. Since C-H1.0 is an intrinsically disordered domain, solution NMR is an excellent approach to characterize the effect of phosphorylation on the structural and dynamic properties of C-H1.0. However, its very repetitive, low-amino acid-diverse and Pro-rich sequence, together with the low signal dispersion observed at the ¹H-¹⁵N HSQC spectra of both non- and tri-phosphorylated C-H1.0 preclude the use of standard ¹H-detected assignment strategies. We have achieved an essentially complete assignment of the heavy backbone atoms (¹⁵N, ¹³C' and ¹³C_α), as well as ¹H^N and ¹³C_β nuclei, of non- and tri-phosphorylated C-H1.0 by applying a novel ¹³C-detected CON-based strategy. No C-H1.0 region with a clear secondary structure tendency was detected by chemical shift analyses, confirming at residue level that C-H1.0 is disordered in aqueous solution. Phosphorylation only affected the chemical shifts of phosphorylated Thr's, and their adjacent residues. Heteronuclear {¹H}-¹⁵N NOEs were also essentially equal in the non- and tri-phosphorylated states. Hence, structural tendencies and dynamic properties of C-H1.0 free in aqueous solution are unmodified by phosphorylation. We propose that the assignment strategy used for C-H1.0, which is based on the acquisition of only a few 3D spectra, is an excellent choice for short-lived intrinsically disordered proteins with repetitive sequences.

TAR DNA-binding protein 43 (TDP-43) liquid-liquid phase separation is mediated by just a few aromatic residues

Eukaryotic cells contain distinct organelles, but not all of these compartments are enclosed by membranes. Some intrinsically disordered proteins mediate membraneless organelle formation through liquid-liquid phase separation (LLPS). LLPS facilitates many biological functions such as regulating RNA stability and ribonucleoprotein assembly, and disruption of LLPS pathways has been implicated in several diseases. Proteins exhibiting LLPS typically have low sequence complexity and specific repeat motifs. These motifs promote multivalent connections with other molecules and the formation of higher-order oligomers, and their removal usually prevents LLPS. The intrinsically disordered C-terminal domain of TAR DNA-binding protein 43 (TDP-43), a protein involved in motor neuron disease and dementia lacks a dominant LLPS motif, however, and how this domain forms condensates is unclear. Using extensive mutagenesis of TDP-43, we demonstrate here that three tryptophan residues and, to a lesser extent, four other aromatic residues are most important for TDP-43 to undergo LLPS. Our results also suggested that only a few residues may be required for TDP-43 LLPS because the α-helical segment (spanning ∼20 residues) in the middle part of the C-terminal domain tends to self-assemble, reducing the number of motifs required for forming a multivalent connection. Our results indicating that a self-associating α-helical element with a few key residues regulates condensate formation highlight a different type of LLPS involving intrinsically disordered regions. The C-terminal domain of TDP-43 contains ∼50 disease-related mutations, with no clear physicochemical link between them. We propose that they may disrupt LLPS indirectly by interfering with the key residues identified here.

Robust and transferable quantification of NMR spectral quality using IROC analysis

Non-Fourier methods are increasingly utilized in NMR spectroscopy because of their ability to handle nonuniformly-sampled data. However, non-Fourier methods present unique challenges due to their nonlinearity, which can produce nonrandom noise and render conventional metrics for spectral quality such as signal-to-noise ratio unreliable. The lack of robust and transferable metrics (i.e. applicable to methods exhibiting different nonlinearities) has hampered comparison of non-Fourier methods and nonuniform sampling schemes, preventing the identification of best practices. We describe a novel method, in situ receiver operating characteristic analysis (IROC), for characterizing spectral quality based on the Receiver Operating Characteristic curve. IROC utilizes synthetic signals added to empirical data as "ground truth", and provides several robust scalar-valued metrics for spectral quality. This approach avoids problems posed by nonlinear spectral estimates, and provides a versatile quantitative means of characterizing many aspects of spectral quality. We demonstrate applications to parameter optimization in Fourier and non-Fourier spectral estimation, critical comparison of different methods for spectrum analysis, and optimization of nonuniform sampling schemes. The approach will accelerate the discovery of optimal approaches to nonuniform sampling experiment design and non-Fourier spectrum analysis for multidimensional NMR.

The physical forces mediating self-association and phase-separation in the C-terminal domain of TDP-43

The TAR DNA-binding protein of 43kDa (TDP-43) has been identified as the main component of amyotrophic lateral sclerosis (ALS) cytoplasmic inclusions. The link between this proteinopathy and TDP-43's intrinsically disordered C-terminal domain is well known, but recently also, this domain has been shown to be involved in the formation of the membraneless organelles that mediate TDP-43's functions. The mechanisms that underpin the liquid-liquid phase separation (LLPS) of these membraneless organelles undergo remain elusive. Crucially though, these factors may be the key to understanding the delicate balance between TDP-43's physiological and pathological functions. In this study, we used nuclear magnetic resonance spectroscopy and optical methods to demonstrate that an α-helical component in the centre (residues 320-340) of the C-terminal domain is related to the protein's self-association and LLPS. Systematically analysing ALS-related TDP-43 mutants (G298S, M337V, and Q331K) in different buffer conditions at different temperatures, we prove that this phase separation is driven by hydrophobic interactions but is inhibited by electrostatic repulsion. Based on these findings, we rationally introduced a mutant, W334G, and demonstrate that this mutant disrupts LLPS without disturbing this α-helical propensity. This tryptophan may serve as a key residue in this protein's LLPS.

Analysis of DNA binding by human factor xeroderma pigmentosum complementation group A (XPA) provides insight into its interactions with nucleotide excision repair substrates

Xeroderma pigmentosum (XP) complementation group A (XPA) is an essential scaffolding protein in the multiprotein nucleotide excision repair (NER) machinery. The interaction of XPA with DNA is a core function of this protein; a number of mutations in the DNA-binding domain (DBD) are associated with XP disease. Although structures of the central globular domain of human XPA and data on binding of DNA substrates have been reported, the structural basis for XPA's DNA-binding activity remains unknown. X-ray crystal structures of the central globular domain of yeast XPA (Rad14) with lesion-containing DNA duplexes have provided valuable insights, but the DNA substrates used for this study do not correspond to the substrates of XPA as it functions within the NER machinery. To better understand the DNA-binding activity of human XPA in NER, we used NMR to investigate the interaction of its DBD with a range of DNA substrates. We found that XPA binds different single-stranded/double-stranded junction DNA substrates with a common surface. Comparisons of our NMR-based mapping of binding residues with the previously reported Rad14-DNA crystal structures revealed similarities and differences in substrate binding between XPA and Rad14. This includes direct evidence for DNA contacts to the residues extending C-terminally from the globular core, which are lacking in the Rad14 construct. Moreover, mutation of the XPA residue corresponding to Phe-262 in Rad14, previously reported as being critical for DNA binding, had only a moderate effect on the DNA-binding activity of XPA. The DNA-binding properties of several disease-associated mutations in the DBD were investigated. These results suggest that for XPA mutants exhibiting altered DNA-binding properties, a correlation exists between the extent of reduction in DNA-binding affinity and the severity of symptoms in XP patients.

Resonance assignment of disordered protein with repetitive and overlapping sequence using combinatorial approach reveals initial structural propensities and local restrictions in the denatured state

NMR resonance assignment of intrinsically disordered proteins poses a challenge because of the limited dispersion of amide proton chemical shifts. This becomes even more complex with the increase in the size of the system. Residue specific selective labeling/unlabeling experiments have been used to resolve the overlap, but require multiple sample preparations. Here, we demonstrate an assignment strategy requiring only a single sample of uniformly labeled (13)C,(15)N-protein. We have used a combinatorial approach, involving 3D-HNN, CC(CO)NH and 2D-MUSIC, which allowed us to assign a denatured centromeric protein Cse4 of 229 residues. Further, we show that even the less sensitive experiments, when used in an efficient manner can lead to the complete assignment of a complex system without the use of specialized probes in a relatively short time frame. The assignment of the amino acids discloses the presence of local structural propensities even in the denatured state accompanied by restricted motion in certain regions that provides insights into the early folding events of the protein.

Convex accelerated maximum entropy reconstruction

Maximum entropy (MaxEnt) spectral reconstruction methods provide a powerful framework for spectral estimation of nonuniformly sampled datasets. Many methods exist within this framework, usually defined based on the magnitude of a Lagrange multiplier in the MaxEnt objective function. An algorithm is presented here that utilizes accelerated first-order convex optimization techniques to rapidly and reliably reconstruct nonuniformly sampled NMR datasets using the principle of maximum entropy. This algorithm - called CAMERA for Convex Accelerated Maximum Entropy Reconstruction Algorithm - is a new approach to spectral reconstruction that exhibits fast, tunable convergence in both constant-aim and constant-lambda modes. A high-performance, open source NMR data processing tool is described that implements CAMERA, and brief comparisons to existing reconstruction methods are made on several example spectra.

Fast acquisition of multidimensional NMR spectra of solids and mesophases using alternative sampling methods

Unique information about the atom-level structure and dynamics of solids and mesophases can be obtained by the use of multidimensional nuclear magnetic resonance (NMR) experiments. Nevertheless, the acquisition of these experiments often requires long acquisition times. We review here alternative sampling methods, which have been proposed to circumvent this issue in the case of solids and mesophases. Compared to the spectra of solutions, those of solids and mesophases present some specificities because they usually display lower signal-to-noise ratios, non-Lorentzian line shapes, lower spectral resolutions and wider spectral widths. We highlight herein the advantages and limitations of these alternative sampling methods. A first route to accelerate the acquisition time of multidimensional NMR spectra consists in the use of sparse sampling schemes, such as truncated, radial or random sampling ones. These sparsely sampled datasets are generally processed by reconstruction methods differing from the Discrete Fourier Transform (DFT). A host of non-DFT methods have been applied for solids and mesophases, including the G-matrix Fourier transform, the linear least-square procedures, the covariance transform, the maximum entropy and the compressed sensing. A second class of alternative sampling consists in departing from the Jeener paradigm for multidimensional NMR experiments. These non-Jeener methods include Hadamard spectroscopy as well as spatial or orientational encoding of the evolution frequencies. The increasing number of high field NMR magnets and the development of techniques to enhance NMR sensitivity will contribute to widen the use of these alternative sampling methods for the study of solids and mesophases in the coming years.

Multidimensional NMR spectroscopy in a single scan

Multidimensional NMR has become one of the most widespread spectroscopic tools available to study diverse structural and functional aspects of organic and biomolecules. A main feature of multidimensional NMR is the relatively long acquisition times that these experiments demand. For decades, scientists have been working on a variety of alternatives that would enable NMR to overcome this limitation, and deliver its data in shorter acquisition times. Counting among these methodologies is the so-called ultrafast (UF) NMR approach, which in principle allows one to collect arbitrary multidimensional correlations in a single sub-second transient. By contrast to conventional acquisitions, a main feature of UF NMR is a spatiotemporal manipulation of the spins that imprints the chemical shift and/or J-coupling evolutions being sought, into a spatial pattern. Subsequent gradient-based manipulations enable the reading out of this information and its multidimensional correlation into patterns that are identical to those afforded by conventional techniques. The current review focuses on the fundamental principles of this spatiotemporal UF NMR manipulation, and on a few of the methodological extensions that this form of spectroscopy has undergone during the years.

Resolution-enhanced 2D NMR of complex mixtures by non-uniform sampling

NMR is a powerful tool for the analysis of complex mixtures and the identification of individual components. Two-dimensional (2D) NMR potentially offers a wealth of information, but resolution is often sacrificed in order to contain experimental times. We explore the use of non-uniform sampling (NUS) to increase substantially the resolution of 2D NMR spectra of complex mixtures of small molecules, with no increase in experimental time. Two common pulse sequences for metabolomics applications are analysed, HSQC and TOCSY. Specific attention is paid to sensitivity in resolution-enhanced NUS spectra, using the signal-to-maximum-noise ratio as a metric. With a careful choice of sampling schedule and reconstruction algorithm, resolution in the (13) C dimension for HSQC is increased by a factor of at least 32, with no loss in sensitivity and no spurious peaks. For TOCSY, multiplets can be resolved in the indirect dimension in a reasonable experimental time. These properties should increase the usefulness of 2D NMR for metabolomics applications by, for example, increasing the chances of metabolite identification.

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.

Magic angle spinning NMR spectroscopy: a versatile technique for structural and dynamic analysis of solid-phase systems

Magic Angle Spinning (MAS) NMR spectroscopy is a powerful method for analysis of a broad range of systems, including inorganic materials, pharmaceuticals, and biomacromolecules. The recent developments in MAS NMR instrumentation and methodologies opened new vistas to atomic-level characterization of a plethora of chemical environments previously inaccessible to analysis, with unprecedented sensitivity and resolution.

Efficient and generalized processing of multidimensional NUS NMR data: the NESTA algorithm and comparison of regularization terms

The advantages of non-uniform sampling (NUS) in offering time savings and resolution enhancement in NMR experiments have been increasingly recognized. The possibility of sensitivity gain by NUS has also been demonstrated. Application of NUS to multidimensional NMR experiments requires the selection of a sampling scheme and a reconstruction scheme to generate uniformly sampled time domain data. In this report, an efficient reconstruction scheme is presented and used to evaluate a range of regularization algorithms that collectively yield a generalized solution to processing NUS data in multidimensional NMR experiments. We compare l1-norm (L1), iterative re-weighted l1-norm (IRL1), and Gaussian smoothed l0-norm (Gaussian-SL0) regularization for processing multidimensional NUS NMR data. Based on the reconstruction of different multidimensional NUS NMR data sets, L1 is demonstrated to be a fast and accurate reconstruction method for both quantitative, high dynamic range applications (e.g. NOESY) and for all J-coupled correlation experiments. Compared to L1, both IRL1 and Gaussian-SL0 are shown to produce slightly higher quality reconstructions with improved linearity in peak intensities, albeit with a computational cost. Finally, a generalized processing system, NESTA-NMR, is described that utilizes a fast and accurate first-order gradient descent algorithm (NESTA) recently developed in the compressed sensing field. NESTA-NMR incorporates L1, IRL1, and Gaussian-SL0 regularization. NESTA-NMR is demonstrated to provide an efficient, streamlined approach to handling all types of multidimensional NMR data using proteins ranging in size from 8 to 32 kDa.

Development of a method for reconstruction of crowded NMR spectra from undersampled time-domain data

NMR is a unique methodology for obtaining information about the conformational dynamics of proteins in heterogeneous biomolecular systems. In various NMR methods, such as transferred cross-saturation, relaxation dispersion, and paramagnetic relaxation enhancement experiments, fast determination of the signal intensity ratios in the NMR spectra with high accuracy is required for analyses of targets with low yields and stabilities. However, conventional methods for the reconstruction of spectra from undersampled time-domain data, such as linear prediction, spectroscopy with integration of frequency and time domain, and analysis of Fourier, and compressed sensing were not effective for the accurate determination of the signal intensity ratios of the crowded two-dimensional spectra of proteins. Here, we developed an NMR spectra reconstruction method, "conservation of experimental data in analysis of Fourier" (Co-ANAFOR), to reconstruct the crowded spectra from the undersampled time-domain data. The number of sampling points required for the transferred cross-saturation experiments between membrane proteins, photosystem I and cytochrome b 6 f, and their ligand, plastocyanin, with Co-ANAFOR was half of that needed for linear prediction, and the peak height reduction ratios of the spectra reconstructed from truncated time-domain data by Co-ANAFOR were more accurate than those reconstructed from non-uniformly sampled data by compressed sensing.

Magic angle spinning NMR of viruses

Viruses, relatively simple pathogens, are able to replicate in many living organisms and to adapt to various environments. Conventional atomic-resolution structural biology techniques, X-ray crystallography and solution NMR spectroscopy provided abundant information on the structures of individual proteins and nucleic acids comprising viruses; however, viral assemblies are not amenable to analysis by these techniques because of their large size, insolubility, and inherent lack of long-range order. In this article, we review the recent advances in magic angle spinning NMR spectroscopy that enabled atomic-resolution analysis of structure and dynamics of large viral systems and give examples of several exciting case studies.

Evaluation of the reliability of the maximum entropy method for reconstructing 3D and 4D NOESY-type NMR spectra of proteins

Despite their advantages in analysis, 4D NMR experiments are still infrequently used as a routine tool in protein NMR projects due to the long duration of the measurement and limited digital resolution. Recently, new acquisition techniques for speeding up multidimensional NMR experiments, such as nonlinear sampling, in combination with non-Fourier transform data processing methods have been proposed to be beneficial for 4D NMR experiments. Maximum entropy (MaxEnt) methods have been utilised for reconstructing nonlinearly sampled multi-dimensional NMR data. However, the artefacts arising from MaxEnt processing, particularly, in NOESY spectra have not yet been clearly assessed in comparison with other methods, such as quantitative maximum entropy, multidimensional decomposition, and compressed sensing. We compared MaxEnt with other methods in reconstructing 3D NOESY data acquired with variously reduced sparse sampling schedules and found that MaxEnt is robust, quick and competitive with other methods. Next, nonlinear sampling and MaxEnt processing were applied to 4D NOESY experiments, and the effect of the artefacts of MaxEnt was evaluated by calculating 3D structures from the NOE-derived distance restraints. Our results demonstrated that sufficiently converged and accurate structures (RMSD of 0.91Å to the mean and 1.36Å to the reference structures) were obtained even with NOESY spectra reconstructed from 1.6% randomly selected sampling points for indirect dimensions. This suggests that 3D MaxEnt processing in combination with nonlinear sampling schedules is still a useful and advantageous option for rapid acquisition of high-resolution 4D NOESY spectra of proteins.

Nonuniform sampling and non-Fourier signal processing methods in multidimensional NMR

Beginning with the introduction of Fourier Transform NMR by Ernst and Anderson in 1966, time domain measurement of the impulse response (the free induction decay, FID) consisted of sampling the signal at a series of discrete intervals. For compatibility with the discrete Fourier transform (DFT), the intervals are kept uniform, and the Nyquist theorem dictates the largest value of the interval sufficient to avoid aliasing. With the proposal by Jeener of parametric sampling along an indirect time dimension, extension to multidimensional experiments employed the same sampling techniques used in one dimension, similarly subject to the Nyquist condition and suitable for processing via the discrete Fourier transform. The challenges of obtaining high-resolution spectral estimates from short data records using the DFT were already well understood, however. Despite techniques such as linear prediction extrapolation, the achievable resolution in the indirect dimensions is limited by practical constraints on measuring time. The advent of non-Fourier methods of spectrum analysis capable of processing nonuniformly sampled data has led to an explosion in the development of novel sampling strategies that avoid the limits on resolution and measurement time imposed by uniform sampling. The first part of this review discusses the many approaches to data sampling in multidimensional NMR, the second part highlights commonly used methods for signal processing of such data, and the review concludes with a discussion of other approaches to speeding up data acquisition in NMR.

Nonuniform sampling and non-Fourier signal processing methods in multidimensional NMR

Beginning with the introduction of Fourier Transform NMR by Ernst and Anderson in 1966, time domain measurement of the impulse response (the free induction decay, FID) consisted of sampling the signal at a series of discrete intervals. For compatibility with the discrete Fourier transform (DFT), the intervals are kept uniform, and the Nyquist theorem dictates the largest value of the interval sufficient to avoid aliasing. With the proposal by Jeener of parametric sampling along an indirect time dimension, extension to multidimensional experiments employed the same sampling techniques used in one dimension, similarly subject to the Nyquist condition and suitable for processing via the discrete Fourier transform. The challenges of obtaining high-resolution spectral estimates from short data records using the DFT were already well understood, however. Despite techniques such as linear prediction extrapolation, the achievable resolution in the indirect dimensions is limited by practical constraints on measuring time. The advent of non-Fourier methods of spectrum analysis capable of processing nonuniformly sampled data has led to an explosion in the development of novel sampling strategies that avoid the limits on resolution and measurement time imposed by uniform sampling. The first part of this review discusses the many approaches to data sampling in multidimensional NMR, the second part highlights commonly used methods for signal processing of such data, and the review concludes with a discussion of other approaches to speeding up data acquisition in NMR.

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.

Sensitivity gains, linearity, and spectral reproducibility in nonuniformly sampled multidimensional MAS NMR spectra of high dynamic range

Recently, we have demonstrated that considerable inherent sensitivity gains are attained in MAS NMR spectra acquired by nonuniform sampling (NUS) and introduced maximum entropy interpolation (MINT) processing that assures the linearity of transformation between the time and frequency domains. In this report, we examine the utility of the NUS/MINT approach in multidimensional datasets possessing high dynamic range, such as homonuclear (13)C-(13)C correlation spectra. We demonstrate on model compounds and on 1-73-(U-(13)C,(15)N)/74-108-(U-(15)N) E. coli thioredoxin reassembly, that with appropriately constructed 50% NUS schedules inherent sensitivity gains of 1.7-2.1-fold are readily reached in such datasets. We show that both linearity and line width are retained under these experimental conditions throughout the entire dynamic range of the signals. Furthermore, we demonstrate that the reproducibility of the peak intensities is excellent in the NUS/MINT approach when experiments are repeated multiple times and identical experimental and processing conditions are employed. Finally, we discuss the principles for design and implementation of random exponentially biased NUS sampling schedules for homonuclear (13)C-(13)C MAS correlation experiments that yield high-quality artifact-free datasets.

Sparsely-sampled, high-resolution 4-D omit spectra for detection and assignment of intermolecular NOEs of protein complexes

Unambiguous detection and assignment of intermolecular NOEs are essential for structure determination of protein complexes by NMR. Such information has traditionally been obtained with 3-D half-filtered experiments, where scalar coupling-based purging of intramolecular signals allows for selective detection of intermolecular NOEs. However, due to the large variation of (1)JHC scalar couplings and limited chemical shift dispersion in the indirect proton dimension, it is difficult to obtain reliable and complete assignments of interfacial NOEs. Here, we demonstrate a strategy that combines selective labeling and high-resolution 4-D NOE spectroscopy with sparse sampling for reliable identification and assignment of intermolecular NOEs. Spectral subtraction of component-labeled complexes from a uniformly-labeled protein complex yields an "omit" spectrum containing positive intermolecular NOEs with little signal degeneracy. Such a strategy can be broadly applied to unbiased detection, assignment and presentation of intermolecular NOEs of protein complexes.

Practical aspects of NMR signal assignment in larger and challenging proteins

NMR has matured into a technique routinely employed for studying proteins in near physiological conditions. However, applications to larger proteins are impeded by the complexity of the various correlation maps necessary to assign NMR signals. This article reviews the data analysis techniques traditionally employed for resonance assignment and describes alternative protocols necessary for overcoming challenges in large protein spectra. In particular, simultaneous analysis of multiple spectra may help overcome ambiguities or may reveal correlations in an indirect manner. Similarly, visualization of orthogonal planes in a multidimensional spectrum can provide alternative assignment procedures. We describe examples of such strategies for assignment of backbone, methyl, and nOe resonances. We describe experimental aspects of data acquisition for the related experiments and provide guidelines for preliminary studies. Focus is placed on large folded monomeric proteins and examples are provided for 37, 48, 53, and 81 kDa proteins.

Perspectives in magnetic resonance: NMR in the post-FFT era

Multi-dimensional NMR spectra have traditionally been processed with the fast Fourier transformation (FFT). The availability of high field instruments, the complexity of spectra of large proteins, the narrow signal dispersion of some unstructured proteins, and the time needed to record the necessary increments in the indirect dimensions to exploit the resolution of the highfield instruments make this traditional approach unsatisfactory. New procedures need to be developed beyond uniform sampling of the indirect dimensions and reconstruction methods other than the straight FFT are necessary. Here we discuss approaches of non-uniform sampling (NUS) and suitable reconstruction methods. We expect that such methods will become standard for multi-dimensional NMR data acquisition with complex biological macromolecules and will dramatically enhance the power of modern biological NMR.

The causality principle in the reconstruction of sparse NMR spectra

Improved quality of NMR spectra reconstruction from non-uniformly sampled data by harnessing the causality principle and prior knowledge about the signal phase.

In situ study of reaction kinetics using compressed sensing NMR

CS-NMR improves the temporal resolution of conventional multi-dimensional NMR for species identification and study of reaction kinetics.

Sampling scheme and compressed sensing applied to solid-state NMR spectroscopy

We describe the incorporation of non-uniform sampling (NUS) compressed sensing (CS) into oriented sample (OS) solid-state NMR for stationary aligned samples and magic angle spinning (MAS) Solid-state NMR for unoriented 'powder' samples. Both simulated and experimental results indicate that 25-33% of a full linearly sampled data set is required to reconstruct two- and three-dimensional solid-state NMR spectra with high fidelity. A modest increase in signal-to-noise ratio accompanies the reconstruction.

NMR methods for structural studies of large monomeric and multimeric proteins

NMR structural studies of large monomeric and multimeric proteins face distinct challenges. In large monomeric proteins, the common occurrence of frequency degeneracies between residues impedes unambiguous assignment of NMR signals. To overcome this barrier, nonuniform sampling (NUS) is used to measure spectra with optimal resolution within reasonable time, new correlation maps resolve previous impasses in assignment strategies, and novel selective methyl labeling schemes provide additional structural probes without cluttering NMR spectra. These advances push the limits of NMR studies of large monomeric proteins. Large multimeric and multidomain proteins are studied by NMR when individual components can also be studied by NMR and have known structures. The structural properties of large assemblies are obtained by identifying binding surfaces, by orienting domains, and employing limited distance constraints. Segmental labeling and the combination of NMR with other methods have helped popularize NMR studies of such systems.

4D Non-uniformly sampled C,C-NOESY experiment for sequential assignment of 13C, 15N-labeled RNAs

A 4D (13)C(aromatic),(13)C(ribose)-edited NOESY experiment is introduced to improve sequential assignment of non-coding RNA, often hampered by a limited dispersion of (1)H and (13)C chemical shifts. The (13)C-labeling of RNA is fully utilized by inclusion of two (13)C evolution periods. These dimensions provide enhanced dispersion of resonances in the 4D spectrum. High spectral resolution is obtained using random non-uniform sampling in three indirect dimensions. The autocorrelation peaks are efficiently suppressed using band-selective pulses. Since the dynamic range of observed resonances is significantly decreased, the reconstruction of the 4D spectrum is greatly simplified. The experiment can replace two conventionally sampled 3D NOESY spectra (either ribose-(13)C- or aromatic-(13)C-separated), and remove most ambiguities encountered during sequential walks. The assignment strategy based on a homonuclear and 4D C,C-edited NOESY experiments is proposed and verified on a 34-nt RNA showing typical structure elements.

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 time-saving strategy for MAS NMR spectroscopy by combining nonuniform sampling and paramagnetic relaxation assisted condensed data collection

We present a time-saving strategy for acquiring 3D magic angle spinning NMR spectra for chemical shift assignments in proteins and protein assemblies in the solid state. By simultaneous application of nonuniform sampling (NUS) and paramagnetic-relaxation-assisted condensed data collection (PACC), we can attain 16-fold time reduction in the 3D experiments without sacrificing the signal-to-noise ratio or the resolution. We demonstrate that with appropriate concentration of paramagnetic dopant introduced into the sample the overwhelming majority of chemical shifts are not perturbed, with the exception of a limited number of shifts corresponding to residues located at the surface of the protein, which exhibit small perturbations. This approach enables multidimensional MAS spectroscopy in samples of intrinsically low sensitivity and/or high spectral congestion where traditional experiments fail, and is especially beneficial for structural and dynamics studies of large proteins and protein assemblies.

Fast Forward Maximum entropy reconstruction of sparsely sampled data

We present an analytical algorithm using fast Fourier transformations (FTs) for deriving the gradient needed as part of the iterative reconstruction of sparsely sampled datasets using the forward maximum entropy reconstruction (FM) procedure by Hyberts and Wagner [J. Am. Chem. Soc. 129 (2007) 5108]. The major drawback of the original algorithm is that it required one FT and one evaluation of the entropy per missing datapoint to establish the gradient. In the present study, we demonstrate that the entire gradient may be obtained using only two FT's and one evaluation of the entropy derivative, thus achieving impressive time savings compared to the original procedure. An example: A 2D dataset with sparse sampling of the indirect dimension, with sampling of only 75 out of 512 complex points (15% sampling) would lack (512-75)×2=874 points per ν(2) slice. The original FM algorithm would require 874 FT's and entropy function evaluations to setup the gradient, while the present algorithm is ∼450 times faster in this case, since it requires only two FT's. This allows reduction of the computational time from several hours to less than a minute. Even more impressive time savings may be achieved with 2D reconstructions of 3D datasets, where the original algorithm required days of CPU time on high-performance computing clusters only require few minutes of calculation on regular laptop computers with the new algorithm.

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.

Enhanced sensitivity by nonuniform sampling enables multidimensional MAS NMR spectroscopy of protein assemblies

We report dramatic sensitivity enhancements in multidimensional MAS NMR spectra by the use of nonuniform sampling (NUS) and introduce maximum entropy interpolation (MINT) processing that assures the linearity between the time and frequency domains of the NUS acquired data sets. A systematic analysis of sensitivity and resolution in 2D and 3D NUS spectra reveals that with NUS, at least 1.5- to 2-fold sensitivity enhancement can be attained in each indirect dimension without compromising the spectral resolution. These enhancements are similar to or higher than those attained by the newest-generation commercial cryogenic probes. We explore the benefits of this NUS/MaxEnt approach in proteins and protein assemblies using 1-73-(U-(13)C,(15)N)/74-108-(U-(15)N) Escherichia coli thioredoxin reassembly. We demonstrate that in thioredoxin reassembly, NUS permits acquisition of high-quality 3D-NCACX spectra, which are inaccessible with conventional sampling due to prohibitively long experiment times. Of critical importance, issues that hinder NUS-based SNR enhancement in 3D-NMR of liquids are mitigated in the study of solid samples in which theoretical enhancements on the order of 3-4 fold are accessible by compounding the NUS-based SNR enhancement of each indirect dimension. NUS/MINT is anticipated to be widely applicable and advantageous for multidimensional heteronuclear MAS NMR spectroscopy of proteins, protein assemblies, and other biological systems.

Efficient acquisition of high-resolution 4-D diagonal-suppressed methyl-methyl NOESY for large proteins

The methyl-methyl NOESY experiment plays an important role in determining the global folds of large proteins. Despite the high sensitivity of this experiment, the analysis of methyl-methyl NOEs is frequently hindered by the limited chemical shift dispersion of methyl groups, particularly methyl protons. This makes it difficult to unambiguously assign all of the methyl-methyl NOE crosspeaks using 3-D spectroscopy. The recent development of sparse sampling methods enables highly efficient acquisition of high-resolution 4-D spectra, which provides an excellent solution to resolving the degeneracy of methyl signals. However, many reconstruction algorithms for processing sparsely-sampled NMR data do not provide adequate suppression of aliasing artifacts in the presence of strong NOE diagonal signals. In order to overcome this limitation, we present a 4-D diagonal-suppressed methyl-methyl NOESY experiment specifically optimized for ultrasparse sampling and evaluate it using a deuterated, ILV methyl-protonated sample of the 42 kDa Escherichia coli maltose binding protein (MBP). Suppression of diagonal signals removes the dynamic range barrier of the methyl-methyl NOESY experiment such that residual aliasing artifacts in the CLEAN-reconstructed high-resolution 4-D spectrum can be further reduced. At an ultrasparse sampling rate of less than 1%, we were able to identify and unambiguously assign the vast majority of expected NOE crosspeaks between methyl groups separated by less than 5 Å and to detect very weak NOE crosspeaks from methyl groups that are over 7 Å apart.

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.

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.

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.

Sensitivity enhancement for maximally resolved two-dimensional NMR by nonuniform sampling

Resolving NMR signals which are separated in frequency on the order of their line widths requires obtaining the time domain free induction decay for a maximum time tmax = πT2 , where T2 is the transverse relaxation time of the given signals. Unfortunately, samples acquired beyond ∼1.26T2 contribute more noise than signal to the data; and samples in the range of about (0.75-1.26)× T2 have a negligible effect on the signal-to-noise ratio (SNR). Therefore, one must sacrifice SNR to reach evolution times of πT2 . One can preserve resolution in a shorter total experimental time by selecting a reduced set of samples from the Nyquist grid according to an exponential probability density which is on the order of the T2 of the signals. This practice is widely termed nonuniform sampling (NUS). We derive analytic theory for the enhancement of the intrinsic SNR of NUS time domain data compared with uniformly sampled data when the total experimental times are equivalent. This theory is general for any tmax and exponential weighting and is further carefully validated with simulations. Enhancements of SNR in the time domain on the order of twofold are routinely available when tmax ∼ πT2 and are reflected in the subsequent maximum entropy reconstructed spectra. SNR enhancement by NUS is demonstrated to be helpful in enabling the acquisition of HMQC spectra of dilute bile salts in which high resolution in the indirect carbon dimension is required.