Knowledge-based nonuniform sampling in multidimensional NMR

Fisher information for smart sampling in time-domain spectroscopy

Time-domain spectroscopy encompasses a wide range of techniques, such as Fourier-transform infrared, pump-probe, Fourier-transform Raman, and two-dimensional electronic spectroscopies. These methods enable various applications, such as molecule characterization, excited state dynamics studies, or spectral classification. Typically, these techniques rarely use sampling schemes that exploit the prior knowledge scientists typically have before the actual experiment. Indeed, not all sampling coordinates carry the same amount of information, and a careful selection of the sampling points may notably affect the resulting performance. In this work, we rationalize, with examples, the various advantages of using an optimal sampling scheme tailored to the specific experimental characteristics and/or expected results. We show that using a sampling scheme optimizing the Fisher information minimizes the variance of the desired parameters. This can greatly improve, for example, spectral classifications and multidimensional spectroscopy. We demonstrate how smart sampling may reduce the acquisition time of an experiment by one to two orders of magnitude, while still providing a similar level of information.

nus-tool: A unified program for generating and analyzing sample schedules for nonuniformly sampled NMR experiments

Increases in digital resolution achieved by high-field NMR require increases in spectral width. Additionally, the ability to resolve two overlapping peaks requires a sufficiently long acquisition time. These constraints combine, so that achieving high resolution spectra on high-field magnets requires long experiment times when employing uniform sampling and Fourier Transform processing. These limitations may be addressed by using nonuniform sampling (NUS), but the complexity of the parameter space across the variety of available NUS schemes greatly hinders the establishment of optimal approaches and best practices. We address these challenges with nus-tool, which is a software package for generating and analyzing NUS schedules. The nus-tool software internally implements random sampling and exponentially biased sampling. Through pre-configured plug-ins, it also provides access to quantile sampling and Poisson gap sampling. The software computes the relative sensitivity, mean evolution time, point spread function, and peak-to-sidelobe ratio; all of which can be determined for a candidate sample schedule prior to running an experiment to verify expected sensitivity, resolution, and artifact suppression. The nus-tool package is freely available on the NMRbox platform through an interactive GUI and via the command line, which is especially useful for scripted workflows that investigate the effectiveness of various NUS schemes.

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.

The influence of the probability density function on spectral quality in nonuniformly sampled multidimensional NMR

The goal of nonuniform sampling (NUS) is to select a subset of free induction decays (FIDs) from the conventional, uniform grid in a manner that sufficiently samples short evolution times needed for improved sensitivity and long evolution times needed for enhanced resolution. In addition to specifying the number of FIDs to be collected from a uniform grid, NUS schemes also specify the distribution of the selected FIDs, which directly impacts sampling-induced artifacts. Sampling schemes typically address these heuristic guidelines by utilizing a probability density function (PDF) to bias the distribution of sampled evolution times. Given this common approach, schemes differentiate themselves by how the evolution times are distributed within the envelope of the PDF. Here, we employ maximum entropy reconstruction and utilize in situ receiver operating characteristic (IROC) to conduct a critical comparison of the sensitivity and resolution that can be achieved by three types of biased sampling schemes: exponential (PDF is exponentially decaying), Poisson-gap (PDF derived from a sine function), and quantile-directed (PDF defined by simple polynomial decay). This methodology reveals practical insights and trends regarding how the sampling schemes and bias can provide the highest sensitivity and resolution for two nonuniformly sampled dimensions in a three-dimensional biomolecular NMR experiment. The IROC analysis circumvents the limitations of common metrics when used with nonlinear spectral estimation (a characteristic of all methods used with NUS) by quantifying the spectral quality via synthetic signals that are added to the empirical dataset. Recovery of these synthetic signals provides a proxy for the quality of the empirical portion of the spectrum. The central finding is that differences in spectral quality are primarily driven by the strength of bias in the PDF. In addition, a sampling coverage threshold is observed that appears to be connected to the dependence of each NUS method on its random seed. The differences between sampling schemes and biases are most relevant below 20% coverage where seed-dependence is high, whereas at higher coverages, the performance metrics for all of the sampling schemes begin to converge and approach a seed-independent regime. The results presented here show that aggressive sampling at low coverage can produce high-quality spectra by employing a sampling scheme that adheres to a decaying PDF with a bias to a broad range of short evolution times and includes relatively few FIDs at long evolution times.

Non-uniform HYSCORE: Measurement, processing and analysis with Hyscorean

Non-uniform sampling (NUS) provides a considerable reduction of measurement time especially for multi-dimensional experiments. This comes at the cost of additional signal processing steps to reconstruct the complete signal from the experimental data points. Despite being routinely employed in NMR for many experiments, EPR applications have not benefited from NUS due to the lack of a straightforward implementation to perform NUS in common commercial spectrometers. In this work we present a novel method to perform NUS HYSCORE experiments on commercial Bruker EPR spectrometers, along with a benchmark of modern reconstruction methods, and new processing software tools for NUS HYSCORE signals. All of this comes in the form of a free-software package: Hyscorean. Experimental NUS spectra are measured and processed with this package using different reconstruction methods and compared to their uniform sampled counterparts, thereby showcasing the method's potential for EPR spectroscopy.

Improving the sensitivity of FT-NMR spectroscopy by apodization weighted sampling

Apodization weighted acquisition is a simple approach to enhance the sensitivity of multidimensional NMR spectra by scaling the number of scans during acquisition of the indirect dimension(s). The signal content of the resulting spectra is identical to conventionally sampled data, yet the spectra show improved signal-to-noise ratios. There are no special requirements for data acquisition and processing: the time-domain data can be transformed with the same schemes used for conventionally recorded spectra, including Fourier transformation. The method is of general use in multidimensional liquid and solid state NMR experiments if the number of recorded transients per sampling point is bigger than the minimum required phase cycle of the pulse sequence.

Framework for and evaluation of bursts in random sampling of multidimensional NMR experiments

The grouping of data in bursts, also referred to as clusters, spikes or clumps, is a common phenomenon in stochastic sampling. There have been several reports that suggest that in NMR, the presence of such bursts is beneficial to spectral reconstruction where data are sampled nonuniformly. In this work, we seek to define a mode of sampling that produces bursts of randomly distributed data in a controlled manner. An algorithm is described for achieving this where the burst length and its uniformity is controlled - we refer to this type of sampling mode as clustered sampling. Measures are introduced for assessing the "burstiness" of nonuniformly sampled data in multiple dimensions and properties of the point-spread-function of these schedules are assessed. The clustered sampling method is applied to samples drawn from an exponentially weighted distribution either distributed randomly or pseudo-randomly by use of a jittering algorithm. The results reveal that bursts introduce characteristic sampling artifacts that are shifted to low frequencies (red shifted), with respect to the signal frequency, and that they produce artifact-reduced regions at frequencies related to the burst length. This observation is contrary to that observed for sampling methods that seek to evenly distribute NUS data, such as jittered or Poisson sampling. Extensive evaluation of simulated data with comparable inherent sensitivity, reveals that at high sampling coverage (25% in 1D), the distribution of the data has little impact on common spectral quality measures. Application of the introduced clustered sampling method to an experimental 3D NOESY experiment showed results consistent with that found for the simulated 1D data. However, in the extremes of very sparse sampling, the results suggest that there may be some advantages associated with incorporation of bursts in nonuniform sampling. The tools and theory presented will serve as a starting point to further explore this novel mode of sampling in NMR.

Systematic Evaluation of Non-Uniform Sampling Parameters in the Targeted Analysis of Urine Metabolites by 1H,1H 2D NMR Spectroscopy

Non-uniform sampling (NUS) allows the accelerated acquisition of multidimensional NMR spectra. The aim of this contribution was the systematic evaluation of the impact of various quantitative NUS parameters on the accuracy and precision of 2D NMR measurements of urinary metabolites. Urine aliquots spiked with varying concentrations (15.6-500.0 µM) of tryptophan, tyrosine, glutamine, glutamic acid, lactic acid, and threonine, which can only be resolved fully by 2D NMR, were used to assess the influence of the sampling scheme, reconstruction algorithm, amount of omitted data points, and seed value on the quantitative performance of NUS in 1H,1H-TOCSY and 1H,1H-COSY45 NMR spectroscopy. Sinusoidal Poisson-gap sampling and a compressed sensing approach employing the iterative re-weighted least squares method for spectral reconstruction allowed a 50% reduction in measurement time while maintaining sufficient quantitative accuracy and precision for both types of homonuclear 2D NMR spectroscopy. Together with other advances in instrument design, such as state-of-the-art cryogenic probes, use of 2D NMR spectroscopy in large biomedical cohort studies seems feasible.

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.

Practical Nonuniform Sampling and Non-Fourier Spectral Reconstruction for Multidimensional NMR

A general approach to accelerating multidimensional NMR experiments via nonuniform sampling and maximum entropy spectral reconstruction was first demonstrated by Laue and colleagues in 1987. Following decades of continual improvements involving dozens of software packages for non-Fourier spectral analysis and many different schemes for nonuniform sampling, we still lack a clear consensus on best practices for sampling or spectral reconstruction, and programs for processing nonuniformly sampled data are not particularly user-friendly. Nevertheless, it is possible to discern conservative and general guidelines for nonuniform sampling and spectral reconstruction. Here, we describe a robust semi-automated workflow that employs these guidelines for simplifying the selection of a sampling schedule and the processing of the resulting nonuniformly sampled multidimensional NMR data. Our approach is based on NMRbox, a shared platform for NMR software that facilitates workflow development and execution, and enables rapid comparison of alternate approaches.

A non-uniform sampling approach enables studies of dilute and unstable proteins

NMR spectroscopy is a powerful method in structural and functional analysis of macromolecules and has become particularly prevalent in studies of protein structure, function and dynamics. Unique to NMR spectroscopy is the relatively low constraints on sample preparation and the high level of control of sample conditions. Proteins can be studied in a wide range of buffer conditions, e.g. different pHs and variable temperatures, allowing studies of proteins under conditions that are closer to their native environment compared to other structural methods such as X-ray crystallography and electron microscopy. The key disadvantage of NMR is the relatively low sensitivity of the method, requiring either concentrated samples or very lengthy data-acquisition times. Thus, proteins that are unstable or can only be studied in dilute solutions are often considered practically unfeasible for NMR studies. Here, we describe a general method, where non-uniform sampling (NUS) allows for signal averaging to be monitored in an iterative manner, enabling efficient use of spectrometer time, ultimately leading to savings in costs associated with instrument and isotope-labelled protein use. The method requires preparation of multiple aliquots of the protein sample that are flash-frozen and thawed just before acquisition of a short NMR experiments carried out while the protein is stable (12 h in the presented case). Non-uniform sampling enables sufficient resolution to be acquired for each short experiment. Identical NMR datasets are acquired and sensitivity is monitored after each co-added spectrum is reconstructed. The procedure is repeated until sufficient signal-to-noise is obtained. We discuss how maximum entropy reconstruction is used to process the data, and propose a variation on the previously described method of automated parameter selection. We conclude that combining NUS with iterative co-addition is a general approach, and particularly powerful when applied to unstable proteins.

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.

Non-uniform sampling: post-Fourier era of NMR data collection and processing

The invention of multidimensional techniques in the 1970s revolutionized NMR, making it the general tool of structural analysis of molecules and materials. In the most straightforward approach, the signal sampling in the indirect dimensions of a multidimensional experiment is performed in the same manner as in the direct dimension, i.e. with a grid of equally spaced points. This results in lengthy experiments with a resolution often far from optimum. To circumvent this problem, numerous sparse-sampling techniques have been developed in the last three decades, including two traditionally distinct approaches: the radial sampling and non-uniform sampling. This mini review discusses the sparse signal sampling and reconstruction techniques from the point of view of an underdetermined linear algebra problem that arises when a full, equally spaced set of sampled points is replaced with sparse sampling. Additional assumptions that are introduced to solve the problem, as well as the shape of the undersampled Fourier transform operator (visualized as so-called point spread function), are shown to be the main differences between various sparse-sampling methods.

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.

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.

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.

Fast multidimensional NMR spectroscopy for sparse spectra

Multidimensional NMR spectroscopy is widely used for studies of molecular and biomolecular structure. A major disadvantage of multidimensional NMR is the long acquisition time which, regardless of sensitivity considerations, may be needed to obtain the final multidimensional frequency domain coefficients. In this article, a method for under-sampling multidimensional NMR acquisition of sparse spectra is presented. The approach is presented in the case of two-dimensional NMR acquisitions. It relies on prior knowledge about the support in the two-dimensional frequency domain to recover an over-determined system from the under-determined system induced in the linear acquisition model when under-sampled acquisitions are performed. This over-determined system can then be solved with linear least squares. The prior knowledge is obtained efficiently at a low cost from the one-dimensional NMR acquisition, which is generally acquired as a first step in multidimensional NMR. If this one-dimensional acquisition is intrinsically sparse, it is possible to reconstruct the corresponding two-dimensional acquisition from far fewer observations than those imposed by the Nyquist criterion, and subsequently to reduce the acquisition time. Further improvements are obtained by optimizing the sampling procedure for the least-squares reconstruction using the sequential backward selection algorithm. Theoretical and experimental results are given in the case of a traditional acquisition scheme, which demonstrate reliable and fast reconstructions with acceleration factors in the range 3-6. The proposed method outperforms the CS methods (OMP, L1) in terms of the reconstruction performance, implementation and computation time. The approach can be easily extended to higher dimensions and spectroscopic imaging.

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.

Nonuniform sampling and maximum entropy reconstruction in multidimensional NMR

NMR spectroscopy is one of the most powerful and versatile analytic tools available to chemists. The discrete Fourier transform (DFT) played a seminal role in the development of modern NMR, including the multidimensional methods that are essential for characterizing complex biomolecules. However, it suffers from well-known limitations: chiefly the difficulty in obtaining high-resolution spectral estimates from short data records. Because the time required to perform an experiment is proportional to the number of data samples, this problem imposes a sampling burden for multidimensional NMR experiments. At high magnetic field, where spectral dispersion is greatest, the problem becomes particularly acute. Consequently multidimensional NMR experiments that rely on the DFT must either sacrifice resolution in order to be completed in reasonable time or use inordinate amounts of time to achieve the potential resolution afforded by high-field magnets. Maximum entropy (MaxEnt) reconstruction is a non-Fourier method of spectrum analysis that can provide high-resolution spectral estimates from short data records. It can also be used with nonuniformly sampled data sets. Since resolution is substantially determined by the largest evolution time sampled, nonuniform sampling enables high resolution while avoiding the need to uniformly sample at large numbers of evolution times. The Nyquist sampling theorem does not apply to nonuniformly sampled data, and artifacts that occur with the use of nonuniform sampling can be viewed as frequency-aliased signals. Strategies for suppressing nonuniform sampling artifacts include the careful design of the sampling scheme and special methods for computing the spectrum. Researchers now routinely report that they can complete an N-dimensional NMR experiment 3(N-1) times faster (a 3D experiment in one ninth of the time). As a result, high-resolution three- and four-dimensional experiments that were prohibitively time consuming are now practical. Conversely, tailored sampling in the indirect dimensions has led to improved sensitivity. Further advances in nonuniform sampling strategies could enable further reductions in sampling requirements for high resolution NMR spectra, and the combination of these strategies with robust non-Fourier methods of spectrum analysis (such as MaxEnt) represent a profound change in the way researchers conduct multidimensional experiments. The potential benefits will enable more advanced applications of multidimensional NMR spectroscopy to study biological macromolecules, metabolomics, natural products, dynamic systems, and other areas where resolution, sensitivity, or experiment time are limiting. Just as the development of multidimensional NMR methods presaged multidimensional methods in other areas of spectroscopy, we anticipate that nonuniform sampling approaches will find applications in other forms of spectroscopy.

A gel-state 2D-NMR method for plant cell wall profiling and analysis: a model study with the amorphous cellulose and xylan from ball-milled cotton linters

Amorphous cellulose and xylan structures were analyzed using high-resolution 2D-NMR, and the NMR data were obtained in a DMSO-d₆/pyridine-d₅ (4 : 1) solvent system.

Effects of NMR spectral resolution on protein structure calculation

Adequate digital resolution and signal sensitivity are two critical factors for protein structure determinations by solution NMR spectroscopy. The prime objective for obtaining high digital resolution is to resolve peak overlap, especially in NOESY spectra with thousands of signals where the signal analysis needs to be performed on a large scale. Achieving maximum digital resolution is usually limited by the practically available measurement time. We developed a method utilizing non-uniform sampling for balancing digital resolution and signal sensitivity, and performed a large-scale analysis of the effect of the digital resolution on the accuracy of the resulting protein structures. Structure calculations were performed as a function of digital resolution for about 400 proteins with molecular sizes ranging between 5 and 33 kDa. The structural accuracy was assessed by atomic coordinate RMSD values from the reference structures of the proteins. In addition, we monitored also the number of assigned NOESY cross peaks, the average signal sensitivity, and the chemical shift spectral overlap. We show that high resolution is equally important for proteins of every molecular size. The chemical shift spectral overlap depends strongly on the corresponding spectral digital resolution. Thus, knowing the extent of overlap can be a predictor of the resulting structural accuracy. Our results show that for every molecular size a minimal digital resolution, corresponding to the natural linewidth, needs to be achieved for obtaining the highest accuracy possible for the given protein size using state-of-the-art automated NOESY assignment and structure calculation methods.

Formalism for hypercomplex multidimensional NMR employing partial-component subsampling

Multidimensional NMR spectroscopy typically employs phase-sensitive detection, which results in hypercomplex data (and spectra) when utilized in more than one dimension. Nonuniform sampling approaches have become commonplace in multidimensional NMR, enabling dramatic reductions in experiment time, increases in sensitivity and/or increases in resolution. In order to utilize nonuniform sampling optimally, it is necessary to characterize the relationship between the spectrum of a uniformly sampled data set and the spectrum of a subsampled data set. In this work we construct an algebra of hypercomplex numbers suitable for representing multidimensional NMR data along with partial-component nonuniform sampling (i.e. the hypercomplex components of data points are subsampled). This formalism leads to a modified DFT-Convolution relationship involving a partial-component, hypercomplex point-spread function set. The framework presented here is essential for the continued development and appropriate characterization of partial-component nonuniform sampling.

Sparse sampling methods in multidimensional NMR

Although the discrete Fourier transform played an enabling role in the development of modern NMR spectroscopy, it suffers from a well-known difficulty providing high-resolution spectra from short data records. In multidimensional NMR experiments, so-called indirect time dimensions are sampled parametrically, with each instance of evolution times along the indirect dimensions sampled via separate one-dimensional experiments. The time required to conduct multidimensional experiments is directly proportional to the number of indirect evolution times sampled. Despite remarkable advances in resolution with increasing magnetic field strength, multiple dimensions remain essential for resolving individual resonances in NMR spectra of biological macromolecues. Conventional Fourier-based methods of spectrum analysis limit the resolution that can be practically achieved in the indirect dimensions. Nonuniform or sparse data collection strategies, together with suitable non-Fourier methods of spectrum analysis, enable high-resolution multidimensional spectra to be obtained. Although some of these approaches were first employed in NMR more than two decades ago, it is only relatively recently that they have been widely adopted. Here we describe the current practice of sparse sampling methods and prospects for further development of the approach to improve resolution and sensitivity and shorten experiment time in multidimensional NMR. While sparse sampling is particularly promising for multidimensional NMR, the basic principles could apply to other forms of multidimensional spectroscopy.