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Nonuniform sampling was first proposed more than 40 years ago as an alternate method for sampling two-dimensional NMR data was initially pursued by only a small number of scientists. However, it has been gradually adopted after it was shown that major gains in measuring time and spectrum resolution can be obtained. Furthermore, migration of NMR software to the Unix environment facilitated development of new processing tools, and there is now a selection of programs available that yield high-quality reconstructions of NUS data. Moreover, it became obvious that recording high-resolution 3D and 4D protein NMR spectra at the resolution provided by modern high-field instruments was not possible with uniform sampling. It has become apparent that sparse, low dynamic-range NMR spectra, in particular, the triple resonance experiments are all best recorded with NUS. Optimal sampling schedules can yield benefits with respect to detecting weak peaks in high dynamic-range spectra and therefore a careful use of 2D HSQC-like spectra of mixed concentrations of small molecules is feasible. It is not yet clear whether crowded high dynamic-range spectra, such as NOESYs with many cross-peaks, benefit from NUS. On the other hand, NUS appears to be the best option for recording such high dynamic-range spectra if crowding can be reduced by shorten NOESY mixing times and/or by reducing overlap through isotopic labeling in high-resolution 3D and 4D experiments. Here we discuss principals and applications of the method, trying to provide a wide overview, but are biased by our own approaches. Thus, we will obviously miss important developments elsewhere and do not claim to be comprehensive., (© 2019 Elsevier Inc. All rights reserved.)

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

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., (Copyright © 2013 Elsevier Inc. All rights reserved.)

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.

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.

The Fourier transform has been the gold standard for transforming data from the time domain to the frequency domain in many spectroscopic methods, including NMR spectroscopy. While reliable, it has the drawback that it requires a grid of uniformely sampled data points, which is not efficient for decaying signals, and it also suffers from artifacts when dealing with nondecaying signals. Over several decades, many alternative sampling and transformation schemes have been proposed. Their common problem is that relative signal amplitudes are not well-preserved. Here we demonstrate the superior performance of a sine-weighted Poisson-gap distribution sparse-sampling scheme combined with forward maximum entropy (FM) reconstruction. While the relative signal amplitudes are well-preserved, we also find that the signal-to-noise ratio is enhanced up to 4-fold per unit of data acquisition time relative to traditional linear sampling.

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

We describe an efficient NMR triple resonance approach that correlates, at high resolution, protein side-chain and backbone resonances. It relies on the combination of two strategies: joint evolution of aliphatic side-chain proton/carbon coherences using a backbone N-H based HCcoNH reduced dimensionality (RD) experiment and non-uniform sampling (NUS) in two indirect dimensions. A typical data set containing such correlation information can be acquired in 2 days, at very high resolution unfeasible for conventional 4D HCcoNH-TOCSY experiments. The resonances of the aliphatic side-chain protons are unambiguously assigned to their attached carbons through the analysis of the 'sum' and 'difference' spectra. This approach circumvents the tedious process of manual resonance assignments using HCcH-TOCSY data, while providing additional resolving power of backbone N-H signals. A simple peak-list based algorithm has been implemented in the IBIS software for rapid automated backbone and side-chain assignments.

We have developed a tool for computer-assisted assignments of protein NMR spectra from triple resonance data. The program is designed to resemble established manual assignment procedures as closely as possible. IBIS exports its results in XEASY format. Thus, using IBIS the operator has continuous visual and accounting control over the progress of the assignment procedure. IBIS achieves complete assignments for those residues that exhibit sequential triple resonance connectivities within a few hours or days.