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.     

Analytical solution to the coupled evolution of multidimensional NMR data

A substantial time savings in the collection of multidimensional NMR data can be achieved by coupling the evolution of nuclei in the indirect dimensions. In order to save time, the sampling of the indirect dimensions is inherently incomplete. Therefore, many algorithms and samplings schemes have been developed aimed at separating the coevolved frequencies into analyzable data with limited artifacts. This paper extends the use of circulant matrices to describe coupled evolution with convolutions. By understanding the data in terms of convolutions, there is an exact solution to the inversion problem of extracting the orthogonal vectors from the coupled dimensions. Previously, this inversion problem has been solved using peak coordinates extracted from spectra. In contrast, the method described here uses spectra directly. This solution suggests a simple sampling scheme of collecting N orthogonal spectra, and N + 1 projections at specific projection angles, however, the theory developed can be extended generally to arbitrary projection angles. The circulant matrix methodology is demonstrated for simulated and real data. Further, an algorithm for separating overlapped signals in the detected dimension is presented. The algorithm involves the forward calculation of the coupled spectra from the orthogonal spectra, followed by back calculation of the orthogonal spectra from the coupled spectra, thus permitting rigorous cross-validation. This algorithm is shown to be robust in that erroneous solutions give rise to large artifacts. Electronic supplementary material  The online version of this article (doi:) contains supplementary material, which is available to authorized users.     

NMR spectroscopy reveals unexpected structural variation at the protein–protein interface in MHC class I molecules

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.     

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

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.     

GFT NMR Experiments for Polypeptide Backbone and 13Cβ Chemical Shift Assignment

(4,3)D, (5,3)D and (5,2)D GFT triple resonance NMR experiments are presented for polypeptide backbone and (13)C(beta) resonance assignment of (15)N/(13)C labeled proteins. The joint sampling of m = 2, 3 or 4 indirect chemical shift evolution periods of 4D and 5D NMR experiments yields the measurement of 2(m) - 1 linear combinations of shifts. To obtain sequential assignments, these are matched in corresponding experiments delineating either intra or interresidue correlations. Hence, an increased set of matches is registered when compared to conventional approaches, and the 4D or 5D information allows one to efficiently break chemical shift degeneracy. Moreover, comparison of single-quantum chemical shifts obtained after a least squares fit using either the intra or the interresidue data demonstrates that GFT NMR warrants highly accurate shift measurements. The new features of GFT NMR based resonance assignment strategies promise to be of particular value for establishing automated protocols.     

Performance tuning non-uniform sampling for sensitivity enhancement of signal-limited biological NMR

Non-uniform sampling (NUS) has been established as a route to obtaining true sensitivity enhancements when recording indirect dimensions of decaying signals in the same total experimental time as traditional uniform incrementation of the indirect evolution period. Theory and experiments have shown that NUS can yield up to two-fold improvements in the intrinsic signal-to-noise ratio (SNR) of each dimension, while even conservative protocols can yield 20–40 % improvements in the intrinsic SNR of NMR data. Applications of biological NMR that can benefit from these improvements are emerging, and in this work we develop some practical aspects of applying NUS nD-NMR to studies that approach the traditional detection limit of nD-NMR spectroscopy. Conditions for obtaining high NUS sensitivity enhancements are considered here in the context of enabling ¹H,¹⁵N-HSQC experiments on natural abundance protein samples and ¹H,¹³C-HMBC experiments on a challenging natural product. Through systematic studies we arrive at more precise guidelines to contrast sensitivity enhancements with reduced line shape constraints, and report an alternative sampling density based on a quarter-wave sinusoidal distribution that returns the highest fidelity we have seen to date in line shapes obtained by maximum entropy processing of non-uniformly sampled data.     

Removal of a time barrier for high-resolution multidimensional NMR spectroscopy

We introduce the recursive multidimensional decomposition (R-MDD) method to speed recording of high-resolution NMR spectra. The measurement time is logarithmically dependent on the sizes of indirect spectral dimensions. R-MDD has the sensitivity and resolution advantages of optimized nonuniform acquisition schemes and is applicable to all types of biomolecular spectra. We demonstrated it for triple resonance experiments on three globular proteins (ubiquitin, azurin and the barstar-barnase complex) of 8-22 kDa.     

Al NMR: a novel NMR data processing program optimized for sparse sampling

Sparse sampling in biomolecular multidimensional NMR offers increased acquisition speed and resolution and, if appropriate conditions are met, an increase in sensitivity. Sparse sampling of indirectly detected time domains combined with the direct truly multidimensional Fourier transform has elicited particular attention because of the ability to generate a final spectrum amenable to traditional analysis techniques. A number of sparse sampling schemes have been described including radial sampling, random sampling, concentric sampling and variations thereof. A fundamental feature of these sampling schemes is that the resulting time domain data array is not amenable to traditional Fourier transform based processing and phasing correction techniques. In addition, radial sampling approaches offer a number of advantages and capabilities that are also not accessible using standard NMR processing techniques. These include sensitivity enhancement, sub-matrix processing and determination of minimal sets of sampling angles. Here we describe a new software package (Al NMR) that enables these capabilities in the context of a general NMR data processing environment.     

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 ¹³C–¹³C correlation spectra. We demonstrate on model compounds and on 1–73-(U-¹³C,¹⁵N)/74–108-(U-¹⁵N) 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 ¹³C–¹³C MAS correlation experiments that yield high-quality artifact-free datasets.     

GFT projection NMR spectroscopy for proteins in the solid state

Recording of four-dimensional (4D) spectra for proteins in the solid state has opened new avenues to obtain virtually complete resonance assignments and three-dimensional (3D) structures of proteins. As in solution state NMR, the sampling of three indirect dimensions leads per se to long minimal measurement time. Furthermore, artifact suppression in solid state NMR relies primarily on radio-frequency pulse phase cycling. For an n-step phase cycle, the minimal measurement times of both 3D and 4D spectra are increased n times. To tackle the associated ‘sampling problem’ and to avoid sampling limited data acquisition, solid state G-Matrix Fourier Transform (SS GFT) projection NMR is introduced to rapidly acquire 3D and 4D spectral information. Specifically, (4,3)D (HA)CANCOCX and (3,2)D (HACA)NCOCX were implemented and recorded for the 6 kDa protein GB1 within about 10% of the time required for acquiring the conventional congeners with the same maximal evolution times and spectral widths in the indirect dimensions. Spectral analysis was complemented by comparative analysis of expected spectral congestion in conventional and GFT NMR experiments, demonstrating that high spectral resolution of the GFT NMR experiments enables one to efficiently obtain nearly complete resonance assignments even for large proteins.     

Pitfalls in compressed sensing reconstruction and how to avoid them

Multidimensional NMR can provide unmatched spectral resolution, which is crucial when dealing with samples of biological macromolecules. The resolution, however, comes at the high price of long experimental time. Non-uniform sampling (NUS) of the evolution time domain allows to suppress this limitation by sampling only a small fraction of the data, but requires sophisticated algorithms to reconstruct omitted data points. A significant group of such algorithms known as compressed sensing (CS) is based on the assumption of sparsity of a reconstructed spectrum. Several papers on the application of CS in multidimensional NMR have been published in the last years, and the developed methods have been implemented in most spectral processing software. However, the publications rarely show the cases when NUS reconstruction does not work perfectly or explain how to solve the problem. On the other hand, every-day users of NUS develop their rules-of-thumb, which help to set up the processing in an optimal way, but often without a deeper insight. In this paper, we discuss several sources of problems faced in CS reconstructions: low sampling level, missassumption of spectral sparsity, wrong stopping criterion and attempts to extrapolate the signal too much. As an appendix, we provide MATLAB codes of several CS algorithms used in NMR. We hope that this work will explain the mechanism of NUS reconstructions and help readers to set up acquisition and processing parameters. Also, we believe that it might be helpful for algorithm developers.     

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.

     

Strategy for complete NMR assignment of disordered proteins with highly repetitive sequences based on resolution-enhanced 5D experiments

A strategy for complete backbone and side-chain resonance assignment of disordered proteins with highly repetitive sequence is presented. The protocol is based on three resolution-enhanced NMR experiments: 5D HN(CA)CONH provides sequential connectivity, 5D HabCabCONH is utilized to identify amino acid types, and 5D HC(CC-TOCSY)CONH is used to assign the side-chain resonances. The improved resolution was achieved by a combination of high dimensionality and long evolution times, allowed by non-uniform sampling in the indirect dimensions. Random distribution of the data points and Sparse Multidimensional Fourier Transform processing were used. Successful application of the assignment procedure to a particularly difficult protein, δ subunit of RNA polymerase from Bacillus subtilis, is shown to prove the efficiency of the strategy. The studied protein contains a disordered C-terminal region of 81 amino acids with a highly repetitive sequence. While the conventional assignment methods completely failed due to a very small differences in chemical shifts, the presented strategy provided a complete backbone and side-chain assignment.     

4D experiments measured with APSY for automated backbone resonance assignments of large proteins

Detailed structural and functional characterization of proteins by solution NMR requires sequence-specific resonance assignment. We present a set of transverse relaxation optimization (TROSY) based four-dimensional automated projection spectroscopy (APSY) experiments which are designed for resonance assignments of proteins with a size up to 40 kDa, namely HNCACO, HNCOCA, HNCACB and HN(CO)CACB. These higher-dimensional experiments include several sensitivity-optimizing features such as multiple quantum parallel evolution in a ‘just-in-time’ manner, aliased off-resonance evolution, evolution-time optimized APSY acquisition, selective water-handling and TROSY. The experiments were acquired within the concept of APSY, but they can also be used within the framework of sparsely sampled experiments. The multidimensional peak lists derived with APSY provided chemical shifts with an approximately 20 times higher precision than conventional methods usually do, and allowed the assignment of 90 % of the backbone resonances of the perdeuterated primase-polymerase ORF904, which contains 331 amino acid residues and has a molecular weight of 38.4 kDa.     

Resolution enhancement by homonuclear J-decoupling: application to three-dimensional solid-state magic angle spinning NMR spectroscopy

We describe a simple protocol to achieve homonuclear J-decoupling in the indirect dimensions of multidimensional experiments, and to enhance spectral resolution of the backbone Calpha carbons in the 3D NCACX experiment. In the proposed protocol, the refocusing of the Calpha-CO homonuclear J-couplings is achieved by applying an off-resonance selective pi pulse to the CO spectral region in the middle of Calpha chemical shift evolution. As is commonly used in solution NMR, a compensatory echo period is used to refocus the unwanted chemical shift evolution of Calpha spins, which takes place during the off-resonance selective pulse. The experiments were carried out on the beta1 immunoglobulin binding domain of protein G (GB1). In GB1, such implementation results in significantly reduced line widths, and leads to an overall sensitivity enhancement.     

High-resolution aliphatic side-chain assignments in 3D HCcoNH experiments with joint H–C evolution and non-uniform sampling

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.     

NMR experiments for the rapid identification of P=O···H–X type hydrogen bonds in nucleic acids

Implementation of a new algorithm, SMILE, is described for reconstruction of non-uniformly sampled two-, three- and four-dimensional NMR data, which takes advantage of the known phases of the NMR spectrum and the exponential decay of underlying time domain signals. The method is very robust with respect to the chosen sampling protocol and, in its default mode, also extends the truncated time domain signals by a modest amount of non-sampled zeros. SMILE can likewise be used to extend conventional uniformly sampled data, as an effective multidimensional alternative to linear prediction. The program is provided as a plug-in to the widely used NMRPipe software suite, and can be used with default parameters for mainstream application, or with user control over the iterative process to possibly further improve reconstruction quality and to lower the demand on computational resources. For large data sets, the method is robust and demonstrated for sparsities down to ca 1%, and final all-real spectral sizes as large as 300 Gb. Comparison between fully sampled, conventionally processed spectra and randomly selected NUS subsets of this data shows that the reconstruction quality approaches the theoretical limit in terms of peak position fidelity and intensity. SMILE essentially removes the noise-like appearance associated with the point-spread function of signals that are a default of five-fold above the noise level, but impacts the actual thermal noise in the NMR spectra only minimally. Therefore, the appearance and interpretation of SMILE-reconstructed spectra is very similar to that of fully sampled spectra generated by Fourier transformation.     

Prediction of peak overlap in NMR spectra

Peak overlap is one of the major factors complicating the analysis of biomolecular NMR spectra. We present a general method for predicting the extent of peak overlap in multidimensional NMR spectra and its validation using both, experimental data sets and Monte Carlo simulation. The method is based on knowledge of the magnetization transfer pathways of the NMR experiments and chemical shift statistics from the Biological Magnetic Resonance Data Bank. Assuming a normal distribution with characteristic mean value and standard deviation for the chemical shift of each observable atom, an analytic expression was derived for the expected overlap probability of the cross peaks. The analytical approach was verified to agree with the average peak overlap in a large number of individual peak lists simulated using the same chemical shift statistics. The method was applied to eight proteins, including an intrinsically disordered one, for which the prediction results could be compared with the actual overlap based on the experimentally measured chemical shifts. The extent of overlap predicted using only statistical chemical shift information was in good agreement with the overlap that was observed when the measured shifts were used in the virtual spectrum, except for the intrinsically disordered protein. Since the spectral complexity of a protein NMR spectrum is a crucial factor for protein structure determination, analytical overlap prediction can be used to identify potentially difficult proteins before conducting NMR experiments. Overlap predictions can be tailored to particular classes of proteins by preparing statistics from corresponding protein databases. The method is also suitable for optimizing recording parameters and labeling schemes for NMR experiments and improving the reliability of automated spectra analysis and protein structure determination.     

Solution Conformation and Dynamics of the HIV-1 Integrase Core Domain

The human immunodeficiency virus type 1 (HIV-1) integrase (IN) is a critical enzyme involved in infection. It catalyzes two reactions to integrate the viral cDNA into the host genome, 3′ processing and strand transfer, but the dynamic behavior of the active site during catalysis of these two processes remains poorly characterized. NMR spectroscopy can reveal important structural details about enzyme mechanisms, but to date the IN catalytic core domain has proven resistant to such an analysis. Here, we present the first NMR studies of a soluble variant of the catalytic core domain. The NMR chemical shifts are found to corroborate structures observed in crystals, and confirm prior studies suggesting that the α4 helix extends toward the active site. We also observe a dramatic improvement in NMR spectra with increasing MgCl₂ concentration. This improvement suggests a structural transition not only near the active site residues but also throughout the entire molecule as IN binds Mg²⁺. In particular, the stability of the core domain is linked to the conformation of its C-terminal helix, which has implications for relative domain orientation in the full-length enzyme. ¹⁵N relaxation experiments further show that, although conformationally flexible, the catalytic loop of IN is not fully disordered in the absence of DNA. Indeed, automated chemical shift-based modeling of the active site loop reveals several stable clusters that show striking similarity to a recent crystal structure of prototype foamy virus IN bound to DNA.