Compressed sensing reconstruction of undersampled 3D NOESY spectra: application to large membrane proteins

Backbone and side‑chain 1H, 13C and 15N resonance assignments and secondary structure determination of the rhizobial FixJ

The symbiotic nitrogen-fixing bacterium Bradyrhizobium japonicum (B.japonicum) enables high soybean yields with little or no nitrogen fertiliser. A two component regulatory system comprising FixL, a histidine kinase with O2-sensing activity, and FixJ, a response regulator, controls the expression of genes involved in nitrogen fixation, such as fixK and nifA. Only under anaerobic conditions, the monophosphate group is transferred from FixL to the N-terminal receiver domain of FixJ (FixJN), which eventually promote the association of the C-terminal effector domain (FixJC) to the promoter regions of the nitrogen-fixation-related genes. Structural biological analyses carried out so far for rhizobial FixJ molecules have proposed a solution structure for FixJ that differs from the crystal structures, in which the two domains are extended. To understand the FixJ activation caused by phosphorylation of the N-terminal domain, which presumably regulates through the interactions between FixJN and FixJC, here we have performed backbone and sidechain resonance assignments of the unphosphorylated state of B. japonicum FixJ.

A DNA condensation code for linker histones

Linker histones play an essential role in chromatin packaging by facilitating compaction of the 11-nm fiber of nucleosomal "beads on a string." The result is a heterogeneous condensed state with local properties that range from dynamic, irregular, and liquid-like to stable and regular structures (the 30-nm fiber), which in turn impact chromatin-dependent activities at a fundamental level. The properties of the condensed state depend on the type of linker histone, particularly on the highly disordered C-terminal tail, which is the most variable region of the protein, both between species, and within the various subtypes and cell-type specific variants of a given organism. We have developed an in vitro model system comprising linker histone tail and linker DNA, which although very minimal, displays surprisingly complex behavior, and is sufficient to model the known states of linker histone-condensed chromatin: disordered "fuzzy" complexes ("open" chromatin), dense liquid-like assemblies (dynamic condensates), and higher-order structures (organized 30-nm fibers). A crucial advantage of such a simple model is that it allows the study of the various condensed states by NMR, circular dichroism, and scattering methods. Moreover, it allows capture of the thermodynamics underpinning the transitions between states through calorimetry. We have leveraged this to rationalize the distinct condensing properties of linker histone subtypes and variants across species that are encoded by the amino acid content of their C-terminal tails. Three properties emerge as key to defining the condensed state: charge density, lysine/arginine ratio, and proline-free regions, and we evaluate each separately using a strategic mutagenesis approach.

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.

Phosphorylation of the smooth muscle master splicing regulator RBPMS regulates its splicing activity

We previously identified RBPMS as a master regulator of alternative splicing in differentiated smooth muscle cells (SMCs). RBPMS is transcriptionally downregulated during SMC dedifferentiation, but we hypothesized that RBPMS protein activity might be acutely downregulated by post-translational modifications. Publicly available phosphoproteomic datasets reveal that Thr113 and Thr118 immediately adjacent to the RRM domain are commonly both phosphorylated. An RBPMS T113/118 phosphomimetic T/E mutant showed decreased splicing regulatory activity both in transfected cells and in a cell-free in vitro assay, while a non-phosphorylatable T/A mutant retained full activity. Loss of splicing activity was associated with a modest reduction in RNA affinity but significantly reduced RNA binding in nuclear extract. A lower degree of oligomerization of the T/E mutant might cause lower avidity of multivalent RNA binding. However, NMR analysis also revealed that the T113/118E peptide acts as an RNA mimic which can loop back and antagonize RNA-binding by the RRM domain. Finally, we identified ERK2 as the most likely kinase responsible for phosphorylation at Thr113 and Thr118. Collectively, our data identify a potential mechanism for rapid modulation of the SMC splicing program in response to external signals during the vascular injury response and atherogenesis.

Hidden Multivalency in Phosphatase Recruitment by a Disordered AKAP Scaffold

Disordered scaffold proteins provide multivalent landing pads that, via a series of embedded Short Linear Motifs (SLiMs), bring together the components of a complex to orchestrate precise spatial and temporal regulation of cellular processes. One such protein is AKAP5 (previously AKAP79), which contains SLiMs that anchor PKA and Calcineurin, and recruit substrate (the TRPV1 receptor). Calcineurin is anchored to AKAP5 by a well-characterised PxIxIT SLiM. Here we show, using a combination of biochemical and biophysical approaches, that the Calcineurin PxIxIT-binding groove also recognises several hitherto unknown lower-affinity SLiMs in addition to the PxIxIT motif. We demonstrate that the assembly is in reality a complex system with conserved SLiMs spanning a wide affinity range. The capture is analogous to that seen for many DNA-binding proteins that have a weak non-specific affinity for DNA outside the canonical binding site, but different in that it involves (i) two proteins, and (ii) hydrophobic rather than electrostatic interactions. It is also compatible with the requirement for both stable anchoring of the enzyme and responsive downstream signalling. We conclude that the AKAP5 C-terminus is enriched in lower-affinity/mini-SLiMs that, together with the canonical SLiM, maintain a structurally disordered but tightly regulated signalosome.

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.

NUScon: a community-driven platform for quantitative evaluation of nonuniform sampling in NMR

Although the concepts of nonuniform sampling (NUS​​​​​​​) and non-Fourier spectral reconstruction in multidimensional NMR began to emerge 4 decades ago , it is only relatively recently that NUS has become more commonplace. Advantages of NUS include the ability to tailor experiments to reduce data collection time and to improve spectral quality, whether through detection of closely spaced peaks (i.e., "resolution") or peaks of weak intensity (i.e., "sensitivity"). Wider adoption of these methods is the result of improvements in computational performance, a growing abundance and flexibility of software, support from NMR spectrometer vendors, and the increased data sampling demands imposed by higher magnetic fields. However, the identification of best practices still remains a significant and unmet challenge. Unlike the discrete Fourier transform, non-Fourier methods used to reconstruct spectra from NUS data are nonlinear, depend on the complexity and nature of the signals, and lack quantitative or formal theory describing their performance. Seemingly subtle algorithmic differences may lead to significant variabilities in spectral qualities and artifacts. A community-based critical assessment of NUS challenge problems has been initiated, called the "Nonuniform Sampling Contest" (NUScon), with the objective of determining best practices for processing and analyzing NUS experiments. We address this objective by constructing challenges from NMR experiments that we inject with synthetic signals, and we process these challenges using workflows submitted by the community. In the initial rounds of NUScon our aim is to establish objective criteria for evaluating the quality of spectral reconstructions. We present here a software package for performing the quantitative analyses, and we present the results from the first two rounds of NUScon. We discuss the challenges that remain and present a roadmap for continued community-driven development with the ultimate aim of providing best practices in this rapidly evolving field. The NUScon software package and all data from evaluating the challenge problems are hosted on the NMRbox platform.

A simple approach for reconstruction of non-uniformly sampled pseudo-3D NMR data for accurate measurement of spin relaxation parameters

We explain how to conduct a pseudo-3D relaxation series NUS measurement so that it can be reconstructed by existing 3D NUS reconstruction methods to give accurate relaxation values. We demonstrate using reconstruction algorithms IST and SMILE that this 3D approach allows lower sampling densities than for independent 2D reconstructions. This is in keeping with the common finding that higher dimensionality increases signal sparsity, enabling lower sampling density. The approach treats the relaxation series as ordinary 3D time-domain data whose imaginary part in the pseudo-dimension is zero, and applies any suitably linear 3D NUS reconstruction method accordingly. Best results on measured and simulated data were achieved using acquisitions with 9 to 12 planes and exponential spacing in the pseudo-dimension out to ~ 2 times the inverse decay time. Given these criteria, in typical cases where 2D reconstructions require 50% sampling, the new 3D approach generates spectra reliably at sampling densities of 25%.

Software for reconstruction of nonuniformly sampled NMR data

Nonuniform sampling (NUS) of multidimensional NMR experiments is a powerful tool to obtain high-resolution spectra with less instrument time. With NUS, only a subset of the points needed for conventional Fourier transformation is recorded, and sophisticated algorithms are needed to reconstruct the missing data points. During the last decade, several software packages implementing the reconstruction algorithms have emerged and been refined and now result in spectra of almost similar quality as spectra from conventionally recorded and processed data. However, from the number of literature references to the reconstruction methods, many more multidimensional NMR spectra could presumably be recorded with NUS. To help researchers considering to start using NUS for their NMR experiments, we here review 13 different reconstruction methods found in five software packages (CambridgeCS, hmsIST, MddNMR, NESTA-NMR, and SMILE). We have compared how the methods run with the provided example scripts for reconstructing a nonuniform sampled three-dimensional ¹⁵ N-NOESY-HSQC at sampling densities from 5% to 50%. Overall, the spectra are all of similar quality above 20% sampling density. Thus, without any particular knowledge about the details of the reconstruction algorithms, significant reduction in the experiment time can be achieved. Below 20% sampling density, the intensities of particular weak peaks start being affected. MddNMR's IST with virtual echo and the SMILE algorithms still reproduced the spectra with the highest accuracy of peak intensities.

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.

Structure and Molecular Recognition Mechanism of IMP-13 Metallo-β-Lactamase

Multidrug resistance among Gram-negative bacteria is a major global public health threat. Metallo-β-lactamases (MBLs) target the most widely used antibiotic class, the β-lactams, including the most recent generation of carbapenems. Interspecies spread renders these enzymes a serious clinical threat, and there are no clinically available inhibitors. We present the crystal structures of IMP-13, a structurally uncharacterized MBL from the Gram-negative bacterium Pseudomonas aeruginosa found in clinical outbreaks globally, and characterize the binding using solution nuclear magnetic resonance spectroscopy and molecular dynamics simulations. The crystal structures of apo IMP-13 and IMP-13 bound to four clinically relevant carbapenem antibiotics (doripenem, ertapenem, imipenem, and meropenem) are presented. Active-site plasticity and the active-site loop, where a tryptophan residue stabilizes the antibiotic core scaffold, are essential to the substrate-binding mechanism. The conserved carbapenem scaffold plays the most significant role in IMP-13 binding, explaining the broad substrate specificity. The observed plasticity and substrate-locking mechanism provide opportunities for rational drug design of novel metallo-β-lactamase inhibitors, essential in the fight against antibiotic resistance.

Compressed Sensing for Reconstructing Coherent Multidimensional Spectra

We apply two sparse reconstruction techniques, the least absolute shrinkage and selection operator (LASSO) and the sparse exponential mode analysis (SEMA), to two-dimensional (2D) spectroscopy. The algorithms are first tested on model data, showing that both are able to reconstruct the spectra using only a fraction of the data required by the traditional Fourier-based estimator. Through the analysis of the sparsely sampled experimental fluorescence-detected 2D spectra of LH2 complexes, we conclude that both SEMA and LASSO can be used to significantly reduce the required data, still allowing one to reconstruct the multidimensional spectra. Of the two techniques, it is shown that SEMA offers preferable performance, providing more accurate estimation of the spectral line widths and their positions. Furthermore, SEMA allows for off-grid components, enabling the use of a much smaller dictionary than that of the LASSO, thereby improving both the performance and the lowering of the computational complexity for reconstructing coherent multidimensional spectra.

Amino-acid selective isotope labeling enables simultaneous overlapping signal decomposition and information extraction from NMR spectra

Signal overlapping is a major bottleneck for protein NMR analysis. We propose a new method, stable-isotope-assisted parameter extraction (SiPex), to resolve overlapping signals by a combination of amino-acid selective isotope labeling (AASIL) and tensor decomposition. The basic idea of Sipex is that overlapping signals can be decomposed with the help of intensity patterns derived from quantitative fractional AASIL, which also provides amino-acid information. In SiPex, spectra for protein characterization, such as ¹⁵N relaxation measurements, are assembled with those for amino-acid information to form a four-order tensor, where the intensity patterns from AASIL contribute to high decomposition performance even if the signals share similar chemical shift values or characterization profiles, such as relaxation curves. The loading vectors of each decomposed component, corresponding to an amide group, represent both the amino-acid and relaxation information. This information link provides an alternative protein analysis method that does not require "assignments" in a general sense; i.e., chemical shift determinations, since the amino-acid information for some of the residues allows unambiguous assignment according to the dual selective labeling. SiPex can also decompose signals in time-domain raw data without Fourier transform, even in non-uniformly sampled data without spectral reconstruction. These features of SiPex should expand biological NMR applications by overcoming their overlapping and assignment problems.

Importance of time-ordered non-uniform sampling of multi-dimensional NMR spectra of Aβ1-42 peptide under aggregating conditions

Although the order of the time steps in which the non-uniform sampling (NUS) schedule is implemented when acquiring multi-dimensional NMR spectra is of limited importance when sample conditions remain unchanged over the course of the experiment, it is shown to have major impact when samples are unstable. In the latter case, time-ordering of the NUS data points by the normalized radial length yields a reduction of sampling artifacts, regardless of the spectral reconstruction algorithm. The disadvantage of time-ordered NUS sampling is that halting the experiment prior to its completion will result in lower spectral resolution, rather than a sparser data matrix. Alternatively, digitally correcting for sample decay prior to reconstruction of randomly ordered NUS data points can mitigate reconstruction artifacts, at the cost of somewhat lower sensitivity. Application of these sampling schemes to the Alzheimer's amyloid beta (Aβ1-42) peptide at an elevated concentration, low temperature, and 3 kbar of pressure, where approximately 75% of the peptide reverts to an NMR-invisible state during the collection of a 3D 15N-separated NOESY spectrum, highlights the improvement in artifact suppression and reveals weak medium-range NOE contacts in several regions, including the C-terminal region of the peptide.

Insight into partial agonism by observing multiple equilibria for ligand-bound and Gs-mimetic nanobody-bound β1-adrenergic receptor

A complex conformational energy landscape determines G-protein-coupled receptor (GPCR) signalling via intracellular binding partners (IBPs), e.g., G_s and β-arrestin. Using ¹³C methyl methionine NMR for the β₁-adrenergic receptor, we identify ligand efficacy-dependent equilibria between an inactive and pre-active state and, in complex with G_s-mimetic nanobody, between more and less active ternary complexes. Formation of a basal activity complex through ligand-free nanobody–receptor interaction reveals structural differences on the cytoplasmic receptor side compared to the full agonist-bound nanobody-coupled form, suggesting that ligand-induced variations in G-protein interaction underpin partial agonism. Significant differences in receptor dynamics are observed ranging from rigid nanobody-coupled states to extensive μs-to-ms timescale dynamics when bound to a full agonist. We suggest that the mobility of the full agonist-bound form primes the GPCR to couple to IBPs. On formation of the ternary complex, ligand efficacy determines the quality of the interaction between the rigidified receptor and an IBP and consequently the signalling level.

β₁-adrenergic receptors are expressed in cardiac tissue and stimulated by the sympathetic nervous system. Here, the authors use NMR spectroscopy to unravel the conformational diversity upon β₁-adrenergic receptor activation and provide structural insights into partial agonism and basal activity.

Improving resolution in multidimensional NMR using random quadrature detection with compressed sensing reconstruction

NMR spectroscopy is central to atomic resolution studies in biology and chemistry. Key to this approach are multidimensional experiments. Obtaining such experiments with sufficient resolution, however, is a slow process, in part since each time increment in every indirect dimension needs to be recorded twice, in quadrature. We introduce a modified compressed sensing (CS) algorithm enabling reconstruction of data acquired with random acquisition of quadrature components in gradient-selection NMR. We name this approach random quadrature detection (RQD). Gradient-selection experiments are essential to the success of modern NMR and with RQD, a 50 % reduction in the number of data points per indirect dimension is possible, by only acquiring one quadrature component per time point. Using our algorithm (CS_RQD), high quality reconstructions are achieved. RQD is modular and combined with non-uniform sampling we show that this provides increased flexibility in designing sampling schedules leading to improved resolution with increasing benefits as dimensionality of experiments increases, with particular advantages for 4- and higher dimensional experiments.

Sparse multidimensional iterative lineshape-enhanced (SMILE) reconstruction of both non-uniformly sampled and conventional NMR data

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.

Correlation between k-space sampling pattern and MTF in compressed sensing MRSI

PURPOSE

To investigate the relationship between the k-space sampling patterns used for compressed sensing MR spectroscopic imaging (CS-MRSI) and the modulation transfer function (MTF) of the metabolite maps. This relationship may allow the desired frequency content of the metabolite maps to be quantitatively tailored when designing an undersampling pattern.

METHODS

Simulations of a phantom were used to calculate the MTF of Nyquist sampled (NS) 32 × 32 MRSI, and four-times undersampled CS-MRSI reconstructions. The dependence of the CS-MTF on the k-space sampling pattern was evaluated for three sets of k-space sampling patterns generated using different probability distribution functions (PDFs). CS-MTFs were also evaluated for three more sets of patterns generated using a modified algorithm where the sampling ratios are constrained to adhere to PDFs.

RESULTS

Strong visual correlation as well as high R² was found between the MTF of CS-MRSI and the product of the frequency-dependant sampling ratio and the NS 32 × 32 MTF. Also, PDF-constrained sampling patterns led to higher reproducibility of the CS-MTF, and stronger correlations to the above-mentioned product.

CONCLUSIONS

The relationship established in this work provides the user with a theoretical solution for the MTF of CS MRSI that is both predictable and customizable to the user's needs.

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.

A new approach to compressed sensing for NMR

Compressed sensing (CS) has attracted a great deal of recent interest as an approach for spectrum analysis of nonuniformly sampled NMR data. Although theoretical justification for the method is abundant, it suffers from several weaknesses, among them poor convergence of some algorithms, and it remains an open question whether NMR spectra satisfy the sparsity requirements of CS theorems. The versions of CS used in NMR involve minimizing the l1 norm of the spectrum. They bear similarity to maximum entropy (MaxEnt) reconstruction, but critical comparison of the methods can be difficult. Here we describe a formalism that places CS and MaxEnt reconstruction on equal footing, enabling critical comparison of the two methods. We also describe a new algorithm for CS that restricts the computation of the l1 norm to the real channel for complex spectra and ensures causality. Preliminary 1D results demonstrate that this approach ameliorates some artifacts that can occur when using the l1 norm of the complex spectrum.

The experimental accuracy of the uni-directional exact NOE

We have established protocols to calculate exact NOEs (eNOE) from NOE data. eNOEs lend unprecedented precision to the calculation of distance restraints used for structure calculation. Moreover, as eNOEs are averaged quantities over all conformations of a molecule, they may contain accessible information of the sampled conformational space. In practice, a prerequisite for an exact interpretation is the evaluation of both NOESY cross-peak buildups. For large molecular sizes, the fraction of NOEs which can only be obtained from one cross peak typically increases. Distance restraints derived from such NOEs must be used with a tolerance for errors associated with the broken symmetry of the individual magnetization transfer pathways. The correct choice of upper and lower limits is particularly important for multiple-state ensemble calculation, where too narrow tolerances may lead to incorrect spatial sampling. In order to dissect these pathways in heavy-atom resolved 3D NOESY experiments, we analyze 2D [(1)H, (1)H]-NOESY experiments, which are the fundamental building blocks of the former. In combination with an analysis of excitation and inversion profiles of pulses on heavy atoms and relaxation effects during HXQC elements, we derive a rule for the correct choice of upper and lower distance limits derived from such uni-directional NOEs. We show that normalization of the cross- to the diagonal-peak intensities of the spins of magnetization destination rather than origin leads to similar errors of the distance restraints. This opens up the prospect of extended collection of unidirectional eNOEs.

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.

Reducing seed dependent variability of non-uniformly sampled multidimensional NMR data

The application of NMR spectroscopy to study the structure, dynamics and function of macromolecules requires the acquisition of several multidimensional spectra. The one-dimensional NMR time-response from the spectrometer is extended to additional dimensions by introducing incremented delays in the experiment that cause oscillation of the signal along "indirect" dimensions. For a given dimension the delay is incremented at twice the rate of the maximum frequency (Nyquist rate). To achieve high-resolution requires acquisition of long data records sampled at the Nyquist rate. This is typically a prohibitive step due to time constraints, resulting in sub-optimal data records to the detriment of subsequent analyses. The multidimensional NMR spectrum itself is typically sparse, and it has been shown that in such cases it is possible to use non-Fourier methods to reconstruct a high-resolution multidimensional spectrum from a random subset of non-uniformly sampled (NUS) data. For a given acquisition time, NUS has the potential to improve the sensitivity and resolution of a multidimensional spectrum, compared to traditional uniform sampling. The improvements in sensitivity and/or resolution achieved by NUS are heavily dependent on the distribution of points in the random subset acquired. Typically, random points are selected from a probability density function (PDF) weighted according to the NMR signal envelope. In extreme cases as little as 1% of the data is subsampled. The heavy under-sampling can result in poor reproducibility, i.e. when two experiments are carried out where the same number of random samples is selected from the same PDF but using different random seeds. Here, a jittered sampling approach is introduced that is shown to improve random seed dependent reproducibility of multidimensional spectra generated from NUS data, compared to commonly applied NUS methods. It is shown that this is achieved due to the low variability of the inherent sensitivity of the random subset chosen from a given PDF. Finally, it is demonstrated that metrics used to find optimal NUS distributions are heavily dependent on the inherent sensitivity of the random subset, and such optimisation is therefore less critical when using the proposed sampling scheme.

Compressed sensing to accelerate magnetic resonance spectroscopic imaging: evaluation and application to 23Na-imaging of mouse hearts

BACKGROUND

Magnetic Resonance Spectroscopic Imaging (MRSI) has wide applicability for non-invasive biochemical assessment in clinical and pre-clinical applications but suffers from long scan times. Compressed sensing (CS) has been successfully applied to clinical 1H MRSI, however a detailed evaluation of CS for conventional chemical shift imaging is lacking. Here we evaluate the performance of CS accelerated MRSI, and specifically apply it to accelerate 23Na-MRSI on mouse hearts in vivo at 9.4 T.

METHODS

Synthetic phantom data representing a simplified section across a mouse thorax were used to evaluate the fidelity of the CS reconstruction for varying levels of under-sampling, resolution and signal-to-noise ratios (SNR). The amplitude of signals arising from within a compartment, and signal contamination arising from outside the compartment relative to noise-free Fourier-transformed (FT) data were determined. Simulation results were subsequently verified experimentally in phantoms and in three mouse hearts in vivo.

RESULTS

CS reconstructed MRSI data are scaled linearly relative to absolute signal intensities from the fully-sampled FT reconstructed case (R(2) > 0.8, p-value < 0.001). Higher acceleration factors resulted in a denoising of the reconstructed spectra, but also in an increased blurring of compartment boundaries, particularly at lower spatial resolutions. Increasing resolution and SNR decreased cross-compartment contamination and yielded signal amplitudes closer to the FT data. Proof-of-concept high-resolution, 3-fold accelerated 23Na-amplitude maps of murine myocardium could be obtained within ~23 mins.

CONCLUSIONS

Relative signal amplitudes (i.e. metabolite ratios) and absolute quantification of metabolite concentrations can be accurately determined with up to 5-fold under-sampled, CS-reconstructed MRSI. Although this work focused on murine cardiac 23Na-MRSI, the results are equally applicable to other nuclei and tissues (e.g., 1H MRSI in brain). Significant reduction in MRSI scan time will reduce the burden on the subject, increase scanner throughput, and may open new avenues for (pre-) clinical metabolic studies.

Sensitivity of nonuniform sampling NMR

Many information-rich multidimensional experiments in nuclear magnetic resonance spectroscopy can benefit from a signal-to-noise ratio (SNR) enhancement of up to about 2-fold if a decaying signal in an indirect dimension is sampled with nonconsecutive increments, termed nonuniform sampling (NUS). This work provides formal theoretical results and applications to resolve major questions about the scope of the NUS enhancement. First, we introduce the NUS Sensitivity Theorem in which any decreasing sampling density applied to any exponentially decaying signal always results in higher sensitivity (SNR per square root of measurement time) than uniform sampling (US). Several cases will illustrate this theorem and show that even conservative applications of NUS improve sensitivity by useful amounts. Next, we turn to a serious limitation of uniform sampling: the SNR by US decreases for extending evolution times, and thus total experimental times, beyond 1.26T2 (T2 = signal decay constant). Thus, SNR and resolution cannot be simultaneously improved by extending US beyond 1.26T2. We find that NUS can eliminate this constraint, and we introduce the matched NUS SNR Theorem: an exponential sampling density matched to the signal decay always improves the SNR with additional evolution time. Though proved for a specific case, broader classes of NUS densities also improve SNR with evolution time. Applications of these theoretical results are given for a soluble plant natural product and a solid tripeptide (u-(13)C,(15)N-MLF). These formal results clearly demonstrate the inadequacies of applying US to decaying signals in indirect nD-NMR dimensions, supporting a broader adoption of NUS.

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.

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.

Towards a true protein movie: a perspective on the potential impact of the ensemble-based structure determination using exact NOEs

Confined by the Boltzmann distribution of the energies of the states, a multitude of structural states are inherent to biomolecules. For a detailed understanding of a protein's function, its entire structural landscape at atomic resolution and insight into the interconversion between all the structural states (i.e. dynamics) are required. Whereas dedicated trickery with NMR relaxation provides aspects of local dynamics, and 3D structure determination by NMR is well established, only recently have several attempts been made to formulate a more comprehensive description of the dynamics and the structural landscape of a protein. Here, a perspective is given on the use of exact NOEs (eNOEs) for the elucidation of structural ensembles of a protein describing the covered conformational space.

The nuclear Overhauser effect from a quantitative perspective

The nuclear Overhauser enhancement or effect (NOE) is the most important measure in liquid-state NMR with macromolecules. Thus, the NOE is the subject of numerous reviews and books. Here, the NOE is revisited in light of our recently introduced measurements of exact nuclear Overhauser enhancements (eNOEs), which enabled the determination of multiple-state 3D protein structures. This review encompasses all relevant facets from the theoretical considerations to the use of eNOEs in multiple-state structure calculation. Important aspects include a detailed presentation of the relaxation theory relevant for the nuclear Overhauser effect, the estimation of the correction for spin diffusion, the experimental determination of the eNOEs, the conversion of eNOE rates into distances and validation of their quality, the distance-restraint classification and the protocols for calculation of structures and ensembles.

Low-field permanent magnets for industrial process and quality control

In this review we focus on the technology associated with low-field NMR. We present the current state-of-the-art in low-field NMR hardware and experiments, considering general magnet designs, rf performance, data processing and interpretation. We provide guidance on obtaining the optimum results from these instruments, along with an introduction for those new to low-field NMR. The applications of lowfield NMR are now many and diverse. Furthermore, niche applications have spawned unique magnet designs to accommodate the extremes of operating environment or sample geometry. Trying to capture all the applications, methods, and hardware encompassed by low-field NMR would be a daunting task and likely of little interest to researchers or industrialists working in specific subject areas. Instead we discuss only a few applications to highlight uses of the hardware and experiments in an industrial environment. For details on more particular methods and applications, we provide citations to specialized review articles.

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.

Phase reconstruction from velocity-encoded MRI measurements--a survey of sparsity-promoting variational approaches

In recent years there has been significant developments in the reconstruction of magnetic resonance velocity images from sub-sampled k-space data. While showing a strong improvement in reconstruction quality compared to classical approaches, the vast number of different methods, and the challenges in setting them up, often leaves the user with the difficult task of choosing the correct approach, or more importantly, not selecting a poor approach. In this paper, we survey variational approaches for the reconstruction of phase-encoded magnetic resonance velocity images from sub-sampled k-space data. We are particularly interested in regularisers that correctly treat both smooth and geometric features of the image. These features are common to velocity imaging, where the flow field will be smooth but interfaces between the fluid and surrounding material will be sharp, but are challenging to represent sparsely. As an example we demonstrate the variational approaches on velocity imaging of water flowing through a packed bed of solid particles. We evaluate Wavelet regularisation against Total Variation and the relatively recent second order Total Generalised Variation regularisation. We combine these regularisation schemes with a contrast enhancement approach called Bregman iteration. We verify for a variety of sampling patterns that Morozov's discrepancy principle provides a good criterion for stopping the iterations. Therefore, given only the noise level, we present a robust guideline for setting up a variational reconstruction scheme for MR velocity imaging.

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.

Structure determination of α-helical membrane proteins by solution-state NMR: emphasis on retinal proteins

The biochemical processes of living cells involve a numerous series of reactions that work with exceptional specificity and efficiency. The tight control of this intricate reaction network stems from the architecture of the proteins that drive the chemical reactions and mediate protein-protein interactions. Indeed, the structure of these proteins will determine both their function and interaction partners. A detailed understanding of the proximity and orientation of pivotal functional groups can reveal the molecular mechanistic basis for the activity of a protein. Together with X-ray crystallography and electron microscopy, NMR spectroscopy plays an important role in solving three-dimensional structures of proteins at atomic resolution. In the challenging field of membrane proteins, retinal-binding proteins are often employed as model systems and prototypes to develop biophysical techniques for the study of structural and functional mechanistic aspects. The recent determination of two 3D structures of seven-helical trans-membrane retinal proteins by solution-state NMR spectroscopy highlights the potential of solution NMR techniques in contributing to our understanding of membrane proteins. This review summarizes the multiple strategies available for expression of isotopically labeled membrane proteins. Different environments for mimicking lipid bilayers will be presented, along with the most important NMR methods and labeling schemes used to generate high-quality NMR spectra. The article concludes with an overview of types of conformational restraints used for generation of high-resolution structures of membrane proteins. This article is part of a Special Issue entitled: Retinal Proteins - You can teach an old dog new tricks.

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

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

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

It is well established that non-uniform sampling (NUS) allows acquisition of multi-dimensional NMR spectra at a resolution that cannot be obtained with traditional uniform acquisition through the indirect dimensions. However, the impact of NUS on the signal-to-noise ratio (SNR) and sensitivity are less well documented. SNR and sensitivity are essential aspects of NMR experiments as they define the quality and extent of data that can be obtained. This is particularly important for spectroscopy with low concentration samples of biological macromolecules. There are different ways of defining the SNR depending on how to measure the noise, and the distinction between SNR and sensitivity is often not clear. While there are defined procedures for measuring sensitivity with high concentration NMR standards, such as sucrose, there is no clear or generally accepted definition of sensitivity when comparing different acquisition and processing methods for spectra of biological macromolecules with many weak signals close to the level of noise. Here we propose tools for estimating the SNR and sensitivity of NUS spectra with respect to sampling schedule and reconstruction method. We compare uniformly acquired spectra with NUS spectra obtained in the same total measuring time. The time saving obtained when only 1/k of the Nyquist grid points are sampled is used to measure k-fold more scans per increment. We show that judiciously chosen NUS schedules together with suitable reconstruction methods can yield a significant increase of the SNR within the same total measurement time. Furthermore, we propose to define the sensitivity as the probability to detect weak peaks and show that time-equivalent NUS can considerably increase this detection sensitivity. The sensitivity gain increases with the number of NUS indirect dimensions. Thus, well-chosen NUS schedules and reconstruction methods can significantly increase the information content of multidimensional NMR spectra of challenging biological macromolecules.