A Recursive Non-Uniform Sampling Estimator for Asynchronous Nonlinear Systems

In this paper, we consider the problem of asynchronous estimation in the presence of packet losses for the randomly sampling nonlinear system. Packet losses occur at the control input and at the measurement side. Firstly, the synchronization of the asynchronous sampling system is realized by weighting the state of the adjacent state update points. Secondly, the projection theorem is used to estimate the system state at the sampling time. Due to modeling errors and unmodeled dynamics, obtaining an accurate dynamic model is challenging. Therefore, observation inference based on interpolation techniques is proposed to solve the asynchronous estimation problem. Furthermore, the algorithm is extended to multi-sensor systems to obtain a distributed fusion estimator. Finally, simulation experiments are conducted to validate the effectiveness of the algorithm.

Distributed State Estimation for Flapping-Wing Micro Air Vehicles with Information Fusion Correction

In this paper, we explore a nonlinear interactive network system comprising nodalized flapping-wing micro air vehicles (FMAVs) to address the distributed H∞ state estimation problem associated with FMAVs. We enhance the model by introducing an information fusion function, leading to an information-fusionized estimator model. This model ensures both estimation accuracy and the completeness of FMAV topological information within a unified framework. To facilitate the analysis, each FMAV's received signal is individually sampled using independent and time-varying samplers. Transforming the received signals into equivalent bounded time-varying delays through the input delay method yields a more manageable and analyzable time-varying nonlinear network error system. Subsequently, we construct a Lyapunov-Krasovskii functional (LKF) and integrate it with the refined Wirtinger and relaxed integral inequalities to derive design conditions for the FMAVs' distributed H∞ state estimator, minimizing conservatism. Finally, we validate the effectiveness and superiority of the designed estimator through simulations.

Application of accelerated heteronuclear single quantum coherence experiments to the rapid quantification of monosaccharides and disaccharides in dairy products

Monosaccharides and disaccharides are important dietary components, but if insufficiently metabolized by some population subgroups, they are also linked to disease patterns. Thus, the correct analytical identification, quantification, and labeling of these food components are crucial to inform and potentially protect consumers. Enzymatic assays and high-performance anion-exchange chromatography with pulsed amperometric detection are established methods for the quantification of monosaccharides and disaccharides that, however, require long measuring times (60-180 min). Accelerated methods for the identification and quantification of the nutritionally relevant monosaccharides and disaccharides d-glucose, d-galactose, d-fructose, sucrose, lactose, and maltose were therefore developed. To realize this goal, the NMR experiments HSQC (heteronuclear single quantum coherence) and acceleration by sharing adjacent polarization (ASAP)-HSQC were applied. Measurement times were reduced to 27 and 6 min, respectively, by optimizing the interscan delay and applying non-uniform sampling. The optimized methods were used to quantify d-glucose, d-galactose, d-fructose, sucrose, and lactose in various dairy products. Results of the HSQC and ASAP-HSQC methods are equivalent to the results of the reference methods in terms of both precision and accuracy, demonstrating that these methods can be used to correctly analyze nutritionally relevant monosaccharides and disaccharides in short times.

Distorted Acquisition of Dynamic Events Sensed by Frequency-Scanning Fiber-Optic Interrogators and a Mitigation Strategy

Fiber-optic dynamic interrogators, which use periodic frequency scanning, actually sample a time-varying measurand on a non-uniform time grid. Commonly, however, the sampled values are reported on a uniform time grid, synchronized with the periodic scanning. It is the novel and noteworthy message of this paper that this artificial assignment may give rise to significant distortions in the recovered signal. These distortions increase with both the signal frequency and measurand dynamic range for a given sampling rate and frequency scanning span of the interrogator. They may reach disturbing values in dynamic interrogators, which trade-off scanning speed with scanning span. The paper also calls for manufacturers of such interrogators to report the sampled values along with their instants of acquisition, allowing interpolation algorithms to substantially reduce the distortion. Experimental verification of a simulative analysis includes: (i) a commercial dynamic interrogator of ‘continuous’ FBG fibers that attributes the measurand values to a uniform time grid; as well as (ii) a dynamic Brillouin Optical time Domain (BOTDA) laboratory setup, which provides the sampled measurand values together with the sampling instants. Here, using the available measurand-dependent sampling instants, we demonstrate a significantly cleaner signal recovery using spline interpolation.

Staring Spotlight SAR with Nonlinear Frequency Modulation Signal and Azimuth Non-Uniform Sampling for Low Sidelobe Imaging

In synthetic aperture radar (SAR) imaging, geometric resolution, sidelobe level (SLL) and signal-to-noise ratio (SNR) are the most important parameters for measuring the SAR image quality. The staring spotlight mode continuously transmits signals to a fixed area by steering the azimuth beam to acquire azimuth high geometric resolution, and its two-dimensional (2D) impulse response with the low SLL is usually obtained from the 2D weighted power spectral density (PSD) by the selected weighting window function. However, this results in the SNR reduction due to 2D amplitude window weighting. In this paper, the staring spotlight SAR with nonlinear frequency modulation (NLFM) signal and azimuth non-uniform sampling (ANUS) is proposed to obtain high geometric resolution SAR images with the low SLL and almost without any SNR reduction. The NLFM signal obtains non-equal interval frequency sampling points under uniform time sampling by adjusting the instantaneous chirp rate. Its corresponding PSD is similar to the weighting window function, and its pulse compression result without amplitude window weighting has low sidelobes. To obtain a similar Doppler frequency distribution for low sidelobe imaging in azimuth, the received SAR echoes are designed to be non-uniformly sampled in azimuth, in which the sampling sequence is dense in middle and sparse in both ends, and azimuth compression result with window weighting would also have low sidelobes. According to the echo model of the proposed imaging mode, both the back projection algorithm (BPA) and range migration algorithm (RMA) are modified and presented to handle the raw data of the proposed imaging mode. Both imaging results on simulated targets and experimental real SAR data processing results of a ground-based radar validate the proposed low sidelobe imaging mode.

Remaining Useful Life Estimation for Engineered Systems Operating under Uncertainty with Causal GraphNets

In this work, a novel approach, termed GNN-tCNN, is presented for the construction and training of Remaining Useful Life (RUL) models. The method exploits Graph Neural Networks (GNNs) and deals with the problem of efficiently learning from time series with non-equidistant observations, which may span multiple temporal scales. The efficacy of the method is demonstrated on a simulated stochastic degradation dataset and on a real-world accelerated life testing dataset for ball-bearings. The proposed method learns a model that describes the evolution of the system implicitly rather than at the raw observation level and is based on message-passing neural networks, which encode the irregularly sampled causal structure. The proposed approach is compared to a recurrent network with a temporal convolutional feature extractor head (LSTM-tCNN), which forms a viable alternative for the problem considered. Finally, by taking advantage of recent advances in the computation of reparametrization gradients for learning probability distributions, a simple, yet efficient, technique is employed for representing prediction uncertainty as a gamma distribution over RUL predictions.

Toward an Estimation of the Optical Feedback Factor C on the Fly for Displacement Sensing

In this paper, a method based on the inherent event-based sampling capability of laser optical feedback interferometry (OFI) is proposed to assess the optical feedback factor C when the laser operates in the moderate and strong feedback regimes. Most of the phase unwrapping open-loop OFI algorithms rely on the estimation of C to retrieve the displacement with nanometric precision. Here, the proposed method operates in open-loop configuration and relies only on OFI’s fringe detection, thereby improving its robustness and ease of use. The proposed method is able to estimate C with a precision of <5%. The obtained performances are compared to three different approaches previously published and the impacts of phase noise and sampling frequency are reported. We also show that this method can assess C on the fly even when C is varying due to speckle. To the best of the authors’ knowledge, these are the first reported results of time-varying C estimation. In addition, through C estimation over time, it could pave the way not only to higher performance phase unwrapping algorithms but also to a better control of the optical feedback level via the use of an adaptive lens and thus to better displacement retrieval performances.

Diagnostic Interpretation of Non-Uniformly Sampled Electrocardiogram

We present a set of three fundamental methods for electrocardiogram (ECG) diagnostic interpretation adapted to process non-uniformly sampled signal. The growing volume of ECGs recorded daily all over the world (roughly estimated to be 600 TB) and the expectance of long persistence of these data (on the order of 40 years) motivated us to challenge the feasibility of medical-grade diagnostics directly based on arbitrary non-uniform (i.e., storage-efficient) ECG representation. We used a refined time-independent QRS detection method based on a moving shape matching technique. We applied a graph data representation to quantify the similarity of asynchronously sampled heartbeats. Finally, we applied a correlation-based non-uniform to time-scale transform to get a multiresolution ECG representation on a regular dyadic grid and to find precise P, QRS and T wave delimitation points. The whole processing chain was implemented and tested with MIT-BIH Database (probably the most referenced cardiac database) and CSE Multilead Database (used for conformance testing of medical instruments) signals arbitrarily sampled accordingly to a perceptual model (set for variable sampling frequency of 100-500 Hz, compression ratio 3.1). The QRS detection shows an accuracy of 99.93% with false detection ratio of only 0.18%. The classification shows an accuracy of 99.27% for 14 most frequent MIT-BIH beat types and 99.37% according to AAMI beat labels. The wave delineation shows cumulative (i.e., sampling model and non-uniform processing) errors of: 9.7 ms for P wave duration, 3.4 ms for QRS, 6.7 ms for P-Q segment and 17.7 ms for Q-T segment, all the values being acceptable for medical-grade interpretive software.

A comparison of non-uniform sampling and model-based analysis of NMR spectra for reaction monitoring

Nuclear magnetic resonance (NMR) spectroscopy is widely used for applications in the field of reaction and process monitoring. When complex reaction mixtures are studied, NMR spectra often suffer from low resolution and overlapping peaks, which places high demands on the method used to acquire or to analyse the NMR spectra. This work presents two NMR methods that help overcome these challenges: 2D non-uniform sampling (NUS) and a recently proposed model-based fitting approach for the analysis of 1D NMR spectra. We use the reaction of glycerol with acetic acid as it produces five reaction products that are all chemically similar and, hence, challenging to distinguish. The reaction was measured on a high-field 400 MHz NMR spectrometer with a 2D NUS-heteronuclear single quantum coherence (HSQC) and a conventional 1D ¹ H NMR sequence. We show that comparable results can be obtained using both 2D and 1D methods, if the 2D volume integrals of the 2D NUS-HSQC NMR spectra are calibrated. Further, we monitor the same reaction on a low-field 43 MHz benchtop NMR spectrometer and analyse the acquired 1D ¹ H NMR spectra with the model-based approach and with partial least-squares regression (PLS-R), both trained using a single, calibrated data set. Both methods achieve results that are in good quantitative agreement with the high-field data. However, the model-based method was found to be less sensitive to the training data set used than PLS-R and, hence, was more robust when the reaction conditions differed from that of the training data.

Non-Stationary Complementary Non-Uniform Sampling (NOSCO NUS) for Fast Acquisition of Serial 2D NMR Titration Data

NMR spectroscopy offers unique benefits for ligand binding studies on isotopically labelled target proteins. These benefits include atomic resolution, direct distinction of binding sites and modes, a lowest detectable affinity limit, and function independent setup. Yet, retracing protein signal assignments from apo to holo states to derive exact dissociation constants and chemical shift perturbation amplitudes (for ligand docking and structure‐based optimization) requires lengthy titration series of 2D heteronuclear correlation spectra at variable ligand concentration that may exceed the protein's lifetime and available spectrometer time. We present a novel method to overcome this critical limitation, based on non‐stationary complementary non‐uniform sampling (NOSCO NUS) combined with a robust particle swarm optimization algorithm. We illustrate its potential in two challenging studies with very distinct protein sizes and binding affinities, showing that NOSCO NUS can reduce measurement times by an order of magnitude to make such highly informative NMR titration studies more broadly feasible.

Adaptive Sampling of the Electrocardiogram Based on Generalized Perceptual Features

A non-uniform distribution of diagnostic information in the electrocardiogram (ECG) has been commonly accepted and is the background to several compression, denoising and watermarking methods. Gaze tracking is a widely recognized method for identification of an observer's preferences and interest areas. The statistics of experts' scanpaths were found to be a convenient quantitative estimate of medical information density for each particular component (i.e., wave) of the ECG record. In this paper we propose the application of generalized perceptual features to control the adaptive sampling of a digital ECG. Firstly, based on temporal distribution of the information density, local ECG bandwidth is estimated and projected to the actual positions of components in heartbeat representation. Next, the local sampling frequency is calculated pointwise and the ECG is adaptively low-pass filtered in all simultaneous channels. Finally, sample values are interpolated at new time positions forming a non-uniform time series. In evaluation of perceptual sampling, an inverse transform was used for the reconstruction of regularly sampled ECG with a percent root-mean-square difference (PRD) error of 3-5% (for compression ratios 3.0-4.7, respectively). Nevertheless, tests performed with the use of the CSE Database show good reproducibility of ECG diagnostic features, within the IEC 60601-2-25:2015 requirements, thanks to the occurrence of distortions in less relevant parts of the cardiac cycle.

Emerging solution NMR methods to illuminate the structural and dynamic properties of proteins

The first recognition of protein breathing was more than 50 years ago. Today, we are able to detect the multitude of interaction modes, structural polymorphisms, and binding-induced changes in protein structure that direct function. Solution-state NMR spectroscopy has proved to be a powerful technique, not only to obtain high-resolution structures of proteins, but also to provide unique insights into the functional dynamics of proteins. Here, we summarize recent technical landmarks in solution NMR that have enabled characterization of key biological macromolecular systems. These methods have been fundamental to atomic resolution structure determination and quantitative analysis of dynamics over a wide range of time scales by NMR. The ability of NMR to detect lowly populated protein conformations and transiently formed complexes plays a critical role in its ability to elucidate functionally important structural features of proteins and their dynamics.

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

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

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

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

XLSY: Extra-Large NMR Spectroscopy

NMR studies of intrinsically disordered proteins and other complex biomolecular systems require spectra with the highest resolution and dimensionality. An efficient approach, extra‐large NMR spectroscopy, is presented for experimental data collection, reconstruction, and handling of very large NMR spectra by a combination of the radial and non‐uniform sampling, a new processing algorithm, and rigorous statistical validation. We demonstrate the first high‐quality reconstruction of a full seven‐dimensional HNCOCACONH and two five‐dimensional HACACONH and HN(CA)CONH experiments for a representative intrinsically disordered protein α‐synuclein. XLSY will significantly enhance the NMR toolbox in challenging biomolecular studies.

Compressed NMR: Combining compressive sampling and pure shift NMR techniques

Historically, the resolution of multidimensional nuclear magnetic resonance (NMR) has been orders of magnitude lower than the intrinsic resolution that NMR spectrometers are capable of producing. The slowness of Nyquist sampling as well as the existence of signals as multiplets instead of singlets have been two of the main reasons for this underperformance. Fortunately, two compressive techniques have appeared that can overcome these limitations. Compressive sensing, also known as compressed sampling (CS), avoids the first limitation by exploiting the compressibility of typical NMR spectra, thus allowing sampling at sub-Nyquist rates, and pure shift techniques eliminate the second issue "compressing" multiplets into singlets. This paper explores the possibilities and challenges presented by this combination (compressed NMR). First, a description of the CS framework is given, followed by a description of the importance of combining it with the right pure shift experiment. Second, examples of compressed NMR spectra and how they can be combined with covariance methods will be shown.

Bacteriophage Tail-Tube Assembly Studied by Proton-Detected 4D Solid-State NMR

Obtaining unambiguous resonance assignments remains a major bottleneck in solid-state NMR studies of protein structure and dynamics. Particularly for supramolecular assemblies with large subunits (>150 residues), the analysis of crowded spectral data presents a challenge, even if three-dimensional (3D) spectra are used. Here, we present a proton-detected 4D solid-state NMR assignment procedure that is tailored for large assemblies. The key to recording 4D spectra with three indirect carbon or nitrogen dimensions with their inherently large chemical shift dispersion lies in the use of sparse non-uniform sampling (as low as 2 %). As a proof of principle, we acquired 4D (H)COCANH, (H)CACONH, and (H)CBCANH spectra of the 20 kDa bacteriophage tail-tube protein gp17.1 in a total time of two and a half weeks. These spectra were sufficient to obtain complete resonance assignments in a straightforward manner without use of previous solution NMR data.

Fast 2D NMR Spectroscopy for In vivo Monitoring of Bacterial Metabolism in Complex Mixtures

The biological toolbox is full of techniques developed originally for analytical chemistry. Among them, spectroscopic experiments are very important source of atomic-level structural information. Nuclear magnetic resonance (NMR) spectroscopy, although very advanced in chemical and biophysical applications, has been used in microbiology only in a limited manner. So far, mostly one-dimensional ¹H experiments have been reported in studies of bacterial metabolism monitored in situ. However, low spectral resolution and limited information on molecular topology limits the usability of these methods. These problems are particularly evident in the case of complex mixtures, where spectral peaks originating from many compounds overlap and make the interpretation of changes in a spectrum difficult or even impossible. Often a suite of two-dimensional (2D) NMR experiments is used to improve resolution and extract structural information from internuclear correlations. However, for dynamically changing sample, like bacterial culture, the time-consuming sampling of so-called indirect time dimensions in 2D experiments is inefficient. Here, we propose the technique known from analytical chemistry and structural biology of proteins, i.e., time-resolved non-uniform sampling. The method allows application of 2D (and multi-D) experiments in the case of quickly varying samples. The indirect dimension here is sparsely sampled resulting in significant reduction of experimental time. Compared to conventional approach based on a series of 1D measurements, this method provides extraordinary resolution and is a real-time approach to process monitoring. In this study, we demonstrate the usability of the method on a sample of Escherichia coli culture affected by ampicillin and on a sample of Propionibacterium acnes, an acne causing bacterium, mixed with a dose of face tonic, which is a complicated, multi-component mixture providing complex NMR spectrum. Through our experiments we determine the exact concentration and time at which the anti-bacterial agents affect the bacterial metabolism. We show, that it is worth to extend the NMR toolbox for microbiology by including techniques of 2D z-TOCSY, for total "fingerprinting" of a sample and 2D ¹³C-edited HSQC to monitor changes in concentration of metabolites in selected metabolic pathways.

State Estimation for a Class of Non-Uniform Sampling Systems with Missing Measurements

This paper is concerned with the state estimation problem for a class of non-uniform sampling systems with missing measurements where the state is updated uniformly and the measurements are sampled randomly. A new state model is developed to depict the dynamics at the measurement sampling points within a state update period. A non-augmented state estimator dependent on the missing rate is presented by applying an innovation analysis approach. It can provide the state estimates at the state update points and at the measurement sampling points within a state update period. Compared with the augmented method, the proposed algorithm can reduce the computational burden with the increase of the number of measurement samples within a state update period. It can deal with the optimal estimation problem for single and multi-sensor systems in a unified way. To improve the reliability, a distributed suboptimal fusion estimator at the state update points is also given for multi-sensor systems by using the covariance intersection fusion algorithm. The simulation research verifies the effectiveness of the proposed algorithms.

A Set of Efficient nD NMR Protocols for Resonance Assignments of Intrinsically Disordered Proteins

The RF pulse scheme RN[N-CA HEHAHA]NH, which provides a convenient approach to the acquisition of different multidimensional chemical shift correlation NMR spectra leading to backbone resonance assignments, including those of the proline residues of intrinsically disordered proteins (IDPs), is experimentally demonstrated. Depending on the type of correlation data required, the method involves the generation of in-phase ((15) N)(x) magnetisation via different magnetisation transfer pathways such as H→N→CO→N, HA→CA→CO→N, H→N→CA→N and H→CA→N, the subsequent application of (15) N-(13) C(α) heteronuclear Hartmann-Hahn mixing over a period of ≈100 ms, chemical-shift labelling of relevant nuclei before and after the heteronuclear mixing step and amide proton detection in the acquisition dimension. It makes use of the favourable relaxation properties of IDPs and the presence of (1) JCαN and (2) JCαN couplings to achieve efficient correlation of the backbone resonances of each amino acid residue "i" with the backbone amide resonances of residues "i-1" and "i+1". It can be implemented in a straightforward way through simple modifications of the RF pulse schemes commonly employed in protein NMR studies. The efficacy of the approach is demonstrated using a uniformly ((15) N,(13) C) labelled sample of α-synuclein. The different possibilities for obtaining the amino-acid-type information, simultaneously with the connectivity data between the backbone resonances of sequentially neighbouring residues, have also been outlined.

3D spatially encoded and accelerated TE-averaged echo planar spectroscopic imaging in healthy human brain

Several different pathologies, including many neurodegenerative disorders, affect the energy metabolism of the brain. Glutamate, a neurotransmitter in the brain, can be used as a biomarker to monitor these metabolic processes. One method that is capable of quantifying glutamate concentration reliably in several regions of the brain is TE-averaged (1) H spectroscopic imaging. However, this type of method requires the acquisition of multiple TE lines, resulting in long scan durations. The goal of this experiment was to use non-uniform sampling, compressed sensing reconstruction and an echo planar readout gradient to reduce the scan time by a factor of eight to acquire TE-averaged spectra in three spatial dimensions. Simulation of glutamate and glutamine showed that the 2.2-2.4 ppm spectral region contained 95% glutamate signal using the TE-averaged method. Peak integration of this spectral range and home-developed, prior-knowledge-based fitting were used for quantitation. Gray matter brain phantom measurements were acquired on a Siemens 3 T Trio scanner. Non-uniform sampling was applied retrospectively to these phantom measurements and quantitative results of glutamate with respect to creatine 3.0 (Glu/Cr) ratios showed a coefficient of variance of 16% for peak integration and 9% for peak fitting using eight-fold acceleration. In vivo scans of the human brain were acquired as well and five different brain regions were quantified using the prior-knowledge-based algorithm. Glu/Cr ratios from these regions agreed with previously reported results in the literature. The method described here, called accelerated TE-averaged echo planar spectroscopic imaging (TEA-EPSI), is a significant methodological advancement and may be a useful tool for categorizing glutamate changes in pathologies where affected brain regions are not known a priori. Copyright © 2016 John Wiley & Sons, Ltd.

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

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

Probing Invisible, Excited Protein States by Non-Uniformly Sampled Pseudo-4D CEST Spectroscopy

Chemical exchange saturation transfer (CEST) NMR spectroscopy is a powerful tool for studies of slow timescale protein dynamics. Typical experiments are based on recording a large number of 2D data sets and quantifying peak intensities in each of the resulting planes. A weakness of the method is that peaks must be resolved in 2D spectra, limiting applications to relatively small proteins. Resolution is significantly improved in 3D spectra but recording uniformly sampled data is time-prohibitive. Here we describe non-uniformly sampled HNCO-based pseudo-4D CEST that provides excellent resolution in reasonable measurement times. Data analysis is done through fitting in the time domain, without the need of reconstructing the frequency dimensions, exploiting previously measured accurate peak positions in reference spectra. The methodology is demonstrated on several protein systems, including a nascent form of superoxide dismutase that is implicated in neurodegenerative disease.

ADAPT-NMR 3.0: utilization of BEST-type triple-resonance NMR experiments to accelerate the process of data collection and assignment

ADAPT-NMR (Assignment-directed Data collection Algorithm utilizing a Probabilistic Toolkit in NMR) is a software package whose Bayesian core uses on-the-fly chemical shift assignments to guide data acquisition by non-uniform sampling from a panel of through-bond NMR experiments. The new version of ADAPT-NMR (ADAPT-NMR v3.0) has the option of utilizing 2D tilted-plane versions of 3D fast spectral acquisition with BEST-type pulse sequences, while also retaining the capability of acquiring and processing data from tilted-plane versions of conventional sensitivity-enhanced experiments. The use of BEST experiments significantly reduces data collection times and leads to enhanced performance by ADAPT-NMR.

(13)C-detected NMR experiments for automatic resonance assignment of IDPs and multiple-fixing SMFT processing

Intrinsically disordered proteins (IDPs) have recently attracted much interest, due to their role in many biological processes, including signaling and regulation mechanisms. High-dimensional (13)C direct-detected NMR experiments have proven exceptionally useful in case of IDPs, providing spectra with superior peak dispersion. Here, two such novel experiments recorded with non-uniform sampling are introduced, these are 5D HabCabCO(CA)NCO and 5D HNCO(CA)NCO. Together with the 4D (HACA)CON(CA)NCO, an extension of the previously published 3D experiments (Pantoja-Uceda and Santoro in J Biomol NMR 59:43-50, 2014. doi: 10.1007/s10858-014-9827-1), they form a set allowing for complete and reliable resonance assignment of difficult IDPs. The processing is performed with sparse multidimensional Fourier transform based on the concept of restricting (fixing) some of spectral dimensions to a priori known resonance frequencies. In our study, a multiple-fixing method was developed, that allows easy access to spectral data. The experiments were tested on a resolution-demanding alpha-synuclein sample. Due to superior peak dispersion in high-dimensional spectrum and availability of the sequential connectivities between four consecutive residues, the overwhelming majority of resonances could be assigned automatically using the TSAR program.

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.

Structure refinement and membrane positioning of selectively labeled OmpX in phospholipid nanodiscs

NMR structural studies on membrane proteins are often complicated by their large size, taking into account the contribution of the membrane mimetic. Therefore, classical resonance assignment approaches often fail. The large size of phospholipid nanodiscs, a detergent-free phospholipid bilayer mimetic, prevented their use in high-resolution solution-state NMR spectroscopy so far. We recently introduced smaller nanodiscs that are suitable for NMR structure determination. However, side-chain assignments of a membrane protein in nanodiscs still remain elusive. Here, we utilized a NOE-based approach to assign (stereo-) specifically labeled Ile, Leu, Val and Ala methyl labeled and uniformly (15)N-Phe and (15)N-Tyr labeled OmpX and calculated a refined high-resolution structure. In addition, we were able to obtain residual dipolar couplings (RDCs) of OmpX in nanodiscs using Pf1 phage medium for the induction of weak alignment. Back-calculated NOESY spectra of the obtained NMR structures were compared to experimental NOESYs in order to validate the quality of these structures. We further used NOE information between protonated lipid head groups and side-chain methyls to determine the position of OmpX in the phospholipid bilayer. These data were verified by paramagnetic relaxation enhancement (PRE) experiments obtained with Gd(3+)-modified lipids. Taken together, this study emphasizes the need for the (stereo-) specific labeling of membrane proteins in a highly deuterated background for high-resolution structure determination, particularly in large membrane mimicking systems like phospholipid nanodiscs. Structure validation by NOESY back-calculation will be helpful for the structure determination and validation of membrane proteins where NOE assignment is often difficult. The use of protein to lipid NOEs will be beneficial for the positioning of a membrane protein in the lipid bilayer without the need for preparing multiple protein samples.

C4'/H4' selective, non-uniformly sampled 4D HC(P)CH experiment for sequential assignments of (13)C-labeled RNAs

A through bond, C4'/H4' selective, "out and stay" type 4D HC(P)CH experiment is introduced which provides sequential connectivity via H4'(i)-C4'(i)-C4'(i-1)-H4'(i-1) correlations. The (31)P dimension (used in the conventional 3D HCP experiment) is replaced with evolution of better dispersed C4' dimension. The experiment fully utilizes (13)C-labeling of RNA by inclusion of two C4' evolution periods. An additional evolution of H4' is included to further enhance peak resolution. Band selective (13)C inversion pulses are used to achieve selectivity and prevent signal dephasing due to the of C4'-C3' and C4'-C5' homonuclear couplings. For reasonable resolution, non-uniform sampling is employed in all indirect dimensions. To reduce sensitivity losses, multiple quantum coherences are preserved during shared-time evolution and coherence transfer delays. In the experiment the intra-nucleotide peaks are suppressed whereas inter-nucleotide peaks are enhanced to reduce the ambiguities. The performance of the experiment is verified on a fully (13)C, (15)N-labeled 34-nt hairpin RNA comprising typical structure elements.

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 (1)H,(15)N-HSQC experiments on natural abundance protein samples and (1)H,(13)C-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.

Selective diagonal-free (13)C, (13)C-edited aliphatic-aromatic NOESY experiment with non-uniform sampling

A band-selective aromatic-aliphatic C,C-edited four-dimensional NOESY experiment is proposed here. Its key advantage is the absence of auto-correlation signals which makes it very attractive for joint use with non-uniform sampling. It is demonstrated here that the sensitivity of the experiment is not significantly affected by utilization of selective pulses (for either aromatic-13C or aliphatic-13C spins). The method was applied to the sample of E32Q mutant of human S100A1 protein, a homodimer of total molecular mass ~20 kDa. High-resolution 4D spectra were obtained from ~1.5 % of sampling points required conventionally. It is shown that superior resolution facilitates unambiguous assignment of observed aliphatic-aromatic cross-peaks. Additionally, the addition of aliphatic-13C dimension enables to resolve peaks with degenerated aliphatic (1)H chemical shifts. All observed cross-peaks were validated against previously determined 3D structure of E32Q mutant of S100A1 protein (PDB 2LHL). The increased reliability of structural constraints obtained from the proposed high-resolution 4D 13C(ali),13C(aro)-edited NOESY can be exploited in the automated protocols of structure determination of proteins.

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.

Estimation of time-delayed mutual information and bias for irregularly and sparsely sampled time-series

A method to estimate the time-dependent correlation via an empirical bias estimate of the time-delayed mutual information for a time-series is proposed. In particular, the bias of the time-delayed mutual information is shown to often be equivalent to the mutual information between two distributions of points from the same system separated by infinite time. Thus intuitively, estimation of the bias is reduced to estimation of the mutual information between distributions of data points separated by large time intervals. The proposed bias estimation techniques are shown to work for Lorenz equations data and glucose time series data of three patients from the Columbia University Medical Center database.

High dimensional and high resolution pulse sequences for backbone resonance assignment of intrinsically disordered proteins

Four novel 5D (HACA(N)CONH, HNCOCACB, (HACA)CON(CA)CONH, (H)NCO(NCA)CONH), and one 6D ((H)NCO(N)CACONH) NMR pulse sequences are proposed. The new experiments employ non-uniform sampling that enables achieving high resolution in indirectly detected dimensions. The experiments facilitate resonance assignment of intrinsically disordered proteins. The novel pulse sequences were successfully tested using δ subunit (20 kDa) of Bacillus subtilis RNA polymerase that has an 81-amino acid disordered part containing various repetitive sequences.

Deterministic schedules for robust and reproducible non-uniform sampling in multidimensional NMR

We show that a simple, general, and easily reproducible method for generating non-uniform sampling (NUS) schedules preserves the benefits of random sampling, including inherently reduced sampling artifacts, while removing the pitfalls associated with choosing an arbitrary seed. Sampling schedules are generated from a discrete cumulative distribution function (CDF) that closely fits the continuous CDF of the desired probability density function. We compare random and deterministic sampling using a Gaussian probability density function applied to 2D HSQC spectra. Data are processed using the previously published method of Spectroscopy by Integration of Frequency and Time domain data (SIFT). NUS spectra from deterministic sampling schedules were found to be at least as good as those from random schedules at the SIFT critical sampling density, and significantly better at half that sampling density. The method can be applied to any probability density function and generalized to greater than two dimensions.

Signal enhancement for the sensitivity-limited solid state NMR experiments using a continuous, non-uniform acquisition scheme

We describe a sampling scheme for the two-dimensional (2D) solid state NMR experiments, which can be readily applied to the sensitivity-limited samples. The sampling scheme utilizes continuous, non-uniform sampling profile for the indirect dimension, i.e. the acquisition number decreases as a function of the evolution time (t1) in the indirect dimension. For a beta amyloid (Aβ) fibril sample, we observed overall 40-50% signal enhancement by measuring the cross peak volume, while the cross peak linewidths remained comparable to the linewidths obtained by regular sampling and processing strategies. Both the linear and Gaussian decay functions for the acquisition numbers result in similar percentage of increment in signal. In addition, we demonstrated that this sampling approach can be applied with different dipolar recoupling approaches such as radiofrequency assisted diffusion (RAD) and finite-pulse radio-frequency-driven recoupling (fpRFDR). This sampling scheme is especially suitable for the sensitivity-limited samples which require long signal averaging for each t1 point, for instance the biological membrane proteins where only a small fraction of the sample is isotopically labeled.

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.

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.