Granger causality for compressively sensed sparse signals

Compressed sensing is a scheme that allows for sparse signals to be acquired, transmitted, and stored using far fewer measurements than done by conventional means employing the Nyquist sampling theorem. Since many naturally occurring signals are sparse (in some domain), compressed sensing has rapidly seen popularity in a number of applied physics and engineering applications, particularly in designing signal and image acquisition strategies, e.g., magnetic resonance imaging, quantum state tomography, scanning tunneling microscopy, and analog to digital conversion technologies. Contemporaneously, causal inference has become an important tool for the analysis and understanding of processes and their interactions in many disciplines of science, especially those dealing with complex systems. Direct causal analysis for compressively sensed data is required to avoid the task of reconstructing the compressed data. Also, for some sparse signals, such as for sparse temporal data, it may be difficult to discover causal relations directly using available data-driven or model-free causality estimation techniques. In this work, we provide a mathematical proof that structured compressed sensing matrices, specifically circulant and Toeplitz, preserve causal relationships in the compressed signal domain, as measured by Granger causality (GC). We then verify this theorem on a number of bivariate and multivariate coupled sparse signal simulations which are compressed using these matrices. We also demonstrate a real world application of network causal connectivity estimation from sparse neural spike train recordings from rat prefrontal cortex. In addition to demonstrating the effectiveness of structured matrices for GC estimation from sparse signals, we also show a computational time advantage of the proposed strategy for causal inference from compressed signals of both sparse and regular autoregressive processes as compared to standard GC estimation from original signals.

NMR spectrum reconstruction as a pattern recognition problem

A new deep neural network based on the WaveNet architecture (WNN) is presented, which is designed to grasp specific patterns in the NMR spectra. When trained at a fixed non-uniform sampling (NUS) schedule, the WNN benefits from pattern recognition of the corresponding point spread function (PSF) pattern produced by each spectral peak resulting in the highest quality and robust reconstruction of the NUS spectra as demonstrated in simulations and exemplified in this work on 2D ¹H-¹⁵N correlation spectra of three representative globular proteins with different sizes: Ubiquitin (8.6 kDa), Azurin (14 kDa), and Malt1 (44 kDa). The pattern recognition by WNN is also demonstrated for successful virtual homo-decoupling in a 2D methyl ¹H-¹³C - HMQC spectrum of MALT1. We demonstrate using WNN that prior knowledge about the NUS schedule, which so far was not been fully exploited, can be used for designing new powerful NMR processing techniques that surpass the existing algorithmic methods.

Temperature as an Extra Dimension in Multidimensional Protein NMR Spectroscopy

NMR spectroscopy is a particularly informative method for studying protein structures and dynamics in solution; however, it is also one of the most time-consuming. Modern approaches to biomolecular NMR spectroscopy are based on lengthy multidimensional experiments, the duration of which grows exponentially with the number of dimensions. The experimental time may even be several days in the case of 3D and 4D spectra. Moreover, the experiment often has to be repeated under several different conditions, for example, to measure the temperature-dependent effects in a spectrum (temperature coefficients (TCs)). Herein, a new approach that involves joint sampling of indirect evolution times and temperature is proposed. This allows TCs to be measured through 3D spectra in even less time than that needed to acquire a single spectrum by using the conventional approach. Two signal processing methods that are complementary, in terms of sensitivity and resolution, 1) dividing data into overlapping subsets followed by compressed sensing reconstruction, and 2) treating the complete data set with a variant of the Radon transform, are proposed. The temperature-swept 3D HNCO spectra of two intrinsically disordered proteins, osteopontin and CD44 cytoplasmic tail, show that this new approach makes it possible to determine TCs and their non-linearities effectively. Non-linearities, which indicate the presence of a compact state, are particularly interesting. The complete package of data acquisition and processing software for this new approach are provided.

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.

Review and Prospect: Deep Learning in Nuclear Magnetic Resonance Spectroscopy

Since the concept of deep learning (DL) was formally proposed in 2006, it has had a major impact on academic research and industry. Nowadays, DL provides an unprecedented way to analyze and process data with demonstrated great results in computer vision, medical imaging, natural language processing, and so forth. Herein, applications of DL in NMR spectroscopy are summarized, and a perspective for DL as an entirely new approach that is likely to transform NMR spectroscopy into a much more efficient and powerful technique in chemistry and life sciences is outlined.

Accelerated Nuclear Magnetic Resonance Spectroscopy with Deep Learning

Nuclear magnetic resonance (NMR) spectroscopy serves as an indispensable tool in chemistry and biology but often suffers from long experimental times. We present a proof-of-concept of the application of deep learning and neural networks for high-quality, reliable, and very fast NMR spectra reconstruction from limited experimental data. We show that the neural network training can be achieved using solely synthetic NMR signals, which lifts the prohibiting demand for a large volume of realistic training data usually required for a deep learning approach.

Non-Uniform and Absolute Minimal Sampling for High-Throughput Multidimensional NMR Applications

Many biomolecular NMR applications can benefit from the faster acquisition of multidimensional NMR data with high resolution and their automated analysis and interpretation. In recent years, a number of non-uniform sampling (NUS) approaches have been introduced for the reconstruction of multidimensional NMR spectra, such as compressed sensing, thereby bypassing traditional Fourier-transform processing. Such approaches are applicable to both biomacromolecules and small molecules and their complex mixtures and can be combined with homonuclear decoupling (pure shift) and covariance processing. For homonuclear 2D TOCSY experiments, absolute minimal sampling (AMS) permits the drastic shortening of measurement times necessary for high-throughput applications for identification and quantification of components in complex biological mixtures in the field of metabolomics. Such TOCSY spectra can be comprehensively represented by graphic theoretical maximal cliques for the identification of entire spin systems and their subsequent query against NMR databases. Integration of these methods in webservers permits the rapid and reliable identification of mixture components. Recent progress is reviewed in this Minireview.

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

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

Absolute Minimal Sampling of Homonuclear 2D NMR TOCSY Spectra for High-Throughput Applications of Complex Mixtures

Modern applications of 2D NMR spectroscopy to diagnostic screening, metabolomics, quality control, and other high-throughput applications are often limited by the time-consuming sampling requirements along the indirect time domain t1 . 2D total correlation spectroscopy (TOCSY) provides unique spin connectivity information for the analysis of a large number of compounds in complex mixtures, but standard methods typically require >100 t1 increments for an accurate spectral reconstruction, rendering these experiments ineffective for high-throughput applications. For a complex metabolite mixture it is demonstrated that absolute minimal sampling (AMS), based on direct fitting of resonance frequencies and amplitudes in the time domain, yields an accurate spectral reconstruction of TOCSY spectra using as few as 16 t1 points. This permits the rapid collection of homonuclear 2D NMR experiments at high resolution with measurement times that previously were only the realm of 1D experiments.

Absolute Minimal Sampling in High-Dimensional NMR Spectroscopy

Standard three-dimensional Fourier transform (FT) NMR experiments of molecular systems often involve prolonged measurement times due to extensive sampling required along the indirect time domains to obtain adequate spectral resolution. In recent years, a wealth of alternative sampling methods has been proposed to ease this bottleneck. However, due to their algorithmic complexity, for a given sample and experiment it is often hard to determine the minimal sampling requirement, and hence the maximal achievable experimental speed up. Herein we introduce an absolute minimal sampling (AMS) method that can be applied to common 3D NMR experiments. We show for the proteins ubiquitin and arginine kinase that for widely used experiments, such as 3D HNCO, accurate carbon frequencies can be obtained with a single time increment, while for others, such as 3D HN(CA)CO, all relevant information is obtained with as few as 6 increments amounting to a speed up of a factor 7-50.

Lorentzian sparsity based spectroscopic reconstruction for fast high-dimensional magnetic resonance spectroscopy

Two-dimensional magnetic resonance spectroscopy (2D MRS) is challenging, even with state-of-art compressive sensing methods, such as L1-sparsity method. In this work, using the prior that the 2D MRS can be regarded as a series of Lorentzian functions, we aim to develop a robust Lorentzian-sparsity based spectroscopy reconstruction method for high-dimensional MRS. The proposed method sparsifies 2D MRS in Lorentzian functions. Instead of thousands of pixel-wise variables, this Lorentzian-sparsity method significantly reduces the number of unknowns to several geometric variables, such as the center, magnitude and shape parameters for each Lorentzian function. The spectroscopy reconstruction is formulated as a nonlinear and nonconvex optimization problem, and the simulated annealing algorithm is developed to solve the problem. The proposed method was compared with inverse FFT method and L1-sparsity method, under various undersampling factors. While FFT and L1 results contained severe artifacts, the Lorentzian-sparsity results provided significantly improved spectroscopy. A new 2D MRS reconstruction method is proposed using the Lorentzian sparsity, with significantly improved MRS reconstruction quality, in comparison with standard inverse FFT method or state-of-art L1-sparsity method.

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.

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.

Accelerated NMR spectroscopy with low-rank reconstruction

Accelerated multi-dimensional NMR spectroscopy is a prerequisite for high-throughput applications, studying short-lived molecular systems and monitoring chemical reactions in real time. Non-uniform sampling is a common approach to reduce the measurement time. Here, a new method for high-quality spectra reconstruction from non-uniformly sampled data is introduced, which is based on recent developments in the field of signal processing theory and uses the so far unexploited general property of the NMR signal, its low rank. Using experimental and simulated data, we demonstrate that the low-rank reconstruction is a viable alternative to the current state-of-the-art technique compressed sensing. In particular, the low-rank approach is good in preserving of low-intensity broad peaks, and thus increases the effective sensitivity in the reconstructed spectra.

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.