Correction: Sparse sampling methods in multidimensional NMR.

Correction for 'Sparse sampling methods in multidimensional NMR' by Mehdi Mobli et al., Phys. Chem. Chem. Phys., 2012, 14, 10835-10843.

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.

Nonuniform sampling of hypercomplex multidimensional NMR experiments: Dimensionality, quadrature phase and randomization.

Nonuniform sampling (NUS) in multidimensional NMR permits the exploration of higher dimensional experiments and longer evolution times than the Nyquist Theorem practically allows for uniformly sampled experiments. However, the spectra of NUS data include sampling-induced artifacts and may be subject to distortions imposed by sparse data reconstruction techniques, issues not encountered with the discrete Fourier transform (DFT) applied to uniformly sampled data. The characterization of these NUS-induced artifacts allows for more informed sample schedule design and improved spectral quality. The DFT-Convolution Theorem, via the point-spread function (PSF) for a given sampling scheme, provides a useful framework for exploring the nature of NUS sampling artifacts. In this work, we analyze the PSFs for a set of specially constructed NUS schemes to quantify the interplay between randomization and dimensionality for reducing artifacts relative to uniformly undersampled controls. In particular, we find a synergistic relationship between the indirect time dimensions and the "quadrature phase dimension" (i.e. the hypercomplex components collected for quadrature detection). The quadrature phase dimension provides additional degrees of freedom that enable partial-component NUS (collecting a subset of quadrature components) to further reduce sampling-induced aliases relative to traditional full-component NUS (collecting all quadrature components). The efficacy of artifact reduction is exponentially related to the dimensionality of the sample space. Our results quantify the utility of partial-component NUS as an additional means for introducing decoherence into sampling schemes and reducing sampling artifacts in high dimensional experiments.

Nonuniform sampling and maximum entropy reconstruction in multidimensional NMR.

NMR spectroscopy is one of the most powerful and versatile analytic tools available to chemists. The discrete Fourier transform (DFT) played a seminal role in the development of modern NMR, including the multidimensional methods that are essential for characterizing complex biomolecules. However, it suffers from well-known limitations: chiefly the difficulty in obtaining high-resolution spectral estimates from short data records. Because the time required to perform an experiment is proportional to the number of data samples, this problem imposes a sampling burden for multidimensional NMR experiments. At high magnetic field, where spectral dispersion is greatest, the problem becomes particularly acute. Consequently multidimensional NMR experiments that rely on the DFT must either sacrifice resolution in order to be completed in reasonable time or use inordinate amounts of time to achieve the potential resolution afforded by high-field magnets. Maximum entropy (MaxEnt) reconstruction is a non-Fourier method of spectrum analysis that can provide high-resolution spectral estimates from short data records. It can also be used with nonuniformly sampled data sets. Since resolution is substantially determined by the largest evolution time sampled, nonuniform sampling enables high resolution while avoiding the need to uniformly sample at large numbers of evolution times. The Nyquist sampling theorem does not apply to nonuniformly sampled data, and artifacts that occur with the use of nonuniform sampling can be viewed as frequency-aliased signals. Strategies for suppressing nonuniform sampling artifacts include the careful design of the sampling scheme and special methods for computing the spectrum. Researchers now routinely report that they can complete an N-dimensional NMR experiment 3(N-1) times faster (a 3D experiment in one ninth of the time). As a result, high-resolution three- and four-dimensional experiments that were prohibitively time consuming are now practical. Conversely, tailored sampling in the indirect dimensions has led to improved sensitivity. Further advances in nonuniform sampling strategies could enable further reductions in sampling requirements for high resolution NMR spectra, and the combination of these strategies with robust non-Fourier methods of spectrum analysis (such as MaxEnt) represent a profound change in the way researchers conduct multidimensional experiments. The potential benefits will enable more advanced applications of multidimensional NMR spectroscopy to study biological macromolecules, metabolomics, natural products, dynamic systems, and other areas where resolution, sensitivity, or experiment time are limiting. Just as the development of multidimensional NMR methods presaged multidimensional methods in other areas of spectroscopy, we anticipate that nonuniform sampling approaches will find applications in other forms of spectroscopy.

Single-file diffusion of flagellin in flagellar filaments.

A bacterial flagellar filament is a cylindrical crystal of a protein known as flagellin. Flagellin subunits travel from the cytoplasm through a 2 nm axial pore and polymerize at the filament's distal end. They are supplied by a pump in the cell membrane powered by a proton-motive force. In a recent experiment, it was observed that growth proceeded at a rate of approximately one subunit every 2 s. Here, we asked whether transport of subunits through the pore at this rate could be effected by single-file diffusion, which we simulated by a random walk on a one-dimensional lattice. Assuming that the subunits are α-helical, the answer is yes, by a comfortable margin.

Formalism for hypercomplex multidimensional NMR employing partial-component subsampling.

Multidimensional NMR spectroscopy typically employs phase-sensitive detection, which results in hypercomplex data (and spectra) when utilized in more than one dimension. Nonuniform sampling approaches have become commonplace in multidimensional NMR, enabling dramatic reductions in experiment time, increases in sensitivity and/or increases in resolution. In order to utilize nonuniform sampling optimally, it is necessary to characterize the relationship between the spectrum of a uniformly sampled data set and the spectrum of a subsampled data set. In this work we construct an algebra of hypercomplex numbers suitable for representing multidimensional NMR data along with partial-component nonuniform sampling (i.e. the hypercomplex components of data points are subsampled). This formalism leads to a modified DFT-Convolution relationship involving a partial-component, hypercomplex point-spread function set. The framework presented here is essential for the continued development and appropriate characterization of partial-component nonuniform sampling.

Sparse sampling methods in multidimensional NMR.

Although the discrete Fourier transform played an enabling role in the development of modern NMR spectroscopy, it suffers from a well-known difficulty providing high-resolution spectra from short data records. In multidimensional NMR experiments, so-called indirect time dimensions are sampled parametrically, with each instance of evolution times along the indirect dimensions sampled via separate one-dimensional experiments. The time required to conduct multidimensional experiments is directly proportional to the number of indirect evolution times sampled. Despite remarkable advances in resolution with increasing magnetic field strength, multiple dimensions remain essential for resolving individual resonances in NMR spectra of biological macromolecues. Conventional Fourier-based methods of spectrum analysis limit the resolution that can be practically achieved in the indirect dimensions. Nonuniform or sparse data collection strategies, together with suitable non-Fourier methods of spectrum analysis, enable high-resolution multidimensional spectra to be obtained. Although some of these approaches were first employed in NMR more than two decades ago, it is only relatively recently that they have been widely adopted. Here we describe the current practice of sparse sampling methods and prospects for further development of the approach to improve resolution and sensitivity and shorten experiment time in multidimensional NMR. While sparse sampling is particularly promising for multidimensional NMR, the basic principles could apply to other forms of multidimensional spectroscopy.

Growth of flagellar filaments of Escherichia coli is independent of filament length.

Bacterial flagellar filaments grow at their distal ends, from flagellin that travels through a central channel ∼2 nm in diameter. The flagellin is extruded from the cytoplasm by a pump powered by a proton motive force (PMF). We measured filament growth in cells near the mid-exponential-phase with flagellin bearing a specific cysteine-for-serine substitution, allowing filaments to be labeled with sulfhydryl-specific fluorescent dyes. We labeled filaments first with a green maleimide dye and then, following an additional period of growth, with a red maleimide dye. The contour lengths of the green and red segments were measured. The average lengths of red segments (∼2.3 µm) were the same regardless of the lengths of the green segments from which they grew (ranging from less than 1 to more than 9 µm in length). Thus, flagellar filaments do not grow at a rate that decreases exponentially with length, as formerly supposed. If flagellar filaments were broken by viscous shear, the broken filaments continued to grow. Identical results were obtained whether flagellin was expressed from fliC on the chromosome under the control of its native promoter or on a plasmid under the control of the arabinose promoter.

Data sampling in multidimensional NMR: fundamentals and strategies.

Beginning with the introduction of Fourier Transform NMR by Ernst and Anderson in 1966, time domain measurement of the impulse response (free induction decay) consisted of sampling the signal at a series of discrete intervals. For compatibility with the discrete Fourier transform, 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 were already well understood, and 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 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. In this chapter we review the fundamentals of uniform and nonuniform sampling methods in one- and multidimensional NMR.

Random phase detection in multidimensional NMR.

Despite advances in resolution accompanying the development of high-field superconducting magnets, biomolecular applications of NMR require multiple dimensions in order to resolve individual resonances, and the achievable resolution is typically limited by practical constraints on measuring time. In addition to the need for measuring long evolution times to obtain high resolution, the need to distinguish the sign of the frequency constrains the ability to shorten measuring times. Sign discrimination is typically accomplished by sampling the signal with two different receiver phases or by selecting a reference frequency outside the range of frequencies spanned by the signal and then sampling at a higher rate. In the parametrically sampled (indirect) time dimensions of multidimensional NMR experiments, either method imposes an additional factor of 2 sampling burden for each dimension. We demonstrate that by using a single detector phase at each time sample point, but randomly altering the phase for different points, the sign ambiguity that attends fixed single-phase detection is resolved. Random phase detection enables a reduction in experiment time by a factor of 2 for each indirect dimension, amounting to a factor of 8 for a four-dimensional experiment, albeit at the cost of introducing sampling artifacts. Alternatively, for fixed measuring time, random phase detection can be used to double resolution in each indirect dimension. Random phase detection is complementary to nonuniform sampling methods, and their combination offers the potential for additional benefits. In addition to applications in biomolecular NMR, random phase detection could be useful in magnetic resonance imaging and other signal processing contexts.

A non-uniformly sampled 4D HCC(CO)NH-TOCSY experiment processed using maximum entropy for rapid protein sidechain assignment.

One of the stiffest challenges in structural studies of proteins using NMR is the assignment of sidechain resonances. Typically, a panel of lengthy 3D experiments are acquired in order to establish connectivities and resolve ambiguities due to overlap. We demonstrate that these experiments can be replaced by a single 4D experiment that is time-efficient, yields excellent resolution, and captures unique carbon-proton connectivity information. The approach is made practical by the use of non-uniform sampling in the three indirect time dimensions and maximum entropy reconstruction of the corresponding 3D frequency spectrum. This 4D method will facilitate automated resonance assignment procedures and it should be particularly beneficial for increasing throughput in NMR-based structural genomics initiatives.

NMR data processing using iterative thresholding and minimum l(1)-norm reconstruction.

Iterative thresholding algorithms have a long history of application to signal processing. Although they are intuitive and easy to implement, their development was heuristic and mainly ad hoc. Using a special form of the thresholding operation, called soft thresholding, we show that the fixed point of iterative thresholding is equivalent to minimum l(1)-norm reconstruction. We illustrate the method for spectrum analysis of a time series. This result helps to explain the success of these methods and illuminates connections with maximum entropy and minimum area methods, while also showing that there are more efficient routes to the same result. The power of the l(1)-norm and related functionals as regularizers of solutions to underdetermined systems will likely find numerous useful applications in NMR.

Spectral reconstruction methods in fast NMR: reduced dimensionality, random sampling and maximum entropy.

The need to reduce data acquisition times of multidimensional NMR experiments has fostered considerable interest in novel data acquisition schemes. A recurring theme is that of reduced dimensionality experiments, in which time evolutions in the indirect dimensions are incremented together, rather than independently. Spectral analysis of such data is carried out using methods such as filtered back-projection, GFT, or parametric signal modeling. By using Maximum Entropy reconstruction of reduced-dimensionality data, we show that the artifacts that arise in reduced dimensionality experiments are intrinsic to the data sampling, and are not, in general, the result of the methods used to compute spectra. Our results illustrate that reduced dimensionality is a special case of non-uniform sampling in the time domain. We show that MaxEnt reconstruction yields more accurate spectra for reduced dimensionality data than back-projection reconstruction and that randomly choosing time increments based on an exponentially weighted distribution is more efficient, with fewer artifacts, than the systematic coupling of time increments used in most reduced dimensionality approaches.

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

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

Optimization of 13C direct detection NMR methods.

(13)C-detected experiments are still limited by their inherently lower sensitivity, as compared to the equivalent (1)H-detected experiments. Improving the sensitivity of (13)C detection methods remains a significant area of NMR research that may provide better means for studying large macromolecular systems by NMR. In this communication, we show that (13)C-detected experiments are less sensitive to the salt concentration of the sample solution than (1)H-detected experiments. In addition, acquisition can be started with anti-phase coherence, resulting in higher sensitivity due to the elimination of the final INEPT transfer step.

Accelerated acquisition of high resolution triple-resonance spectra using non-uniform sampling and maximum entropy reconstruction.

Non-uniform sampling is shown to provide significant time savings in the acquisition of a suite of three-dimensional NMR experiments utilized for obtaining backbone assignments of H, N, C', CA, and CB nuclei in proteins : HNCO, HN(CA)CO, HNCA, HN(CO)CA, HNCACB, and HN(CO)CACB. Non-uniform sampling means that data were collected for only a subset of all incremented evolution periods, according to a user-specified sampling schedule. When the suite of six 3D experiments was acquired in a uniform fashion for an 11 kDa cytoplasmic domain of a membrane protein at 1.5 mM concentration, a total of 146 h was consumed. With non-uniform sampling, the same experiments were acquired in 32 h and, through subsequent maximum entropy reconstruction, yielded spectra of similar quality to those obtained by conventional Fourier transform of the uniformly acquired data. The experimental time saved with this methodology can significantly accelerate protein structure determination by NMR, particularly when combined with the use of automated assignment software, and enable the study of samples with poor stability at room temperature. Since it is also possible to use the time savings to acquire a greater numbers of scans to increase sensitivity while maintaining high resolution, this methodology will help extend the size limit of proteins accessible to NMR studies, and open the way to studies of samples that suffer from solubility problems.

Molecular dynamics of the long neurotoxin LSIII.

Long neurotoxins bind tightly and specifically to the nicotinic acetylcholine receptor (AChR) in postsynaptic membranes and are useful for exploring the biology of synapses. In crystallographic studies of long neurotoxins the principal binding loop appears disordered, but the NMR solution structure of the long neurotoxin LSIII revealed significant local order, even though the loop is disordered with respect to the globular core. A possible mechanism for conferring global disorder while preserving local order is rigid-body motion of the loop about a hinge region. Here we report investigations of LSIII dynamics based on (13)C(alpha) magnetic relaxation rates and molecular dynamics simulation. The relaxation rates and MD simulation both confirm the hypothesis of rigid-body motion of the loop and place bounds on the extent and time scale of the motion. The bending motion of the loop is slow compared to the rapid fluctuations of individual dihedral angles, reflecting the collective nature and largely entropic free energy profile for hinge bending. The dynamics of the central binding loop in LSIII illustrates two distinct mechanisms by which molecular dynamics directly impacts biological activity. The relative rigidity of key residues involved in recognition at the tip of the central binding loop lowers the otherwise substantial entropic cost of binding. Large excursions of the loop hinge angle may endow the protein with structural plasticity, allowing it to adapt to conformational changes induced in the receptor.

Multiple-quantum magic-angle spinning spectroscopy using nonlinear sampling.

NMR spectroscopy is a relatively insensitive technique and many biomolecular applications operate near the limits of sensitivity and resolution. A particularly challenging example is detection of the quadrupolar nucleus 17O, due to its low natural abundance, large quadrupole couplings, and low gyromagnetic ratio. Yet the chemical shift of 17O spans almost 1000 ppm in organic molecules and it serves as a potentially unique reporter of hydrogen bonding in peptides, nucleic acids, and water, and as a valuable complement to 13C and 15N NMR. Recent developments including the multiple-quantum magic-angle spinning (MQMAS) experiment have enabled the detection of 17O in biological solids, but very long data acquisitions are required to achieve sufficient sensitivity and resolution. Here, we perform nonlinear sampling in the indirect dimension of MQMAS experiments to substantially reduce the total acquisition time and improve sensitivity and resolution. Nonlinear sampling prevents the use of the discrete Fourier transform; instead, we employ maximum entropy (MaxEnt) reconstruction. Nonlinearly sampled MQMAS spectra are shown to provide high resolution and sensitivity in several systems, including lithium sulfate monohydrate (LiSO(4)-H(2)17O) and L-asparagine monohydrate (H(2)17O). The combination of nonlinear sampling and MaxEnt reconstruction promises to make the application of 17O MQMAS practical in a wider range of biological systems.

Elimination of 13Calpha splitting in protein NMR spectra by deconvolution with maximum entropy reconstruction.

Homonuclear 13C-13C couplings can significantly reduce the sensitivity and resolution of multidimensional NMR experiments. The most important of these couplings is the 13Calpha-13Cbeta coupling, and several different methods have been developed to eliminate its effect from spectra used for backbone assignment, including short or constant-time evolution periods, selectively labeled amino acids, and multiple-band decoupling sequences. In this communication we show that postacquisition deconvolution of the spectra with a maximum entropy algorithm can be superior to experimental decoupling. The method is very robust, does not introduce shifts of the resonance positions, and simplifies the measurement of the most important NMR experiments for protein backbone assignment.

Modern spectrum analysis in multidimensional NMR spectroscopy: comparison of linear-prediction extrapolation and maximum-entropy reconstruction.

NMR spectroscopy is an inherently insensitive technique, and many challenging applications such as biomolecular studies operate at the very limits of sensitivity and resolution. Advances in superconducting magnet, cryogenic probe, and pulse sequence technologies have resulted in dramatic improvements in both sensitivity and resolution in the past decade. Conversely, the signal-processing method used most widely in NMR spectroscopy, extrapolation of the time domain signal by linear prediction (LP) followed by discrete Fourier transformation (DFT), was developed in the early 1980s and has not been subjected to detailed scrutiny for its impact on sensitivity and resolution. Here we report the first systematic investigation of the accuracy and precision of spectra obtained by LP extrapolation followed by DFT. We compare the results to spectra obtained by maximum-entropy (MaxEnt) reconstruction, which was developed contemporaneously to LP extrapolation but is not widely employed in NMR spectroscopy. Although it reduces truncation artifacts and increases the amplitudes of strong peaks, we find that LP extrapolation generates false-positive peaks and introduces frequency errors. These defects of LP extrapolation become less pronounced for longer data records and higher signal-to-noise ratio. MaxEnt generally yields more detectable peaks for a given number of data samples, more accurate peak frequencies, and fewer false-positive peaks than LP extrapolation. MaxEnt also permits the use of nonlinear sampling, which can give dramatic improvements in resolution. These results show that the use of MaxEnt together with nonlinear sampling, rather than LP extrapolation, could reduce the amount of instrument time required for adequate sensitivity and resolution by a factor of 2 or more.