Nonuniform Sampling and Maximum Entropy Reconstruction in Multidimensional NMR

Abstract

CONSPECTUS 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 complex biomolecules, but it suffers from well-known limitations. Chief among these is the difficulty of obtaining high-resolution spectral estimates from short data records. For multidimensional NMR experiments, this imposes a sampling burden, because the time required to perform an experiment is proportional to the number of data samples. At high magnetic field, where spectral dispersion is greatest, the problem becomes particularly acute. Consequently multidimensional NMR experiments that rely on the DFT either must sacrifice resolution in order to be completed in reasonable time, or they must 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 capable of providing 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 attend the use of nonuniform sampling can be viewed as frequency-aliased signals. Strategies for suppressing nonuniform sampling artifacts include careful design of the sampling scheme and special methods for computing the spectrum. Time savings of a factor of three for each of the N-1 indirect dimensions of an N-dimensional NMR experiment are now routinely reported, making practical high-resolution 3- and 4-dimensional experiments that were previously prohibitively time consuming. Conversely, tailored sampling in the indirect dimensions has been utilized to improve sensitivity. Improvements in nonuniform sampling strategies appear poised to 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 multidimensional experiments are conducted. The potential benefits will enable more advanced applications of multidimensional NMR spectroscopy to 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 application in other forms of spectroscopy.

Applications of NMR spectroscopy to systems biochemistry

“…There have been multiple developments that improve this aspect, including non-uniform sampling [154,365] and projection–reconstruction [366,367]. There are also hardware approaches, including multiple receivers enabling 2 or more 2D experiments to be acquired simultaneously [368,369], as well as one-shot 2/3D methods which acquire different “increments” at different slices of the sample [370–376].…”

“…The main impediments to the use of 3D experiments are sensitivity and experiment time, though non-uniform sampling in two indirect dimensions [154], GFT or PR reconstructions [155,156] may make such applications practical in some cases.…”

MultidimensionalNMRspectroscopy in a single scan

“…The latter include relaxation-enhanced acquisitions, [41][42][43] and a variety of nonuniformly sampled experiments that with the aid of advanced data/image processing techniques, can deliver the highdimensional correlations being sought. [44][45][46][47][48][49] Still, even with the help of all these advances, NMR lags behind the outstanding parallelization and fast data acquisition achievements that have augured in the 'omics' age. Also, beyond NMR's ability is the tracking of fast processes like protein folding, chemical dynamics, molecular isomerizations -at least in a real time fashion.…”

Convex accelerated maximum entropy reconstruction

“…Footnote 1: It is worthy of mention that other reconstruction methods, such as CLEAN and SCRUB [11], attempt to directly deconvolve the PSF from the true spectrum without the use of an explicitly defined regularization functional. Contemporary reconstruction methods include multidimensional decomposition (MDD; [12, 13]), iterative soft thresholding (IST; [14-16]), iteratively reweighted least-squares (IRLS; [17]), and maximum entropy (MaxEnt; [18]). In fact, all these methods may be interpreted as involving the optimization (i.e .…”