Krzysztof Kazimierczuk

Optimization of random time domain sampling in multidimensional NMR.

The detailed description of rules for generation of different random sampling schemes is shown and discussed with regard to Multidimensional Fourier Transform (MFT). The influence of different constrained random sampling schedules on FT of constant signal, i.e., Point Spread Function (PSF), is analyzed considering artifacts level and distribution. We found that Poisson disk sampling schedule, which provides a large low-artifact area in the signal vicinity, is the method of choice in the case of nonlinear sampling of time domain in NMR experiments. We have verified the new sampling schemes by application to the 3D HNCACB and 15N-edited NOESY-HSQC spectra acquired for 13C,15N labeled ubiquitin sample.

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.

Narrow peaks and high dimensionalities: exploiting the advantages of random sampling.

Level of artifacts in spectra obtained by Multidimensional Fourier Transform has been studied, considering randomly sampled signals of high dimensionality and long evolution times. It has been shown theoretically and experimentally, that this level is dependent on the number of time domain samples, but not on its relation to the number of points required in appropriate conventional experiment. Independence of the evolution time domain size (in the terms of both: dimensionality and evolution time reached), suggests that random sampling should be used rather to design new techniques with large time domain than to accelerate standard experiments. 5D HC(CC-TOCSY)CONH has been presented as the example of such approach. The feature of Multidimensional Fourier Transform, namely the possibility of calculating spectral values at arbitrary chosen frequency points, allowed easy examination of resulting spectrum. We present the example of such approach, referred to as Sparse Multidimensional Fourier Transform.

Non-uniform frequency domain for optimal exploitation of non-uniform sampling.

Random sampling of NMR signal, not limited by Nyquist Theorem, yields up to thousands-fold gain in the experiment time required to obtain desired spectral resolution. Discrete Fourier transform (DFT), that can be used for processing of randomly sampled datasets, provides rarely exploited possibility to introduce irregular frequency domain. Here we demonstrate how this feature opens an avenue to NMR techniques of ultra-high resolution and dimensionality. We present the application of high resolution 5D experiments for protein backbone assignment and measurements of coupling constants from the 4D E.COSY multiplets. Spectral data acquired with the use of proposed techniques allow easy assignment of protein backbone resonances and precise determination of coupling constants.

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. Copyright

A comparison of convex and non-convex compressed sensing applied to multidimensional NMR

The resolution of multidimensional NMR spectra can be severely limited when regular sampling based on the Nyquist-Shannon theorem is used. The theorem binds the sampling rate with a bandwidth of a sampled signal and thus implicitly creates a dependence between the line width and the time of experiment, often making the latter one very long. Recently, Candès et al. (2006) [25] formulated a non-linear sampling theorem that determines the required number of sampling points to be dependent mostly on the number of peaks in a spectrum and only slightly on the number of spectral points. The result was pivotal for rapid development and broad use of signal processing method called compressed sensing. In our previous work, we have introduced compressed sensing to multidimensional NMR and have shown examples of reconstruction of two-dimensional spectra. In the present paper we discuss in detail the accuracy and robustness of two compressed sensing algorithms: convex (iterative soft thresholding) and non-convex (iteratively re-weighted least squares with local ℓ(0)-norm) in application to two- and three-dimensional datasets. We show that the latter method is in many terms more effective, which is in line with recent works on the theory of compressed sensing. We also present the comparison of both approaches with multidimensional decomposition which is one of the established methods for processing of non-linearly sampled data.

A set of 4D NMR experiments of enhanced resolution for easy resonance assignment in proteins.

This paper presents examples of techniques based on the principle of random sampling that allows acquisition of NMR spectra featuring extraordinary resolution. This is due to increased dimensionality and maximum evolution time reached. The acquired spectra of CsPin protein and maltose binding protein were analyzed statistically with the aim to evaluate each technique. The results presented include exemplary spectral cross-sections. The spectral data provided by the proposed techniques allow easy assignment of backbone and side-chain resonances.

Determination of Spin-Spin Couplings from Ultrahigh Resolution 3D NMR Spectra Obtained by Optimized Random Sampling and Multidimensional Fourier Transformation

An example of precise evaluation of backbone scalar J couplings using random sampling of evolution time space in 3D NMR experiments is presented. The recorded spectrum, due to violation of the Nyquist theorem limitation, exhibits ultrahigh resolution in indirect dimensions compared to standard NMR experiment acquired at the same time. The obtained results enable simple and accurate evaluation of scalar and residual dipolar couplings from a single multidimensional NMR experiment.

Generalized Fourier Transform for Non-Uniform Sampled Data

Fourier transform can be effectively used for processing of sparsely sampled multidimensional data sets. It provides the possibility to acquire NMR spectra of ultra-high dimensionality and/or resolution which allow easy resonance assignment and precise determination of spectral parameters, e.g., coupling constants. In this chapter, the development and applications of non-uniform Fourier transform is presented.

High-Dimensional NMR Spectra for Structural Studies of Biomolecules

Recent developments in the acquisition and processing of NMR data sets facilitate the recording of ultra-high-resolution NMR spectra in a reasonable time. The new experiments allow easy resonance assignment for folded and unfolded proteins, as well as the precise determination of spectral parameters, for example, chemical shifts, NOE contacts, coupling constants or cross-correlated relaxation rates. Owing to exceptional resolution of 4D-6D spectroscopy, detailed studies of biomolecules of unprecedented complexity are now possible. Herein, the principles of acquisition and processing methods are presented. The main applications of high-dimensional NMR experiments, including backbone and side-chain resonance assignment in proteins, as well as heteronuclear edited NOE techniques are reviewed.