Chapter 10: Compressed Sensing ℓ₁-Norm Minimisation in Multidimensional NMR Spectroscopy

Central to studies of biomolecules are multidimensional experiments; with the increasing size limit of NMR enabling studies of larger proteins and protein complexes, high-dimensional experiments, e.g. 4D spectra, are important to identify additional resonance correlations and to simplify the process of spectral assignment. However, high-dimensional experiments are lengthy to record and typically result in a trade-off between resolution and signal-to-noise ratio, which can undermine their usefulness. Non-uniform sampling strategies can achieve significant time savings and allow high resolution, whilst maintaining good signal-to-noise ratio; since such sampling strategies do not sample the full Nyquist grid, it is no longer possible to reconstruct spectra using the Fourier transform. A range of different reconstruction methods has been proposed over the years; recently, approaches termed ‘compressed sensing’ (CS) based on minimisation of an ℓ_p-norm where 0 <p≤1 have been shown to provide the optimal reconstruction under certain conditions. Application of CS to NMR spectra has been demonstrated for a range of different spectra in the literature. We discuss the theory of this approach as well as current applications of the method and present evidence to show the benefit of using CS-reconstructed 4D spectra for gathering structural restraints of large proteins.