Joint non-uniform sampling of all incremented time delays for quicker acquisition in protein relaxation studies

NMR relaxometry plays crucial role in studies of protein dynamics. The measurement of longitudinal and transverse relaxation rates of ¹⁵N is the main source of information on backbone motions. However, even the most basic approach exploiting a series of ¹⁵N HSQC spectra can require several hours of measurement time. Standard non-uniform sampling (NUS), i.e. random under-sampling of indirect time domain, typically cannot reduce this by more than 2–4× due to relatively low “compressibility” of these spectra. In this paper we propose an extension of NUS to relaxation delays. The two-dimensional space of t₁ /t_relax is sampled in a way similar to NUS of t₁/t₂ domain in 3D spectra. The signal is also processed in a way similar to that known from 3D NUS spectra i.e. using one of the most popular compressed sensing algorithms, iterative soft thresholding. The 2D Fourier transform matrix is replaced with mixed inverse Laplace-Fourier transform matrix. The peak positions in resulting 3D spectrum are characterized by two frequency coordinates and relaxation rate and thus no additional fitting of exponential curves is required. The method is tested on three globular proteins, providing satisfactory results in a time corresponding to acquisition of two conventional ¹⁵N HSQC spectra.