Sampling scheme and compressed sensing applied to solid-state NMR spectroscopy

Abstract

We describe the incorporation of non-uniform sampling (NUS) compressed sensing (CS) into oriented sample (OS) solid-state NMR for stationary aligned samples and magic angle spinning (MAS) Solid-state NMR for unoriented 'powder' samples. Both simulated and experimental results indicate that 25-33% of a full linearly sampled data set is required to reconstruct two- and three-dimensional solid-state NMR spectra with high fidelity. A modest increase in signal-to-noise ratio accompanies the reconstruction.

Keywords: Compressed sensing; Membrane proteins; Non-uniform sampling; Oriented samples; Protein NMR; Sensitivity; Solid-state NMR.

Figure 1

Illustrations of sampling schemes. A. Uniform sampling scheme (regular experiments). B. Non-uniform sampling (NUS) scheme consisted by a. the uniform sampling region and b. the sampling region that the sampling probabilities are modulated by an exponential decay weighting function.

Figure 2

The values shown in the figure are the averages of every fifty NUS schemes generated by different conditions that are applied to the simulated spectra. Lower L2 values indicate higher similarities between the reconstructed and noiseless spectra. A. The qualities of reconstructions with various sampling percentages under different signal to noise ratios. It shows that the reconstructions have similar tendencies to the ideal signal if the signals have signal-to-noise ratio ~10, and the errors of reconstructions increase much faster if the sampling percentage lower than 20%. B. The qualities of reconstruction by adding uniform sampling region for 33% and 25% sampling. The reconstructions are improved by adding for 10% to 25% uniform sampling region. The error bars indicate the standard deviations of L2, which are significantly improved by introducing uniform sampling region, implying the qualities reconstructions are more stable. Lagrange multiplier is 0.05 for reconstructing the simulated signals.

Figure 3

Best and worst reconstructed spectra of 25% sampling with 0% to 50% uniform sampling region on the spectra with signal to noise ratios equal 16 and 8. These spectra are chosen from fifty reconstructions of a selected simulated spectrum, and the criterion is based on L2, which is shown on each spectrum. The regular spectra shown in the left column for comparing the qualities of spectra acquired with the same amount of time. 25% sampling with signal to signal-to-noise ratio = 16 should be compared to the regular spectrum with signal-to-noise ratio = 8, and 25% sampling with signal to noise =8 should be compared to the regular spectrum with signal-to-noise ratio = 4. Most of the peaks can be reasonably identified after the improvement.

Figure 4

Illustrations of two-dimensional sampling schemes A. Regular sampling scheme B. Truncated sampling scheme (10% of regular sampling scheme) C. NUS scheme (33% of truncated sampling scheme).

Figure 5

¹⁵N-detected SLF spectra of Pf1 protein in DHPC/DMPC bicelles. A. Fully sampled spectrum with 80 t1 points. Reconstructions from B. 50%, C. 33% and D. 25% sampling. Lagrange multiplier is 0.01. The slices are extracted from 81ppm.

Figure 6

Two-dimensional ¹³C/¹³C correlation spectra of MerF protein in 14-O-PC liposomes with 20ms mixing time. A. Fully sampling Fourier transform spectrum with 64 scans. Reconstructions from B. 50%, C. 33% and D. 25% sampling also with 64 scans. Lagrange multiplier is 0.03.

Figure 7

Slices extracted from 59 ppm and 179 ppm in the t2 dimension of Fig. 5 showing the details of reconstructions: From the bottom to the top, regular experiment, reconstructions from 50%, 33%, and 25% respectively, where the strongest peaks are normalized. The average signal to noise ratios of cross peaks in 50%, 33%, and 25% sampling spectra are 1.48, 0.90, and 0.78 folds relative to the regular spectrum, respectively.

Figure 8

Correlation plots of the intensities between regular and reconstructed spectra of Fig. 5. All the intensities are normalized to their own spectra. A. Correlation plots of diagonal and cross peaks. B. Correlation plots of cross peaks.

Figure 9

Three-dimensional HETCOR-SAMMY spectrum and its reconstructed spectra of ¹⁵N-labeled NAL single crystal. A. Conventional three-dimensional HETCOR/SLF spectrum (32 t1 complex points, 96 t2 points). B. Reconstructed three-dimensional HETCOR/SLF spectrum 614 points (20%). C, and D. are the two-dimensional planes extracted from the 1H chemical shift at 12 ppm in A. and B., respectively. Lagrange multiplier is 0.01.

Figure 10

Three-dimensional HETCOR-SAMMY spectrum and its reconstructed spectra of ¹⁵N-labeled Pf1 coat protein. A. Conventional three-dimensional HETCOR-SAMMY spectrum (20 t1 complex points, 32 t2 points, and 40 scans). B. Reconstructed three-dimensional HETCOR-SAMMY spectrum (212 points, and 40 scans). C. Reconstructed three-dimensional HETCOR-SAMMY spectrum (212 points, and 120 scans). D, E., and F. are the two-dimensional planes extracted from the ¹H chemical shift at 9.5 ppm in A., B., and C., respectively. Lagrange multiplier is 0.05.