Applications of non-uniform sampling and processing

Abstract

Modern high-field NMR instruments provide unprecedented resolution. To make use of the resolving power in multidimensional NMR experiment standard linear sampling through the indirect dimensions to the maximum optimal evolution times (~1.2T (2)) is not practical because it would require extremely long measurement times. Thus, alternative sampling methods have been proposed during the past 20 years. Originally, random nonlinear sampling with an exponentially decreasing sampling density was suggested, and data were transformed with a maximum entropy algorithm (Barna et al., J Magn Reson 73:69-77, 1987). Numerous other procedures have been proposed in the meantime. It has become obvious that the quality of spectra depends crucially on the sampling schedules and the algorithms of data reconstruction. Here we use the forward maximum entropy (FM) reconstruction method to evaluate several alternate sampling schedules. At the current stage, multidimensional NMR spectra that do not have a serious dynamic range problem, such as triple resonance experiments used for sequential assignments, are readily recorded and faithfully reconstructed using non-uniform sampling. Thus, these experiments can all be recorded non-uniformly to utilize the power of modern instruments. On the other hand, for spectra with a large dynamic range, such as 3D and 4D NOESYs, choosing optimal sampling schedules and the best reconstruction method is crucial if one wants to recover very weak peaks. Thus, this chapter is focused on selecting the best sampling schedules and processing methods for high-dynamic range spectra.

Figure 1

Sorted L² norm evaluation of five sampling strategies for non-uniform sampling of NMR data. The L² norm ( $=\sqrt{{(f_i^{\mathit{ref}}-f_i^{\mathit{rec}})}^2}$) in this case describes the deviation between the reference spectrum (a singlet with no line-width) and the FM reconstructed equivalent, when selecting 256 of 1024 complex data points in the time domain based on particular sampling strategy and seed value. 100 seed values were used to create 100 specific sampling schedules for each sampling strategy. The L² values were hence sorted and the “best-of-100” (represented rightmost in the figure) and “worst-of-100” (represented leftmost) sampling schedules were identified for each strategy for further evaluation. (N.B. For practical use, only the “best-of-100” schedule would be of interest.)

Figure 2

Justification of prior selecting seed values and evaluation of sampling strategy based on a simulated spectrum with four signals, each simulated to been acquired to the equivalent of one T₂ and with no noise. (I.e., the simulated FID decays to 1/e.) The intensity of the leftmost signal is tenfold that to each of the three others. (a) FM reconstruction using the sampling schedule based on labelled sampling schedules and seed value found to be “best-of-100” from previous evaluation (figure 1.) and (b) FM reconstruction using the sampling schedule based on labelled sampling schedules and using the seed value found to be “worst-of-100” from previous evaluation. The experience is described in the text.

Figure 3

Demonstration of initial state before FM reconstruction setting all non-obtained data values to zero, effectively using a discrete Fourier transformation (DFT) with the same sampling schedules as in Figure 2a. The residual between reconstructed and reference here describe the combined point spread function from the four signals.

Figure 4

FM reconstruction of the test situation in figure 2a – with noise added. The Gaussian noise was generated 100 times, each time with a different seed value and added in the time-domain of the reference. The data was then sampled according to the various sampling schedules and FM reconstruction applied. In order to compare the situation of uniformly sampled data, the same noise was used, but with four fold rmsd in order to emulate equal time acquisition. As the evaluation is concentrated on the effect of the weaker peaks, only the data points 103 through 1072 were used for L² norm evaluation. (Marked with bars in the reference spectrum in (a)). As no prior knowledge about the noise, the result of the spectral presentation may vary from (a) when the noise is most cooperative, through (b) the average situation (median) to (c) where the noise is least cooperative.

Figure 5

Weaved implementation of NUS Poisson gap sampling schedules in two indirect dimensions. First schedules are created with the selected schedule in the first t₁ column. This is followed by using the first two rows where the time points had not yet been selected. Subsequently, the next two columns are picked as indicated and so on.

Figure 6

Comparison of sampling schedules on a simulated spectrum of 2 indirect dimensions including Gaussian noise. Nine peaks of relative intensities 2 to 10 were created. A: Complete spectrum simulated. The central area containing the peaks is boxed and expanded in the panels of the rest of the figure. B: Contour plots of the central part plotted at two different starting levels of lowest contours. The contours are spaced by a factor of 1.5, and the relative intensity is indicated in the right-hand panel. C and D: Peak recovery with the best (C) and worst (D) of the 100 Gaussian noise sets. As best and worst noise sets we selected those that resulted in the lowest and highest L² values for the selected area containing the peaks. This indicates the quality range of peak recovery one can expect. The six panels in C and D correspond to the same total number of scans. Thus, 16 times more scans per increment can be collected for the NUS data. To take account of this we multiplied the added noise by a factor of four for the US spectra (top left panel). The simulations show that NUS recovers the peaks in the noisy spectrum significantly better than US when equal measuring times are considered. For all NUS schedules the best of 100 seed numbers were used as selected with minimizing the point spread function. (C)

Figure 7

Comparison of alternative processing on a 3D ¹⁵N-NOESY-HSQC spectrum of human translation initiation factor eIF4e. (a) Uniformly sampled reference. The time domain data were acquired as 6400 hyper-complex points sampled in the two indirect dimensions (128 (H_indir) × 50 (¹⁵N)). The spectra were measured on a 700 MHz spectrometer with sweep widths of 9765 Hz and 2270 Hz, respectively. The t_max hence were 0.013 and 0.022 seconds each for the indirect proton and nitrogen dimensions, respectively, representing nearly an optimal situation for the nitrogen dimension, but not for the indirect proton dimension. Data were transformed with the standard FFT procedure after cosine apodization and doubling the time domain by zero filling. (b) Reducing the number of complex points to 42 (32%) (b1) and 13 (10%) (b2) in the indirect proton dimension, cosine apodization and zero filling results in low resolution spectra in the indirect dimensions. (c) FM reconstruction of 2048 (32%) (c1) and 640 (10%) (c2) data points sampled with an exponential weighting schedule in the two indirect dimensions. (d) FM reconstruction of 2048 (32%) (d1) and 640 (10%) (d2) sampled data points according to weaved sinusoidal Poisson gap sampling. For all spectra, equal number of scans were recorded per increment. Thus, the NUS spectra were acquired in one third and one tenth of the time used for the US spectrum, respectively.