Summary

The square root of a covariance spectrum, which offers high spectral resolution along both dimensions requiring only few t1 increments, yields in good approximation the idealized 2D FT spectrum provided that the amount of magnetization exchanged between spins is relatively small. When this condition is violated, 2D FT and covariance peak volumes can differ. A regularization method is presented that produces a modified covariance spectrum with cross-peak volumes that closely match their 2D FT analogues. The method is demonstrated for TOCSY spectra with variable mixing times. [1]

Introduction

In covariance NMR spectroscopy, a covariance matrix is constructed from a set of 1D spectra that belong to different evolution times t1.After application of the matrix square root, a spectrum is obtained that approximates the idealized 2D Fourier transform spectrum. Because the square root is by definition positive, negative eigenvalues present in the 2D FT spectrum will not be retrieved by the matrix square-root operation and, consequently, the resulting covariance peak volumes may differ from their 2D FT counter parts. This problem is addressed here by a regularization procedure that shifts the eigenvalues of the 2D FT matrix to positive values.

The procedure works well for experiments for which the diagonal peaks tend to dominate the cross peaks as is the case for NOESY-type cross-relaxation type experiments and TOCSY experiments at short mixing times (assuming that the spectra are phase-corrected as usual such that the diagonal peaks are positive). At longer mixing times, TOCSY cross peaks may be similar or larger in magnitude than the corresponding diagonal peaks. In such cases, the covariance spectrum can display discrepancies in the peak amplitudes with respect to the 2D FT spectrum.

The problem associated with negative eigenvalues of a general 2D FT spectrum F can be circumvented by the following regularization procedure.By adding a suitable diagonal part to F prior to the covariance calculation, most or all eigenvalues can be moved to positive values.In the simplest case, the diagonal part is the unity matrix 1 times a prefactor a. Thus, the regularized 2D FT matrix is

(1)

where 1 is the unity matrix and a is the regularization factor. Here and in the following it is assumed that F is a (generally non-symmetric) N2xN2 square matrix, which is achieved, for example, by appropriate zero-filling along the indirect dimension. One obtains for the regularized covariance matrix

(2)

where

(3)

This procedure, which is related to Tikhonov regularization or ridge regression, yields in the limit of a very large diagonal, ,

(4)
In this limit the regularized matrix Creg corresponds to the matrix obtained from direct symmetrization.
For what follows it is useful to relate a the sum of the negative real parts of the eigenvalues of F

(5)

where lj is the jth eigenvalue of F and a0 is a scaling factor. The real parts of lj enter because F is generally non-symmetric and therefore has complex eigenvalues. a0 typically covers a range between 0 (no regularization) and 10e6 (very strong regularization).

Figure 1 shows a section of the 2D spectra that contains the cross peak indicated by a rectangle between the two methylene protons 20a and 20b, which systematically differ between the 2D FT and covariance spectrum. The corresponding diagonal peaks are relatively weak in the 2D FT but strong in the covariance spectrum. The two protons represent in first order approximation an isolated 2-spin system to which Eqs. (2), (3) apply. A quantitative comparison between the 2D FT and covariance peak volumes for this cross peak is displayed in Figure 2a. The cross-peak volumes scale in a non-linear way relative to each other for different mixing times tm, which is consistent with the prediction. After regularization using Eqs. (2) and (3) with a0 = 100 the covariance peak volume closely matches the one of the 2D FT spectrum over the whole range of mixing times (Figure 2b). Thus, the regularization procedure removes the discrepancy between 2D FT and covariance NMR for this spin pair.

More subtle differences between 2D FT and covariance NMR spectra can occur in TOCSY spectra of larger spin systems. For the decapeptide antamanide, the 2D FT and covariance peak volumes show a good overall agreement, i.e. no large differences are observed as for the strychnine cross peak described above. Still, the agreement is not perfect as can be seen in Figure 2a and Table 1. Generally, the correlation coefficient between covariance and 2D FT peak volumes decreases slightly when going from N1 = 512 (r = 0.98) to N1 = 64 (r = 0.95). Also, cross peaks that are close to the diagonal show a slightly poorer agreement than cross peaks that are further away.

Regularization leads to a clear improvement of the correlation (r = 0.98 – 1.00) and at the same time the average volume ratio is much closer to 1.0. For example, for N1 = 128 increments the correlation coefficient of the off-diagonal covariance peak volumes with respect to the 2D FT peak volumes is r = 0.965 before and r = 0.996 after regularization, while the average peak volume ratio goes from 1.24 to 1.04.

Generally, cross peaks in close vicinity to the diagonal (‘near diagonal peaks’) exhibit a slightly poorer agreement than cross peaks that lie further away from the diagonal (‘off diagonal peaks’) because for low N1 values the covariance statistics is worse for resonance pairs whose chemical shift difference is small.

a Ratio between 2D FT cross-peak volume and covariance cross-peak volume without regularization.
b Pearson correlation coefficient between 2D FT and covariance cross-peak volume without regularization.
c Ratio between 2D FT and covariance cross-peak volume with regularization.
d Pearson correlation coefficient between 2D FT and covariance cross-peak volume with regularization.
e Off-diagonal cross peaks that are ≥ 0.059 ppm away from diagonal.
f Near-diagonal cross peaks that are between 0.059 and 0.352 ppm away from diagonal.

2D FT spectra with low numbers of t1 increments have intrinsically low resolution along w1 manifested in the form of large line widths along this dimension. By contrast, the corresponding covariance spectrum retains high resolution along both dimensions. When the covariance spectrum is regularized a star-like peak shape emerges as the regularization parameter a0 is increased. This effect is demonstrated in Figure 3 for the TOCSY spectrum of antamanide with N1 = 96 and a0=0, 1.5, 10, and 100. At a0=1.5 the onset of the star shapes is visible (Figure 3b). At a0=10, the star shapes are almost completely formed (Figure 3c) and they are reminiscent of the ‘star effect’ of Lorentzian line shapes in 2D NMR. In the limit of very large a0 the regularized covariance spectrum approaches (FT + F)/2 (Eq. 4). Thus, each cross peak, elongated along w1, is superimposed with its transposed, elongated along w2, which creates the star-like appearance. The star shapes are most pronounced for small N1 values and disappear for a sufficiently large N1 irrespective of a0. Because (FT + F)/2 does not provide resolution enhancement over the original 2D FT spectrum F, for the analysis of densely populated peak regions the scaling parameter a0 should be kept moderately small to minimize spectral overlap. Because of the computational efficiency of covariance processing, a0 can be optimized for different regions of the spectrum if necessary.

Acknowledgments

Reference