Indirect covariance

Key Words 1,1-ADEQUATE, 1,/i-ADEQUATE, /%1-ADEQUATE, /i,/77-ADEQUATE, Structure elucidation, Unsymmetrical indirect covariance, Generalized indirect covariance, Carbon-carbon correlation. 

Finally, retrorsine, 22, has also recently been used as a model compound for a study by the author and co-workers that investigated the 1,1- and 1,h-ADEQUATE spectra of the molecule in conjunction with unsymmetrical and generalized indirect covariance processing to afford the corresponding HSQC-1,1-ADEQUATE and HSQC-1,h-ADEQUATE spectra, respectively.51... 

Covariance processing can also be applied to heteronuclear 2D NMR experiments. As first demonstrated in a report by Zhang and Briishwei-ler,82 an "indirect" covariance spectrum can be obtained by multiplying a spectrum by its transpose according to Equation (2) ... 

The resulting spectrum, Cindirect/ is a symmetric matrix whose axes are defined by the indirect dimension, F1. Applying indirect covariance processing to a non-symmetric data matrix such as heteronuclear 1H-13C HSQC-TOCSY spectrum affords a symmetric homonuclear 13C-13C TOCSY spectrum analogous to that described originally by Turner.22... 

Unsymmetrical indirect covariance and generalized indirect covariance processing using ADEQUATE data... 

Indirect covariance processing has been further extended by the work of Martin and co-workers83 84 and by Kupce and Freeman85 to include the reconstruction of non-symmetric spectra from pairs of spectra, F and G, that share a common frequency domain according to Equation (3) ... 

A third area of development that has affected the speed of obtaining molecular connectivity information from NMR takes advantage of the information inherently present in two separate experiments. Traditionally, an analyst would use the information from a group of separate experiments to draw conclusions about molecular connectivity. In recent years, the projection-reconstruction technique97,98 and indirect covariance NMR99 have allowed information from two separately acquired experiments to be correlated into an additional experiment. Both techniques can increase the dimensionality of NMR data providing information that would otherwise require time-consuming acquisitions. 

The projection-reconstruction approach is a technique unrelated to covariance processing which can provide data typically inaccessible to the natural product chemist. For example, 13C-15N correlation spectra were obtained for vitamin B12 at natural abundance.104 Compared with a conventional three-dimensional 13C-15N correlation experiment, the projection-reconstruction method provides a sensitivity enhancement of two orders of magnitude. The final 13C-15N spectrum was reconstructed from data obtained from ll l5N and H- C correlations acquired using a time-shared evolution pulse sequence that allowed all the information to be obtained in one experiment.104 Martin and co-workers also demonstrated the ability to generate 13C-15N correlation spectra using unsymmetrical indirect covariance NMR with vinblastine as an example.105 In the latter case, 13C-15N correlation spectra were obtained from - C HSQC data and H-1sN HMBC data that were acquired separately. Both methods provide access to correlations that would be inaccessible for most natural products at natural abundance. 

Zhang EL, Bruschweiler-Li L, Bruschweiler R (2010) Simultaneous de novo identification of molecules in chemical mixtures by doubly indirect covariance NMR spectroscopy. J Am Chem Soc 132(47) 16922-16927... 

According to Eq. (5.14), covariance transformations can be applied to any type of spectral data sets that are connected to each other by a common history or domain. The spectra thus generated represent heterospectral correlation maps. In NMR spectroscopy, this concept was taken up as unsymmetrical indirect covariance (UIC) NMR, relating, for example. 

As can be seen, UIC according to Eq. (5.22) does not lead to a symmetric matrix, unless F and G are themselves symmetric. Therefore, the square root operation must not be applied. It has been discussed that the lack of this operation causes a more significant presence of artefacts from the mathematical operations. As a remedy, generalized indirect covariance (GIG) according to Eq. (5.23) was proposed [14]. 

The potential to extend 2-dimensional covariance NMR to higher dimensionality has its foundations in Eq. (5.13). Thus, Snyder et al. [16] laid the basis for the computation of 4D NOESY spectra. In their strategy, the critical entry point consisted of considering a 4D data set an array of 2D data or a plane-of-planes. In order to Ulustrate the calculations, the terms donor and acceptor planes in combination with donor and acceptor pairs were coined. It should be noted that an acceptor plane is associated with each donor pair 01,0)2) at frequencies < i and < 2- Mapping of either the acceptor planes onto the donor planes or vice versa describes the projection of a dimension onto another, which leads to an increase for direct covariance or to a decrease for indirect covariance in dimensionality of resolution. 

The Transition from Indirect to Unsymmetrical Indirect Covariance... 

Doubly indirect covariance (DIG) yields a symmetric correlation map [13]. It can further be considered as a bridge to the generalization of indirect covariance. It implies that correlation spectra, that are used for indirect covariance, themselves may be a creation of a previous covariance transformation. Scheme 5.1 serves as an example to elucidate this perspective. In Scheme 5.1A and B, indirect covariance generates homonuclear spectra. Part (C) leads to the creation of a heteronuclear correlation, which itself can be taken as component spectrum for transformation (A) or (B). It is apparent that Scheme 5.1 represents an oversimplification of the types of indirect covariance treatment. A wider variety of spectra that can be obtained by UIC processing is collected in Table 5.1. 

Scheme 5.1 Indirect covariance is sketched in (A), doubly indirect covariance (DIC) in (B) and unsymmetrical indirect covariance in (C). The powers of 14 and regularization factors have been omitted for simplicity. Common experimental abbreviations were used equivalent to the spectra or data matrices obtained from the corresponding experiment.

Table 5.1 Types of spectra that are obtained after unsymmetrical indirect covariance processing of base spectra...

Indirect covariance processing as depicted in Fig. 5.IB is another type of single spectmm manipulation. Here, a heteronuclear correlation spectmm is transformed into a correlation spectmm of the indirect dimension of the original data. In the current example, a H—C HSQC—TOCSY spectmm is converted into a C—C correlation spectmm, containing only correlations of carbon atoms that share the connectivity information of their attached protons. Quaternary carbons thus remain unobserved due to the principle of the H-C HSQC—TOCSY. While the direct covariance processing leads to an increase in resolution, the indirect covariance processing results in a formal decrease. Yet, the reduction of signals under retention of the information content in the case of the latter transformation compensates for the decrease in dimensionafity. It is also possible to use the indirect covariance procedure on homonuclear spectra to obtain homo-decoupled... 

Another valuable aspect of the matrix calculation is the preservation of the sign of a matrix element, which is the spectral phase [20]. As an example may serve the UIC transformation of an edited H—C HSQC together with an H-H COSY to yield an H-C HSQC—COSY, cf. Fig. 5. IE. An overview of the use of phase-sensitive spectra in covariance processing was given by Aspers et al. [24]. Their collection of useful indirect covariance co-processing is extended and presented in Table 5.1. 

The two large-scale cormnercial vendors of spectrometer-independent NMR software currently offer covariance processing as a feature of their spectral manipulation and interpretation suites. The Spectrus platform by ADC/Labs and the MNova by Mestrelab Research provide stand-alone solutions that allow the import of all spectrometer data formats without the necessity of pre-processing. Covariance processing can be performed according to the direct, indirect, and generalized indirect formalism. The doubly indirect covariance transformation is formd in MNova. While... 

Spectrus ACD/Labs Data format independent direct, indirect, unsymmetrical and generalized indirect covariance processing http //WWW. acdlabs.com/ products/ spectrus/... 

Topspin 3.0 Bruker Biospin AG Bruker data format User http //www. program (au) for direct bruker.com and indirect covariance processing ... 

COLMAR Rafael Briischweiler, Florida State University and National High Magnetic Field Laboratory Metabolomics Web portal NMILPipe, Varian, Bruker data format direct and indirect covariance processing with optional water removal also available DemixC, query, TOCCATA query [55,56] http // spinportal. magnet.fsu.edu... 

A very recent report on the application of DemixC wiU be summarized below, since the spectra were processed via indirect covariance [81]. 

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