Least-Squares Method for Quantitative Determination of Chemical Exchange and Cross-Relaxation Rate Constants from a Series of Two-Dimensional Exchange NMR Spectra Zsolt Zolnai, ² Nenad Juranic ´ , and Slobodan Macura* Department of Biochemistry and Molecular Biology, Mayo Clinic and Foundation, Mayo Graduate School, Rochester, Minnesota 55905 ReceiVed: December 3, 1996; In Final Form: March 13, 1997 X We present a new method, the least-error matrix analysis (LEMA), to quantify the dynamic matrix from a series of 2D NMR exchange spectra. The method is based on a weighted averaging of individual dynamic matrices. The matrices are obtained by full-matrix analysis (FMA) from a series of 2D exchange spectra recorded at different mixing times. The weights, calculated by error propagation analysis, are explicit functions of the mixing time. The principal advantage of LEMA in comparison to FMA is that it uses all the known relationships between the spectral peaks: the peak correlations within 2D spectra, and the mixing time evolution among the spectra. We tested LEMA by analyzing a series of 10 cross-relaxation spectra (NOESY, τ m ) 60 µs-1.28 s) in a rigid 10-spin system (cyclo(L-Pro-Gly) in 3:1 v/v H 2 O/DMSO). At 233 K the dipeptide has a mobility like a small protein with a correlation time of 3.8 ns. While FMA at τ m ) 30 ms could extract only 14 distances in a range 1.75-3 Å, LEMA provided 22 distances, of which the longest was 4 Å. The extension of the available interproton distances from 3 to 4 Å afforded by LEMA is caused by a 10-fold decrease of the lower limit of measurable cross-relaxation rates, from -0.59 to -0.06 s -1 . The most important property of LEMA, provision of accurate average values of magnetization exchange rates from a given set of peak volumes, is verified experimentally on a model system. Introduction The most notable forms of two-dimensional (2D) NMR exchange spectroscopy 1 that contributed immensely to the popularity of 2D NMR method in chemistry 2 are the chemical exchange spectroscopy (EXSY) 3,4 and nuclear Overhauser enhancement spectroscopy (NOESY). 5,6 A particularly impres- sive application of NOESY is the determination of the solution structure of proteins. 7 Although a semiquantitative analysis was typically used for it, it is well recognized that the quantitative analysis of cross-relaxation spectra might be more useful. 8-16 Similarly, the quantification of EXSY spectra improves their utility to study chemically exchanging systems. 17-21 However, only a few attempts have been made to estimate the corre- sponding error limits. 19,22-27 The determination of the magnetization exchange rate con- stants [L] pq is independent from the type of exchange observed (cross-relaxation in NOESY, or chemical exchange in EXSY), since both transfer types are directed by the same master equation. The methods to quantify exchange spectra can be roughly divided into two categories: buildup rate analysis (BU) 6,28 and full-matrix analysis (FMA). 8,11,13 The BU analysis provides the [L] pq ’s from the time evolution of individual cross- peaks (Figure 1a), whereas FMA utilizes all peaks at a single mixing time (Figure 1b). Theoretically, FMA needs only one spectrum at an arbitrary mixing time. However, in the presence of noise the errors are close to minimum only in a limited mixing time interval. 19,25,26 Since the width and position of the interval is not known in advance, several experiments in a broad range of mixing times are needed. Thus, both BU and FMA require a set of exchange spectra at different mixing times. For peak volume normalization, FMA requires an exchange spectrum at zero mixing time as well. The principal weakness of BU analysis is that it ignores the correlations among the cross-peaks within individual 2D spectra (Figure 1a). The calculation of one buildup curve ignores the properties of all other buildup curves. Similarly, the FMA ignores the known dependence of cross peaks on time evolution; the FMA at one mixing time is independent from the FMA at any other mixing time (Figure 1b). A superior method shall use all available correlations within and among the spectra (Figure 1c) and shall minimize the random errors in a least-squares manner. Here we present a method that satisfies these requirements: a least error matrix analysis (LEMA). It provides the best estimates of magnetiza- tion exchange rate constants and their errors, using in a least- squares sense the information from a series of 2D spectra ² On leave from Mathematical Institute, Knez Mihailova 35, Beograd, Yugoslavia. * To whom correspondence should be addressed. X Abstract published in AdVance ACS Abstracts, May 1, 1997. Figure 1. Schematic representation of the connectivities used in different methods for quantification of a series of 2D exchange spectra: (a) Buildup analysis (BU) uses the time evolution of individual cross peaks ignoring their connectivities within the spectra. (b) Full- matrix analysis (FMA) takes into account the connectivities among the peaks within a 2D spectrum but ignores the time evolution of peaks. (c) Least-errors matrix analysis (LEMA) uses all available correlations. It exploits the connectivities within a spectrum as FMA and the connectivities over mixing time as BU. 3707 J. Phys. Chem. A 1997, 101, 3707-3710 S1089-5639(96)03956-4 CCC: $14.00 © 1997 American Chemical Society