Least-squares method for quantitative determination of chemical exchange and cross-relaxation rate constants from a series of two-dimensional exchange NMR spectra

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 H₂O/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.