An algebraic solution for determining overall rotational correlation times from cross-correlated relaxation rates
Accurate rotational correlation times ({tau}c) are critical for quantitative analysis of fast timescale NMR dynamics. As molecular weights increase, the classic derivation of{tau} c using transverse and longitudinal relaxation rates becomes increasingly unsuitable due to the non-trivial contribution of remote dipole-dipole interactions to longitudinal relaxation. Derivations using cross-correlated relaxation experiments, such as TRACT, overcome these limitations but are erroneously calculated in 65% of the citing literature. Herein, we developed an algebraic solutions to the Goldman relationship that facilitate rapid, point-by-point calculations for straightforward identification of appropriate spectral regions where global tumbling is likely to be dominant. The rigid-body approximation of the Goldman relationship has been previously shown to underestimate TRACT-based rotational correlation time estimates. This motivated us to develop a second algebraic solution that employs a simplified model-free spectral density function including an order parameter term that could, in principle, be set to an average backbone S2 {approx} 0.9 to further improve the accuracy of{tau} c estimation. These solutions enabled us to explore the boundaries of the Goldman relationship as a function of the H-N internuclear distance (r), difference of the two principal components of the axially-symmetric 15N CSA tensor ({Delta}{delta}N), and angle of the CSA tensor relative to the N-H bond vector ({theta}). We hope our algebraic solutions and analytical strategies will increase the accuracy and application of the TRACT experiment.