TRACT revisited: an algebraic solution for determining overall rotational correlation times from cross-correlated relaxation rates

Accurate rotational correlation times (\({\tau }_{\text{c}}\)) 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 ≈ 0.9 to further improve the accuracy of \({\tau }_{\text{c}}\) 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.