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

Abstract

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.