Full relaxation matrix analysis of apparent cross-correlated relaxation rates in four-spin systems

Abstract

Cross-correlated relaxation (CCR) rates are an established tool for the extraction of relative bond orientations in biomolecules in solution. CCR between dipolar interactions in four-spin systems is a particularly well-suited mechanism. In this paper, a simple approach to analyze systematic experimental errors is formulated in a subspace of the complete four-spin Hilbert space. It is shown that, contrary to the common assumption, the secular approximation of the relaxation matrix is marginal for the most prominent spin systems. With the main focus on the model protein GB3 at room temperature, it is shown that the apparent experimental CCR rates have errors between −12% and +4% for molecules with a molecular tumbling time of 3.5ns. Although depending on the specific pulse sequence used, the following rule-of-thumb can be established: Judged by absolute values, the errors for Hα–Cα/Hα–Cα, HN–N/Cα–C′, HN–N/Cγ–Cβ and HN–N/Hβ–Cβ CCR rates can safely be neglected. However, errors for HN–N/HN–N and HN–N/Hα–Cα CCR rates are on the order of 0.1–0.3s−1 and must be considered. Tabulated correction factors may be used for their extraction. If larger systems are studied, in most cases the errors cannot be neglected anymore. On the other hand, well-calibrated pulses can safely be assumed to be perfect.