Equivalence between Euler angle conventions for the description of tensorial interactions in liquid NMR: application to different software programs

Long-range orientational restraints derived from alignment or rotational diffusion tensors have greatly contributed to the expansion of applications in biomolecular NMR. The orientation of the principal axis system of these tensors is usually described by the so-called Euler angles. However, no clear consensus has emerged concerning the convention of the associated orthogonal rotations. As a result, the different programs that derive or predict them have adopted different conventions, which make comparison between their results difficult. Moreover, the rotation schemes are seldom completely described. Here, we summarize the different conventions, determine which ones are adopted by commonly used software packages, and establish the formal equivalencies between the different calculated Euler angles.