Structure-independent analysis of the breadth of the positional distribution of disordered groups in macromolecules from order parameters for long, variable-length vectors using nmr paramagnetic relaxation enhancement

Quantitative information regarding structurally disordered groups is crucial for a complete understanding of the relationship between structure, dynamics, and function in biological macromolecules. Experimental analysis, however, of the positional distribution of disordered groups in the macromolecular frame is extremely difficult. While NMR order parameters, S ², for fixed-length bond vectors such as N-H and C-H are commonly used for investigations of conformational dynamics of macromolecules, these order parameters provide only angular information about internal motions and are totally insensitive to translational motions. Although analysis of S ² for bond vectors permits identification of disordered groups in macromolecules, this type of order parameter cannot provide any information about the distribution radii of disordered groups. Here we describe an NMR approach to directly determine the distribution radius of a disordered group independent of any structural knowledge. This approach makes use of order parameters for long, variable-length vectors (including proton-paramagnetic center and proton-proton vectors) between a disordered group and a rigid portion of the macromolecule. We demonstrate the application of this formalism to paramagnetic relaxation enhancement vectors. In addition, the potential utility of the same formalism to ¹H-¹H cross-relaxation rates is considered as an alternative approach for analyzing the breadth of the positional distribution of disordered groups.