16-Fold Degeneracy of Peptide Plane Orientations from Residual Dipolar Couplings: Analytical Treatment and Implications for Protein Structure Determination

Abstract

Residual dipolar couplings (RDCs) measured for internally rigid molecular fragments provide important information about the relative orientations of these fragments. Dependent on the symmetry of the alignment tensor and the symmetry of the molecular fragment, however, there generally exists more than one solution for the fragment orientation that is consistent with the measured RDCs. Analytical solutions are presented that describe the complete set of orientations of internally rigid fragments that are consistent with multiple dipolar couplings measured in a single alignment medium that is rhombic. For the first time, it is shown that for a planar fragment, such as the peptide plane, there generally exist 16 different solutions with their analytical expressions presented explicitly. The presence of these solutions is shown to be highly relevant for standard structure determination protocols using RDCs to refine molecular structures. In particular, when using standard protein structure refinement with RDCs that were measured in a single alignment medium as constraints, it is found that often more than one of the peptide plane solutions is physically viable, i.e. despite being consistent with measured RDCs, the local backbone structure can be incorrect. Based on experimental and simulated examples it is rationalized why protein and nucleic acid structures that are refined against RDCs measured in a single medium can have lower resolution (precision) than one would expect based on the experimental accuracy of the RDCs. Conditions are discussed under which the correct solution can be identified.

Correlated Inter-Domain Motions in Adenylate Kinase

“…S14). [39] A second factor is the presence of structural constraints due to the covalent linkage, where steric clashes between domains restrict the available conformational space;[40], [41] AK e is a well-structured proteins where this effect is particularly important but for multi-domain proteins with flexible linker sequences RDCs measured in multiple alignment media will be required. Alternatively, RDCs could be complemented with structural restraints derived from smFRET or SAXS data.…”

High-resolution protein structure determination starting with a global fold calculated from exact solutions to the RDC equations

“…RDC-EXACT takes as input two RDCs per residue (e.g., assigned NH RDCs in two media or NH and CH RDCs in a single medium). In (Wang & Donald 2004b,a), the authors also showed that given a peptide plane, the orientation of the next peptide plane can have at most 16 possible orientations; a related theorem was shown by (Hus et al 2008). Given NH and CH RDCs in one medium, the associated NH and CH vectors can be solved from the RDC curve equations (see SM Section S2) and the protein backbone geometry (Wang & Donald 2004 a , Wang et al 2006).…”

Paramagnetic-Based NMR Restraints Lift Residual Dipolar Coupling Degeneracy in Multidomain Detergent-Solubilized Membrane Proteins

“…It was originally thought that the RDC equation resulted in eight-fold degeneracy for peptide plane orientations16, but recently, it was shown that the analytical solution of the RDC equation contains a 16-fold degeneracy 17. This degeneracy is reduced to four if one considers the regular patterns of the dipolar couplings (i.e.…”

“…In favorable cases, this RDC inherent degeneracy is resolved using two or more alignments media23. Nonetheless, local geometry can still be ill-defined17 and further confounded by the presence of conformational dynamics 24–27.…”