Exact solutions for chemical bond orientations from residual dipolar couplings

New methods for determining chemical structures from residual dipolar couplings are presented. The fundamental dipolar coupling equation is converted to an elliptical equation in the principal alignment frame. This elliptical equation is then combined with other angular or dipolar coupling constraints to form simple polynomial equations that define discrete solutions for the unit vector(s). The methods are illustrated with residual dipolar coupling data on ubiquitin taken in a single anisotropic medium. The protein backbone is divided into its rigid groups (namely, its peptide planes and Cα frames), which may be solved for independently. A simple procedure for recombining these independent solutions results in backbone dihedral angles φ and ψ that resemble those of the known native structure. Subsequent refinement of these φ-ψ angles by the ROSETTA program produces a structure of ubiquitin that agrees with the known native structure to 1.1 Å Cα rmsd.