A structure-factor least-squares refinement procedure for macromolecular structures using constrained and restrained parameters

A method of structure-factor least-squares refinement of constrained groups linked by distance restraints has been developed for the refinement of macromolecular structures. Each constrained group can have any number of variable dihedral rotation parameters within the group in addition to the rigid-body translational, rotational and thermal parameters. The matrix of normal equations may be either full or sparse and provision is made for solution by matrix inversion or the conjugate-gradient iterative method. This procedure has been successfully used for 3 Å data and should be applicable even for lower-resolution data and especially for cases with a poorer data-per-atom ratio. The structure of yeast phenylalanine tRNA has been refined with this procedure from a starting crystallographic R value of 42% to a final R value of 25% with isotropic 'group" thermal parameters and 22% with isotropic atomic thermal parameters for 8207 independent reflections at 2.7 Å resolution. The proper stereochemistry of bond distances, angles and van der Waals contacts for the restrained atoms was maintained within reasonable limits throughout the refinement. Although originally developed for nucleic acids, this procedure is directly applicable to the refinement of protein structures. In addition, a combination of applying distance restraints between groups and least-squares fitting of these groups to target coordinates has been used purely as an idealization process for imposing proper stereochemistry on an approximate model.