*Acta Cryst.*, 2004, D60:1057-1067. [preprint] [paper]

Molecular replacement (MR) often plays a prominent role in determining initial phase angles for structure determination by X-ray crystallography. In this paper, an efficient quaternion-based algorithm is presented for analyzing peaks from a cross-rotation function to identify model orientations consistent with non-crystallographic symmetry (NCS), and to generate NCS-consistent orientations missing from the list of cross-rotation peaks. Our algorithm, CRANS, analyzes the rotation differences between each pair of cross-rotation peaks to identify finite subgroups of NCS. Sets of rotation differences satisfying the subgroup axioms correspond to orientations compatible with the correct NCS. The CRANS algorithm was first tested using cross-rotation peaks computed from structure factor data for three test systems, and then used to assist in the de novo structure determination of dihydrofolate reductase-thymidylate synthase (DHFR-TS) from Cryptosporidium hominis. In every case, the CRANS algorithm runs in seconds to identify orientations consistent with the observed NCS and to generate missing orientations not present in the cross-rotation peak list. The CRANS algorithm has application in every molecular replacement phasing effort with NCS.