Seminvariant density decomposition and connectivity analysis and their application to very low resolution macromolecular phasing

A low-resolution Fourier synthesis is thought to show a molecule as a compact region of a high electron density. As a consequence, the number of such regions, chosen at a proper cut-off level, should be equal to the number of molecules in the unit cell. This hypothesis may be used as a basis for selection criteria in multisolution ab initio phasing procedures. However, when working with a small number of reflections, this hypothesis may break down. The suggested Fourier-synthesis decomposition explains some reasons for failure and provides a connectivity-based procedure for the determination of macromolecular position in the crystal unit cell and the phasing of several low-resolution reflections. The simplest decomposition consists in separating the reflections into two sets according to whether their phases do or do not depend on a permitted origin shift. It is shown that the partial Fourier syntheses corresponding to these subsets are simply a half-sum and a half-difference of the initial electron-density distribution with its shifted copy. Therefore, they display the true images overlapped with the shifted ones (or with shifted and additionally flipped copies for the latter synthesis). The paper generalizes the decomposition for the case of a finite subgroup of the group of permitted origin shifts and reveals the role of one-phase sem­invariants.