Algebraic direct methods for few-atoms structure models

As a basis for direct-methods phasing at very low resolution for macromolecular crystal structures, normalized structure-factor algebra is presented for few-atoms structure models with N = 1, 2, 3, em leader equal atoms or polyatomic globs per unit cell. Main results include: [see text]. Triplet discriminant Delta(hk) and triplet weight W(hk) parameters, a approximately 4.0 and b approximately 3.0, respectively, were determined empirically in numerical error analyses. Tests with phases calculated for few-atoms 'super-glob' models of the protein apo-D-glyceraldehyde-3-phosphate dehydrogenase (approximately 10000 non-H atoms) showed that low-resolution phases from the new few-atoms tangent formula were much better than conventional tangent formula phases for N = 2 and 3; phases from the two formulae were essentially the same for N > or = 4.