Relaxation of the resolution requirements for direct-methods phasing.

Shake-and-bake phasing methods have permitted the ab initio solution of crystal structures containing more than 1000 independent non-H light atoms (C, N, O). The success of these procedures is critically dependent upon having diffraction data measured to at least 1.2 Å resolution.

Recent advances in direct phasing methods for heavy-atom substructure determination.

Macromolecular crystal structure determination has typically been a two-step process. When diffraction data from multiple chemically isomorphous or anomalously scattering crystals are available, the positions of heavy atoms from amplitude differences arising from native-derivative crystal pairs or an anomalously scattering crystal are first located and phasing of the whole protein structure is then completed using the heavy-atom substructure as a bootstrap. Shake-and-Bake, a direct-methods-based dual-space refinement procedure, provides heavy-atom substructure solutions by finding the constrained global minimum of a probabilistically defined minimal function.

Optimizing statistical Shake-and-Bake for Se-atom substructure determination.

A novel statistical approach to the phase problem in X-ray crystallography was introduced in a recent paper [Xu & Hauptman (2004), Acta Cryst. A60, 153-157]. In this approach, a new minimal function based on the statistical distribution of structure-invariant values serves as the foundation of an optimization procedure called statistical Shake-and-Bake.

Statistical approach to the phase problem.

The minimal function and its minimal principle employed in the traditional Shake-and-Bake algorithm rely on the probabilistic estimates of the cosines of the structure invariants. In this paper, a novel statistical approach to the phase problem, which utilizes statistical properties of the structure invariants, is proposed. The statistical maximal function and its maximal principle are formulated, and the corresponding statistical Shake-and-Bake algorithm and its associated statistical parameter-shift procedure are proposed and tested.

The phase problem in neutron crystallography.

The straightforward solution of the crystal structure of cyclosporin (C(62)H(111)N(11)O(12).H(2)O) by a modified Shake-and-Bake procedure, using experimental neutron diffraction data alone, shows that the positivity of the density function is not a necessary prerequisite for solving the phase problem. The initial applications suggest the intriguing possibility that positivity may actually be a hindrance.

On integrating the techniques of direct methods and SIRAS: the probabilistic theory of doublets and its applications.

The mathematical formalism of direct methods is here applied to the SIRAS (single-isomorphous replacement combined with anomalous scattering) case. Specifically, the joint probability distribution of three structure factors, which plays the central role in the probabilistic theory of the two-phase structure invariants (doublets), is derived. This distribution leads directly to the conditional probability distribution of the two-phase structure invariants, given the values of selected sets of magnitudes.

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.

Sine-enhanced Shake-and-Bake: the theoretical basis and applications to Se-atom substructures.

Shake-and-Bake is a dual-space direct-methods procedure for crystal structure determination capable of providing ab initio solutions for structures containing as many as 1200 independent non-H atoms, as well as for heavy-atom substructures containing as many as 160 Se atoms in the asymmetric unit. In traditional Shake-and-Bake, phase refinement in reciprocal space utilizes the technique of parameter shift to reduce the value of a minimal function that considers only the mean-square differences between the current values of the cosine structure invariants and their expected values. A new type of minimal function, termed the sine-enhanced minimal function, considers both cosine and sine values of the structure invariants.