Abstract: An important stage in macromolecular crystallography is that of phase extension and refinement when initial phase estimates are available from isomorphous replacement or anomalous scattering or other methods. For this purpose, an alternative method called the twin variables (TwiV) method has been proposed. The algorithm is based on alternately transferring the phase information between the twin variable sets. The phase extension and refinement is evaluated with the crystallographic symmetry test by deliberately sacrificing the space-group symmetry in the starting set, then using its re-appearance as a criterion for correctness. Here we present a software program (CrysTwiV) that runs on the web (freely available at: http://btweb.aua.gr/crystwiv/) implementing the above-mentioned method.

Abstract: The crystal structures of 4-chlorophenoxyacetic acid (4CPA) included in beta-cyclodextrin (beta-CD) and heptakis(2,3,6-tri-O-methyl)-beta-cyclodextrin (TMbetaCD) have been studied by X-ray diffraction. The 4CPA/beta-CD complex crystallizes as a head-to-head dimer in the space group C2 in the Tetrad packing mode. The packing modes of some beta-CD dimeric complexes, having unique stackings, are also discussed. The 4CPA/TMbetaCD inclusion complex crystallizes in the space group P2(1) and its asymmetric unit contains two crystallographically independent complexes, complex A and complex B, exhibiting different conformations. The host molecule of complex A is significantly distorted, as a glucosidic residue rotated about the O4'-C1 and C4-O4 bonds forms an aperture where the guest molecule is accommodated. The phenyl moiety of the guest molecule of complex B is nearly perpendicular to the mean plane of the O4n atoms. The conformations of the guest molecules of the two complexes are similar. The crystal packing consists of antiparallel columns as in the majority of the TMbetaCD complexes published so far.

Abstract: The crystal structures of the complexes of heptakis(2,3,6-tri-O-methyl)-beta-cyclodextrin with indole-3-butyric acid and with 2,4-dichlorophenoxyacetic acid were studied by X-ray diffraction. The complexes crystallize in the monoclinic P2(1) space group. The host molecules are elliptically puckered and stacked along the a crystal axis, in a head-to-tail fashion, forming columns. One primary methoxy group of the host molecule of the complex with indole-3-butyric acid has the unusual trans-gauche conformation for permethylated CDs. All the secondary O-3-CH(3) methoxy groups, some secondary O-2-CH(3) and some primary methoxy groups pointing inwards the cavity enclose the indole or the 2,4-dichlorophenoxy moieties of the guest molecules inside the cavity, while the chains of the guests protrude between two adjacent host molecules of the columns. The mean planes of the indole and 2,4-dichlorophenoxy moieties of the guests are nearly perpendicular to the mean planes of the elliptical heptagons defined by the O-4n atoms of the hosts. The carboxyl group of the guests form hydrogen bonds with oxygen atoms of the host molecules or with the water molecules found in the space between the complexes of the same column.

Abstract: One of the most important problems in the application of direct methods for large structures is to establish reliable consistency criteria for the correctness of a phasing trial. The introduction in the twin variables method [Bethanis, Tzamalis, Hountas, Mishnev & Tsoucaris (2000). Acta Cryst. A56, 105-111] of a new criterion based on the crystallographic symmetry consists of testing the phase extension and refinement algorithm by deliberately sacrificing the space-group-symmetry information in the auxiliary variable set then using its gradual re-appearance as a criterion for correctness. In the present paper, the crystallographic symmetry test has been used in the implementation of the twin algorithm in two different ways: (i) as an overall test throughout the iterations that is likely to reflect the correctness of the phasing procedure for each one of the extension trials in a macromolecular phasing environment; (ii) as a convenient criterion to determine the optimum cycle for freeing the initial phases used by the algorithm for the phase-extension procedure.

Abstract: It is shown that the use of the Schrödinger equation may lead to the ab initio determination of the positions of the nuclei in a crystal, given a limited number of diffracted-beam intensities. In particular, it is shown that an extremely simplified Schrödinger equation in physical momentum space provides a sufficiently sound theoretical basis to develop an algorithm using diffraction data alone. This algorithm has been tested with a known 41 atom crystalline structure (not including hydrogen atoms) in space group P1. The extracted information is sufficient to determine the positions of all atoms. In addition, theoretical developments relevant to the connection between momentum space in quantum mechanics and diffraction theory have been formulated.

Abstract: A new development of the TWIN algorithm is described and used for phase extension/refinement in supramolecular complexes. A small number of phased reflections at low resolution are sufficient for the quasi-automated determination of all atomic coordinates, including disordered atoms.

Abstract: Phase extension from lower to higher resolution by using an upgraded TWIN variables algorithm [Hountas & Tsoucaris (1995). Acta Cryst. A51, 754-763] in protein molecules with close to 1,000 non-H atoms is presented. Three points of this procedure are of particular interest. (i) The use of a set of auxiliary variables providing a satisfactory fit for many kinds of constraints: the new algorithm works efficiently despite the extreme 'dilution' of very limited initial phase information into a much larger set of auxiliary variables. (ii) The extension of this auxiliary variables set beyond the resolution of the observed data, which enhances the phase extension in a so-called 'super-resolution' sphere. (iii) The use of the crystallographic symmetry as a new figure of merit and as a reliable test for the correctness of the phase-extension process allows an efficient screening.