Advanced implication theory. Symmetry tables.

The MIF (multiple implication function) group symmetry was assigned to all 230 space groups. Knowledge of MIF symmetry allows the calculation of an asymmetric unit. A more accurate procedure for calculating MIFs has been developed. Extensive tables of MIF symmetry and asymmetric units were computer generated. The development of implication theory for crystal structure determination seems to have reached completion.

A novel algorithm for calculation of Fourier and asymmetric units.

A new method is presented for determining asymmetric and Fourier units based on plane groups for all space groups. These units are specifically designed to improve the calculation of fast Fourier transforms compared with the units derived from asymmetric units in the International Tables for Crystallography, Vol. A. The algorithm can be easily implemented into existing crystallographic programs using a short computer code.

Performance of phased rotation, conformation and translation function: accurate protein model building with tripeptidic and tetrapeptidic fragments.

The automatic building of protein structures with tripeptidic and tetrapeptidic fragments was investigated. The oligopeptidic conformers were positioned in the electron-density map by a phased rotation, conformation and translation function and refined by a real-space refinement. The number of successfully located fragments lay within the interval 75-95% depending on the resolution and phase quality. The overlaps of partially located fragments were analyzed. The correctly positioned fragments were connected into chains. Chains formed in this way were extended directly into the electron density and a sequence was assigned. In the initial stage of the model building the number of located fragments was between 60% and 95%, but this number could be increased by several cycles of reciprocal-space refinement and automatic model rebuilding. A nearly complete structure can be obtained on the condition that the resolution is reasonable. Computer graphics will only be needed for a final check and small corrections.

Building of RNA and DNA double helices into electron density.

A method has been developed that automatically fits double-helical regions into the electron density of nucleic acid structures. Rigid fragments consisting of two Watson-Crick base pairs and three pairs of phosphate groups in the A-type or B-type conformation are positioned into the electron density by phased rotation and translation functions. The position and orientation of the localized double-helical fragments are determined by phased refinement. The method has been tested by building double-helical regions of nine RNA structures of variable crystallographic resolution and polynucleotide length and is available for free use.

Conformation families of protein fragments in multidimensional torsion-angle space.

Protein conformation families for automatic model building were determined for dipeptidic, tripeptidic, tetrapeptidic and pentapeptidic fragments. Mapping in n-dimensional conformational space (n = 2, 4 and 6), a conformation-generator method, a deletion-sorting process and a verification procedure were used to calculate the conformational preferences. Torsion angles were harvested from PDB structures with resolutions better than 1.5 A. Statistical weights were calculated for the conformation families.

Accurate automatic protein models.

A method for automatic building of protein structures has been developed. The method is based on the concept of flexible structure units. A structure unit is a fragment (group) of about ten atoms that is positioned in the electron density by a phased rotation and translation function. The positions, orientations and internal torsion angles of all structure units are refined by a phased flexible refinement. Individual structure units are connected into polyalanine chains. The sequence is aligned by combined marker and rotamer methods. The side chains are built either by the marker method and a full conformational search or by the rotamer method. Side chains are independent structure units. The structure unit represents a generalized atom and the group model can be refined by the least-squares method. The protein model is built with an accuracy of about 0.2 A at resolutions of 1.2-1.9 A. Partial results can be obtained at resolutions of between 2.0 and 2.3 A.

Automatic model building based on flexible fragment formalism. The case of high-resolution protein structures.

A concept of flexible fragments has been developed for automatic building of crystal structures. Six monopeptides were designed as search fragments in a phased rotation and translation function for protein building. Electron density in crystal and in molecular fragments is expanded in spherical harmonics and normalized spherical Bessel functions. A fast rotation function, which is calculated at each grid point of the asymmetric unit, is used to find the fragment orientation. Position, orientation and internal torsion angles are refined. An algorithm for chain building is simplified using generalized atoms and virtual bonds. The structure is built from molecular structure units rather than from individual atoms. A polyalanine model is built with a high accuracy at resolutions 1.2-2.1 A.

[N-(Carboxylatomethyl)aspartato(3-)](ethylenediamine)cobalt(III) trihydrate.

The mononuclear title complex, [Co(C(6)H(6)NO(6))(C(2)H(8)N(2))].3H(2)O, contains an octahedrally coordinated Co(III) atom. The N-(carboxymethyl)aspartate moiety is coordinated as a tetradentate ligand, providing an OONO-donor set and forming two trans five-membered chelate rings and one six-membered chelate ring. A seven-membered chelate ring is also formed, which consists of part of the six-membered chelate ring and part of one of the five-membered chelate rings. The crystal structure of the complex is stabilized by hydrogen bonds with three water molecules.

Methodology and applications of automatic electron-density map interpretation by six-dimensional rotational and translational search for molecular fragments.

The electron density for a search fragment and for the crystal is expanded in the space of spherical harmonics Bessel functions. The fast rotation function is evaluated for each grid point to test if the fragment can be orientated there. For the best scoring points, the six-dimensional coordinates of the fragment are refined by the second-derivative block-diagonal procedure. The method is able to locate fragments precisely over a wide range of resolutions for structure types from small organic molecules to proteins.