Mathematical aspects of molecular replacement. IV. Measure-theoretic decompositions of motion spaces

2017 ◽  
Vol 73 (5) ◽  
pp. 387-402 ◽  
Author(s):  
Gregory S. Chirikjian ◽  
Sajdeh Sajjadi ◽  
Bernard Shiffman ◽  
Steven M. Zucker
Keyword(s):  
General Theory ◽  
Rigid Body ◽  
Three Dimensional ◽  
Molecular Replacement ◽  
Translation Group ◽  
Common Occurrence ◽  
Space Group ◽  
Space Groups ◽  
Relevant Case ◽  
The Subject

In molecular-replacement (MR) searches, spaces of motions are explored for determining the appropriate placement of rigid-body models of macromolecules in crystallographic asymmetric units. The properties of the space of non-redundant motions in an MR search, called a `motion space', are the subject of this series of papers. This paper, the fourth in the series, builds on the others by showing that when the space group of a macromolecular crystal can be decomposed into a product of two space subgroups that share only the lattice translation group, the decomposition of the group provides different decompositions of the corresponding motion spaces. Then an MR search can be implemented by trading off between regions of the translation and rotation subspaces. The results of this paper constrain the allowable shapes and sizes of these subspaces. Special choices result when the space group is decomposed into a product of a normal Bieberbach subgroup and a symmorphic subgroup (which is a common occurrence in the space groups encountered in protein crystallography). Examples of Sohncke space groups are used to illustrate the general theory in the three-dimensional case (which is the relevant case for MR), but the general theory in this paper applies to any dimension.

Intergrowth polytypoids as modulated structures: a superspace description of the Sr n (Nb,Ti) n O3n + 2 compound series

2001 ◽  
Vol 57 (4) ◽  
pp. 471-484 ◽  
Author(s):  
L. Elcoro ◽  
J. M. Perez-Mato ◽  
R. L. Withers
Keyword(s):  
Dimensional Space ◽  
Three Dimensional ◽  
Separation Distance ◽  
Space Group ◽  
Space Groups ◽  
Cell Parameters ◽  
Average Structure ◽  
Layer Stacking ◽  
Compound Series ◽  
Three Dimensional Space

A new, unified superspace approach to the structural characterization of the perovskite-related Sr n (Nb,Ti) n O3n + 2 compound series, strontium niobium/titanium oxide, is presented. To a first approximation, the structure of any member of this compound series can be described in terms of the stacking of (110)-bounded perovskite slabs, the number of atomic layers in a single perovskite slab varying systematically with composition. The various composition-dependent layer-stacking sequences can be interpreted in terms of the structural modulation of a common underlying average structure. The average interlayer separation distance is directly related to the average structure periodicity along the layer stacking direction, while an inherent modulation thereof is produced by the presence of different types of layers (particularly vacant layers) along this stacking direction. The fundamental atomic modulation is therefore occupational and can be described by means of crenel (step-like) functions which define occupational atomic domains in the superspace, similarly to what occurs for quasicrystals. While in a standard crystallographic approach, one must describe each structure (in particular the space group and cell parameters) separately for each composition, the proposed superspace model is essentially common to the whole compound series. The superspace symmetry group is unique, while the primary modulation wavevector and the width of some occupation domains vary linearly with composition. For each rational composition, the corresponding conventional three-dimensional space group can be derived from the common superspace group. The resultant possible three-dimensional space groups are in agreement with all the symmetries reported for members of the series. The symmetry-breaking phase transitions with temperature observed in many compounds can be explained in terms of a change in superspace group, again in common for the whole compound series. Inclusion of the incommensurate phases, present in many compounds of the series, lifts the analysis into a five-dimensional superspace. The various four-dimensional superspace groups reported for this incommensurate phase at different compositions are shown to be predictable from a proposed five-dimensional superspace group apparently common to the whole compound series. A comparison with the scarce number of refined structures in this system and the homologous (Nb,Ca)6Ti6O20 compound demonstrates the suitability of the proposed formalism.

scholarly journals From space to superspace and back: Superspace Group Finder

2008 ◽  
Vol 41 (6) ◽  
pp. 1182-1186 ◽  
Author(s):  
Ivan Orlov ◽  
Lukas Palatinus ◽  
Gervais Chapuis
Keyword(s):  
Crystal Structure ◽  
Dimensional Space ◽  
Flexible Structures ◽  
Three Dimensional ◽  
Real Space ◽  
Space Group ◽  
Space Groups ◽  
Group A ◽  
Modular Composition ◽  
Three Dimensional Space

The symmetry of a commensurately modulated crystal structure can be described in two different ways: in terms of a conventional three-dimensional space group or using the superspace concept in (3 +d) dimensions. The three-dimensional space group is obtained as a real-space section of the (3 +d) superspace group. A complete network was constructed linking (3 + 1) superspace groups and the corresponding three-dimensional space groups derived from rational sections. A database has been established and is available at http://superspace.epfl.ch/finder/. It is particularly useful for finding common superspace groups for various series of modular (`composition-flexible') structures and phase transitions. The use of the database is illustrated with examples from various fields of crystal chemistry.

scholarly journals Isomers and polymorphs of (E,E)-1,4-bis(nitrophenyl)-2,3-diaza-1,3-butadienes

2006 ◽  
Vol 62 (4) ◽  
pp. 666-675 ◽  
Author(s):  
Christopher Glidewell ◽  
John N. Low ◽  
Janet M. S. Skakle ◽  
James L. Wardell
Keyword(s):  
Hydrogen Bond ◽  
Hydrogen Bonds ◽  
Three Dimensional ◽  
Stacking Interactions ◽  
Space Group ◽  
Stacking Interaction ◽  
Space Groups ◽  
Π Stacking ◽  
Π Stacking Interaction ◽  
Polymorphic Forms

The structures of five of the possible six isomers of (E,E)-1,4-bis(nitrophenyl)-2,3-diaza-1,3-butadiene are reported, including two polymorphs of one of the isomers. (E,E)-1,4-Bis(2-nitrophenyl)-2,3-diaza-1,3-butadiene, C14H10N4O4 (I), crystallizes in two polymorphic forms (Ia) and (Ib) in which the molecules lie across centres of inversion in space groups P21/n and P21/c, respectively: the molecules in (Ia) and (Ib) are linked into chains by aromatic π...π stacking interactions and C—H...π(arene) hydrogen bonds, respectively. Molecules of (E,E)-1-(2-nitrophenyl)-4-(3-nitrophenyl)-2,3-diaza-1,3-butadiene (II) are linked into sheets by two independent C—H...O hydrogen bonds. The molecules of (E,E)-1,4-bis(3-nitrophenyl)-2,3-diaza-1,3-butadiene (III) lie across inversion centres in the space group P21/n, and a combination of a C—H...O hydrogen bond and a π...π stacking interaction links the molecules into sheets. A total of four independent C—H...O hydrogen bonds link the molecules of (E,E)-1-(3-nitrophenyl)-4-(4-nitrophenyl)-2,3-diaza-1,3-butadiene (IV) into sheets. In (E,E)-1,4-bis(4-nitrophenyl)-2,3-diaza-1,3-butadiene (V) the molecules, which lie across centres of inversion in the space group P21/n, are linked by just two independent C—H...O hydrogen bonds into a three-dimensional framework.

Mathematical aspects of molecular replacement. III. Properties of space groups preferred by proteins in the Protein Data Bank

2015 ◽  
Vol 71 (2) ◽  
pp. 186-194 ◽  
Author(s):  
G. Chirikjian ◽  
S. Sajjadi ◽  
D. Toptygin ◽  
Y. Yan
Keyword(s):  
Rigid Body ◽  
Protein Data Bank ◽  
Data Bank ◽  
Continuous Group ◽  
Molecular Replacement ◽  
Macromolecular Crystallography ◽  
Coset Space ◽  
Space Groups ◽  
The Third ◽  
Rigid Body Motions

The main goal of molecular replacement in macromolecular crystallography is to find the appropriate rigid-body transformations that situate identical copies of model proteins in the crystallographic unit cell. The search for such transformations can be thought of as taking place in the coset space Γ\Gwhere Γ is the Sohncke group of the macromolecular crystal andGis the continuous group of rigid-body motions in Euclidean space. This paper, the third in a series, is concerned with viewing nonsymmorphic Γ in a new way. These space groups, rather than symmorphic ones, are the most common ones for protein crystals. Moreover, their properties impact the structure of the space Γ\G. In particular, nonsymmorphic space groups contain both Bieberbach subgroups and symmorphic subgroups. A number of new theorems focusing on these subgroups are proven, and it is shown that these concepts are related to the preferences that proteins have for crystallizing in different space groups, as observed in the Protein Data Bank.

The three-dimensional structures of two isoforms of nucleoside diphosphate kinase from bovine retina

1999 ◽  
Vol 55 (6) ◽  
pp. 1127-1135 ◽  
Author(s):  
Jane E. Ladner ◽  
Najmoutin G. Abdulaev ◽  
Dmitri L. Kakuev ◽  
Mária Tordová ◽  
Kevin D. Ridge ◽  
...  
Keyword(s):  
Active Site ◽  
Three Dimensional ◽  
Nucleoside Diphosphate Kinase ◽  
Asymmetric Unit ◽  
Ndp Kinase ◽  
Space Group ◽  
Space Groups ◽  
Bovine Retina ◽  
Biological Sources ◽  
Polypeptide Chains

The crystal structures of two isoforms of nucleoside diphosphate kinase from bovine retina overexpressed in Escherischia coli have been determined to 2.4 Å resolution. Both the isoforms, NBR-A and NBR-B, are hexameric and the fold of the monomer is in agreement with NDP-kinase structures from other biological sources. Although the polypeptide chains of the two isoforms differ by only two residues, they crystallize in different space groups. NBR-A crystallizes in space group P212121 with an entire hexamer in the asymmetric unit, while NBR-B crystallizes in space group P43212 with a trimer in the asymmetric unit. The highly conserved nucleotide-binding site observed in other nucleoside diphosphate kinase structures is also observed here. Both NBR-A and NBR-B were crystallized in the presence of cGMP. The nucleotide is bound with the base in the anti conformation. The NBR-A active site contained both cGMP and GDP each bound at half occupancy. Presumably, NBR-A had retained GDP (or GTP) from the purification process. The NBR-B active site contained only cGMP.

Extension of three-dimensional space-group generators to the (3+1) case

1999 ◽  
Vol 32 (3) ◽  
pp. 452-455
Author(s):  
Kazimierz Stróż
Keyword(s):  
Dimensional Space ◽  
Three Dimensional ◽  
Space Group ◽  
Space Groups ◽  
Symmetry Operations ◽  
Three Dimensional Space

A method of building up the generators of 775 (3+1)-dimensional superspace groups is proposed. The generators are based on the conventional space-group generators selected by Wondratschek and applied in theInternational Tables for Crystallography(1995, Vol. A). By the method, the generation of (3+1) space groups is found to be easier, the description of symmetry operations is closer to that used for the conventional space groups, and ambiguities in the (3+1) group notation are avoided.

SPGGEN: a computer program for retrieval of space-group information in several settings and generator-containing space-group symbols

2016 ◽  
Vol 49 (4) ◽  
pp. 1370-1376 ◽  
Author(s):  
Uri Shmueli
Keyword(s):  
Computer Program ◽  
Dimensional Space ◽  
Three Dimensional ◽  
General Setting ◽  
Detailed Comparison ◽  
Space Group ◽  
Space Groups ◽  
Group Number ◽  
Group Information ◽  
Three Dimensional Space

A brief outline of the algorithm for the derivation of a space group is followed by a detailed description of the explicit space-group symbols here employed. These space-group symbols are unique insofar as they contain explicitly the generators of the space group dealt with. Next, the implementation of the above in a computer program,SPGGEN, is briefly discussed and the options presented by the program are outlined. Briefly, these options are (i) conventional derivation of the space group from an explicit symbol, including a user-defined one; (ii) such derivation from the conventional space-group number only; (iii) introduction of a general setting into the derivation; (iv) introduction of a Cartesian setting into the derivation; and (v) treatment of some non-conventional settings of orthorhombic space groups. This is followed by a detailed comparison withInternational Tables for Crystallography, Vol. A, and by examples of the output ofSPGGEN. A complete tabulation of the explicit three-dimensional space-group symbols is readily accessed.

scholarly journals TakeTwo: an indexing algorithm suited to still images with known crystal parameters

2016 ◽  
Vol 72 (8) ◽  
pp. 956-965 ◽  
Author(s):  
Helen Mary Ginn ◽  
Philip Roedig ◽  
Anling Kuo ◽  
Gwyndaf Evans ◽  
Nicholas K. Sauter ◽  
...  
Keyword(s):  
Three Dimensional ◽  
Image Data ◽  
Group Efficacy ◽  
Still Images ◽  
Still Image ◽  
Space Group ◽  
X Ray ◽  
Space Groups ◽  
Ewald Sphere ◽  
Unit Cell Dimensions

The indexing methods currently used for serial femtosecond crystallography were originally developed for experiments in which crystals are rotated in the X-ray beam, providing significant three-dimensional information. On the other hand, shots from both X-ray free-electron lasers and serial synchrotron crystallography experiments are still images, in which the few three-dimensional data available arise only from the curvature of the Ewald sphere. Traditional synchrotron crystallography methods are thus less well suited to still image data processing. Here, a new indexing method is presented with the aim of maximizing information use from a still image given the known unit-cell dimensions and space group. Efficacy for cubic, hexagonal and orthorhombic space groups is shown, and for those showing some evidence of diffraction the indexing rate ranged from 90% (hexagonal space group) to 151% (cubic space group). Here, the indexing rate refers to the number of lattices indexed per image.

COMPANG: automated comparison of orientations

2002 ◽  
Vol 35 (5) ◽  
pp. 644-647 ◽  
Author(s):  
Ludmila Urzhumtseva ◽  
Alexandre Urzhumtsev
Keyword(s):  
Structure Analysis ◽  
Three Dimensional ◽  
Molecular Replacement ◽  
Interactive Program ◽  
Space Group ◽  
Correct Orientation ◽  
Crystallographic Symmetry ◽  
Dimensional Object ◽  
Symmetry Operations ◽  
Difficult Cases

A search for similar orientations of a three-dimensional object is a usual task in structure analysis. An example is a comparison of peaks of rotation functions in molecular replacement. An automated comparison of orientations defined as a list or several lists of corresponding Eulerian angles can be performed using the interactive programCOMPANG. When calculating the closeness of orientations, this program allows one to take into account the symmetry operations of the space group as well as non-crystallographic symmetry. The similarity of orientations can be considered at a given accuracy, thus allowing a user to identify the groups of close orientations,i.e.their clusters. The size of such clusters can be used as a criterion to choose the correct orientation in difficult cases of molecular replacement.

scholarly journals Multiplicity-weighted Euler's formula for symmetrically arranged space-filling polyhedra

2020 ◽  
Vol 76 (5) ◽  
pp. 580-583
Author(s):  
Zbigniew Dauter ◽  
Mariusz Jaskolski
Keyword(s):  
Three Dimensional ◽  
Group Symmetry ◽  
General Character ◽  
Two Dimensional ◽  
Asymmetric Unit ◽  
Space Filling ◽  
Space Group Symmetry ◽  
Space Group ◽  
Space Groups ◽  
Modified Formula

The famous Euler's rule for three-dimensional polyhedra, F − E + V = 2 (F, E and V are the numbers of faces, edges and vertices, respectively), when extended to many tested cases of space-filling polyhedra such as the asymmetric unit (ASU), takes the form Fn − En + Vn = 1, where Fn, En and Vn enumerate the corresponding elements, normalized by their multiplicity, i.e. by the number of times they are repeated by the space-group symmetry. This modified formula holds for the ASUs of all 230 space groups and 17 two-dimensional planar groups as specified in the International Tables for Crystallography, and for a number of tested Dirichlet domains, suggesting that it may have a general character. The modification of the formula stems from the fact that in a symmetrical space-filling arrangement the polyhedra (such as the ASU) have incomplete bounding elements (faces, edges, vertices), since they are shared (in various degrees) with the space-filling neighbors.

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