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.

Die Röntgenstrukturanalyse von (CO)9Co3CΟΒΒr2Ν(C5)3. X-ray Structure Analysis of (CO)9Co3COBBr2N(C2H5)3

1976 ◽  
Vol 31 (3) ◽  
pp. 342-344 ◽  
Author(s):  
Volker Bätzel
Keyword(s):  
Least Squares ◽  
Structure Analysis ◽  
Reliability Index ◽  
Three Dimensional ◽  
Diagonal Matrix ◽  
Crystal Data ◽  
Block Diagonal Matrix ◽  
Fourier Methods ◽  
Space Group ◽  
X Ray

Using three dimensional X-ray data collected on a four circle diffractometer, the structure of (CO)9Co3COBBr2N(C2H5)3 was solved by Patterson and Fourier methods. Least squares refinement with a block-diagonal matrix leads to a reliability index of R = 10.7%. Crystal data: α = 13.277(6) Å, b = 10.17(1) Å, c = 9.22(2) Å; α = 91.12(6)°, β = 87.61(4)°, γ = 98.79(2)°; space group P1̅; Z = 2; V = 1229,7 Å3; Dx = 1.97 gcm-3.

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.

Mathematical aspects of molecular replacement. V. Isolating feasible regions in motion spaces

2020 ◽  
Vol 76 (2) ◽  
pp. 145-162
Author(s):  
Bernard Shiffman ◽  
Shengnan Lyu ◽  
Gregory S. Chirikjian
Keyword(s):  
Configuration Space ◽  
A Priori ◽  
Rigid Bodies ◽  
Molecular Replacement ◽  
Space Group ◽  
Fundamental Domains ◽  
Crystallographic Symmetry ◽  
Symmetry Constraints ◽  
Translation Space ◽  
The Right

This paper mathematically characterizes the tiny feasible regions within the vast 6D rotation–translation space in a full molecular replacement (MR) search. The capability to a priori isolate such regions is potentially important for enhancing robustness and efficiency in computational phasing in macromolecular crystallography (MX). The previous four papers in this series have concentrated on the properties of the full configuration space of rigid bodies that move relative to each other with crystallographic symmetry constraints. In particular, it was shown that the configuration space of interest in this problem is the right-coset space Γ\G, where Γ is the space group of the chiral macromolecular crystal and G is the group of rigid-body motions, and that fundamental domains F Γ\G can be realized in many ways that have interesting algebraic and geometric properties. The cost function in MR methods can be viewed as a function on these fundamental domains. This, the fifth and final paper in this series, articulates the constraints that bodies packed with crystallographic symmetry must obey. It is shown that these constraints define a thin feasible set inside a motion space and that they fall into two categories: (i) the bodies must not interpenetrate, thereby excluding so-called `collision zones' from consideration in MR searches; (ii) the bodies must be in contact with a sufficient number of neighbors so as to form a rigid network leading to a physically realizable crystal. In this paper, these constraints are applied using ellipsoidal proxies for proteins to bound the feasible regions. It is shown that the volume of these feasible regions is small relative to the total volume of the motion space, which justifies the use of ellipsoids as proxies for complex proteins in MR searches, and this is demonstrated with P1 (the simplest space group) and with P212121 (the most common space group in MX).

scholarly journals Coping with translational non-crystallographic symmetry in molecular replacement

2014 ◽  
Vol 70 (a1) ◽  
pp. C319-C319
Author(s):  
Randy Read ◽  
Paul Adams ◽  
Airlie McCoy
Keyword(s):  
Diffraction Pattern ◽  
Likelihood Function ◽  
Molecular Replacement ◽  
Asymmetric Unit ◽  
Space Group ◽  
New Targets ◽  
Likelihood Functions ◽  
Crystallographic Symmetry ◽  
Modified Likelihood ◽  
Statistical Effects

In translational noncrystallographic symmetry (tNCS), two or more copies of a component are present in a similar orientation in the asymmetric unit of the crystal. This causes systematic modulations of the intensities in the diffraction pattern, leading to problems with methods that assume, either implicitly or explicitly, that the distribution of intensities is a function only of resolution. To characterize the statistical effects of tNCS accurately, it is necessary to determine the translation relating the copies, any small rotational differences in their orientations, and the size of random coordinate differences caused by conformational differences. An algorithm has been developed to estimate these parameters and refine their values against a likelihood function. By accounting for the statistical effects of tNCS, it is possible to unmask the competing statistical effects of twinning and tNCS and to more robustly assess the crystal for the presence of twinning. Modified likelihood functions that account for the statistical effects of tNCS have been developed for use in molecular replacement and implemented in Phaser. With the use of these new targets, it is now possible to solve structures that eluded earlier versions of the program. Pseudosymmetry and space group ambiguities often accompany tNCS, but the new version of Phaser is less likely to fall into the traps that these set.

Space-group and origin ambiguity in macromolecular structures with pseudo-symmetry and its treatment with the programZanuda

2014 ◽  
Vol 70 (9) ◽  
pp. 2430-2443 ◽  
Author(s):  
Andrey A. Lebedev ◽  
Michail N. Isupov
Keyword(s):  
Unit Cell ◽  
Unit Cell Parameters ◽  
Incorrect Choice ◽  
Space Group ◽  
Cell Parameters ◽  
Crystallographic Symmetry ◽  
Symmetry Operations

The presence of pseudo-symmetry in a macromolecular crystal and its interplay with twinning may lead to an incorrect space-group (SG) assignment. Moreover, if the pseudo-symmetry is very close to an exact crystallographic symmetry, the structure can be solved and partially refined in the wrong SG. Typically, in such incorrectly determined structures all or some of the pseudo-symmetry operations are, in effect, taken for crystallographic symmetry operations andvice versa. A mistake only becomes apparent when theRfreeceases to decrease below 0.39 and further model rebuilding and refinement cannot improve the refinement statistics. If pseudo-symmetry includes pseudo-translation, the uncertainty in SG assignment may be associated with an incorrect choice of origin, as demonstrated by the series of examples provided here. The programZanudapresented in this article was developed for the automation of SG validation.Zanudaruns a series of refinements in SGs compatible with the observed unit-cell parameters and chooses the model with the highest symmetry SG from a subset of models that have the best refinement statistics.

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.

SIR2011: a new package for crystal structure determination and refinement

2012 ◽  
Vol 45 (2) ◽  
pp. 357-361 ◽  
Author(s):  
Maria Cristina Burla ◽  
Rocco Caliandro ◽  
Mercedes Camalli ◽  
Benedetta Carrozzini ◽  
Giovanni Luca Cascarano ◽  
...  
Keyword(s):  
Electron Diffraction ◽  
Diffraction Data ◽  
Protein Structures ◽  
Three Dimensional ◽  
Molecular Geometry ◽  
Electron Diffraction Data ◽  
Medium Size ◽  
Molecular Replacement ◽  
Density Maps ◽  
Difficult Cases

SIR2011, the successor ofSIR2004, is the latest program of theSIRsuite. It can solveab initiocrystal structures of small- and medium-size molecules, as well as protein structures, using X-ray or electron diffraction data. With respect to the predecessor the program has several new abilities:e.g.a new phasing method (VLD) has been implemented, it is able to exploit prior knowledge of the molecular geometryviasimulated annealing techniques, it can use molecular replacement methods for solving proteins, it includes new tools like free lunch and new approaches for electron diffraction data, and it visualizes three-dimensional electron density maps. The graphical interface has been further improved and allows the straightforward use of the program even in difficult cases.

scholarly journals Different teaching way to space group and symmetry operation: polyhedral vs ball

2014 ◽  
Vol 70 (a1) ◽  
pp. C1389-C1389
Author(s):  
Jian Zhou ◽  
Hejing Wang
Keyword(s):  
Teaching Practice ◽  
Symmetry Plane ◽  
Symmetry Operation ◽  
The Other ◽  
Space Group ◽  
Fundamental Knowledge ◽  
Crystallographic Symmetry ◽  
Position Shift ◽  
Symmetry Operations

To introduce the regulations of space group combining with a symmetry operation, put an orientation ball at a position shift away from the lattice tops is a good way [1]. However, based on the fundamental knowledge of "lattice", it often occurs that the tops of a lattice "should be" the positions of "atom balls" thought by most beginnings in teaching practice. This "thought" leads them never deduce out those regulations in symmetry operations and often misleads a wrong conclusion. As a beginning one wishes watching movies and pictures instead of mathematical deduction or vector calculation. It easily arises that a lattice has eight tops with atom balls. This "idea" lets the orientation balls shifting away from the lattice tops become difficult to understand. Nevertheless, the balls with a sign of "comma" in the middle are also difficult to understand that they can stand for a certain orientation because ball is circle. "Tops" and "directions" are two troubles in learning crystallographic symmetry and symmetry operations for those beginnings. How to guide them to overcome the two fences is an important step that will lead those beginnings to a never understanding status, on one hand, or let them understand throughout all regulations of space group(s) combining with a symmetry operation on the other. From teaching practice, a polyhedral at lattice tops could overcome both difficulties at position and in orientation. First, a polyhedral is always in orientation, even it is a cubic. This is easily understood. Secondly the centre of a polyhedral could easily meet with the tops of a lattice; it lets students easily understand "a lattice has eight tops occupied – a natural thought by beginnings". This way let them easily understand and deduce all regulations in crystallographic symmetry operations, such as a body-centred lattice combining with a symmetry plane (m) produces n symmetry operation at 1/4t, etc. see figures below.

scholarly journals Color-Coding Point Group & Space Group Diagrams and other Mathematical Journeys

2014 ◽  
Vol 70 (a1) ◽  
pp. C1275-C1275
Author(s):  
Maureen Julian
Keyword(s):  
General Position ◽  
Point Group ◽  
Three Dimensional ◽  
Irreducible Representations ◽  
Color Coding ◽  
Space Group ◽  
Point Groups ◽  
Symmetry Operations

Color clarifies diagrams point group and space group diagrams. For example, consider the general position diagrams and the symbol diagrams. Symmetry operations can be represented by matrices whose determinant is either plus one or minus one. In the former case there is no change of handedness and in the latter case there is a change of handedness. The general position diagrams emphasis this information by color-coding. The symbol diagrams are a little more complicated and will be demonstrated. The second topic is a comparison of the thirty-two three-dimensional point groups with their corresponding 18 abstract mathematical groups. The corresponding trees will be explored. This discussion leads into the topic of irreducible representations.

Meso-structure and modeling of a new three-dimensional braided tubular material

2017 ◽  
Vol 37 (5) ◽  
pp. 310-320 ◽  
Author(s):  
Wensuo Ma ◽  
Zhenyu Ma ◽  
Bingjie Ren ◽  
Weifeng Fan
Keyword(s):  
Mathematical Model ◽  
Structural Properties ◽  
Fiber Volume Fraction ◽  
Volume Fraction ◽  
Three Dimensional ◽  
Geometrical Parameters ◽  
Fiber Volume ◽  
Space Group ◽  
Volume Unit ◽  
Symmetry Operations

A new three-dimensional braided tubular preform was introduced in this study. The new preform structure can be derived from the representative volume unit which was deduced by the symmetry operations of space group P4. The braiding process of the tubular preform has been discussed. A mathematical model was established to analyze the structural properties of the three-dimensional braided tubular preform. The interrelation of geometrical parameters is analyzed. The fiber volume fraction of the preform was predicted. The new tubular preform was obtained in laboratory to verify the feasibility of the braiding process.

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