Topological loops with three-dimensional solvable left translation group

2010 ◽  
Vol 79 (1-2) ◽  
pp. 83-97 ◽  
Author(s):  
Ágota Figula
Keyword(s):  
Three Dimensional ◽  
Left Translation ◽  
Translation Group

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.

scholarly journals Isometries of Riemannian and sub-Riemannian structures on three-dimensional Lie groups

2017 ◽  
Vol 25 (2) ◽  
pp. 99-135
Author(s):  
Rory Biggs
Keyword(s):  
Lie Groups ◽  
Lie Group ◽  
Isometry Group ◽  
Three Dimensional ◽  
Isotropy Subgroup ◽  
Left Translation ◽  
Isometry Groups ◽  
Group Automorphism ◽  
Group Of Automorphisms ◽  
Simply Connected

Abstract We investigate the isometry groups of the left-invariant Riemannian and sub-Riemannian structures on simply connected three-dimensional Lie groups. More specifically, we determine the isometry group for each normalized structure and hence characterize for exactly which structures (and groups) the isotropy subgroup of the identity is contained in the group of automorphisms of the Lie group. It turns out (in both the Riemannian and sub-Riemannian cases) that for most structures any isometry is the composition of a left translation and a Lie group automorphism.

The orbit of Pluto over a long interval of time

1966 ◽  
Vol 25 ◽  
pp. 227-229 ◽  
Author(s):  
D. Brouwer
Keyword(s):  
Periodic Orbit ◽  
Numerical Integration ◽  
Restricted Problem ◽  
Three Dimensional ◽  
The Third ◽  
Circular Restricted Problem ◽  
Resonance Relation

The paper presents a summary of the results obtained by C. J. Cohen and E. C. Hubbard, who established by numerical integration that a resonance relation exists between the orbits of Neptune and Pluto. The problem may be explored further by approximating the motion of Pluto by that of a particle with negligible mass in the three-dimensional (circular) restricted problem. The mass of Pluto and the eccentricity of Neptune's orbit are ignored in this approximation. Significant features of the problem appear to be the presence of two critical arguments and the possibility that the orbit may be related to a periodic orbit of the third kind.

Conformation of Ribosomes from Escherichia Coli

1974 ◽  
Vol 32 ◽  
pp. 210-211
Author(s):  
M. Boublik ◽  
W. Hellmann ◽  
F. Jenkins
Keyword(s):  
Binding Sites ◽  
Ribosomal Proteins ◽  
Fluorescent Dyes ◽  
Protein Biosynthesis ◽  
Three Dimensional ◽  
Spatial Arrangement ◽  
Dimensional Structure ◽  
Complete Understanding ◽  
And Function

The present knowledge of the three-dimensional structure of ribosomes is far too limited to enable a complete understanding of the various roles which ribosomes play in protein biosynthesis. The spatial arrangement of proteins and ribonuclec acids in ribosomes can be analysed in many ways. Determination of binding sites for individual proteins on ribonuclec acid and locations of the mutual positions of proteins on the ribosome using labeling with fluorescent dyes, cross-linking reagents, neutron-diffraction or antibodies against ribosomal proteins seem to be most successful approaches. Structure and function of ribosomes can be correlated be depleting the complete ribosomes of some proteins to the functionally inactive core and by subsequent partial reconstitution in order to regain active ribosomal particles.

Ultrastructural Investigations of the Contractile Filaments of Amoebae

1974 ◽  
Vol 32 ◽  
pp. 188-189
Author(s):  
P.L. Moore
Keyword(s):  
Cytoplasmic Streaming ◽  
Three Dimensional ◽  
Freeze Fracture ◽  
Amoeboid Movement ◽  
Different Types ◽  
Chemical Conditions ◽  
Complete Interpretation

Previous freeze fracture results on the intact giant, amoeba Chaos carolinensis indicated the presence of a fibrillar arrangement of filaments within the cytoplasm. A complete interpretation of the three dimensional ultrastructure of these structures, and their possible role in amoeboid movement was not possible, since comparable results could not be obtained with conventional fixation of intact amoebae. Progress in interpreting the freeze fracture images of amoebae required a more thorough understanding of the different types of filaments present in amoebae, and of the ways in which they could be organized while remaining functional.The recent development of a calcium sensitive, demembranated, amoeboid model of Chaos carolinensis has made it possible to achieve a better understanding of such functional arrangements of amoeboid filaments. In these models the motility of demembranated cytoplasm can be controlled in vitro, and the chemical conditions necessary for contractility, and cytoplasmic streaming can be investigated. It is clear from these studies that “fibrils” exist in amoeboid models, and that they are capable of contracting along their length under conditions similar to those which cause contraction in vertebrate muscles.

Structural organization of escherichia coli ribosomes as determined by immuno electron microscopy

1978 ◽  
Vol 36 (2) ◽  
pp. 166-167 ◽  
Author(s):  
G. Stöffler ◽  
R.W. Bald ◽  
J. Dieckhoff ◽  
H. Eckhard ◽  
R. Lührmann ◽  
...  
Keyword(s):  
Electron Microscopy ◽  
Escherichia Coli ◽  
Ribosomal Proteins ◽  
Structural Organization ◽  
Three Dimensional ◽  
Ribosomal Subunit ◽  
Spatial Arrangement ◽  
Small Subunit ◽  
Antibody Binding ◽  
Immuno Electron Microscopy

A central step towards an understanding of the structure and function of the Escherichia coli ribosome, a large multicomponent assembly, is the elucidation of the spatial arrangement of its 54 proteins and its three rRNA molecules. The structural organization of ribosomal components has been investigated by a number of experimental approaches. Specific antibodies directed against each of the 54 ribosomal proteins of Escherichia coli have been performed to examine antibody-subunit complexes by electron microscopy. The position of the bound antibody, specific for a particular protein, can be determined; it indicates the location of the corresponding protein on the ribosomal surface.The three-dimensional distribution of each of the 21 small subunit proteins on the ribosomal surface has been determined by immuno electron microscopy: the 21 proteins have been found exposed with altogether 43 antibody binding sites. Each one of 12 proteins showed antibody binding at remote positions on the subunit surface, indicating highly extended conformations of the proteins concerned within the 30S ribosomal subunit; the remaining proteins are, however, not necessarily globular in shape (Fig. 1).

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