Projective Symmetry Research Articles

Glutamate affects the development of the thalamofugal visual projection of the chick

Examination of the thalamofugal visual projections of the chick following treatment of the hyperstriatal regions with glutamate was undertaken to investigate whether structural change underlies slowed learning known to follow such treatment. On the second day after hatching from eggs exposed to light, chicks were injected bilaterally with 5 μl of 100 mM glutamate or physiological saline. On post-hatching day 8, retrograde fluorescence labelling of the thalamofugal projections with True Blue and FluoroGold led to the labelling of thalamic cells which project to the visual hyperstriatum. The number of contralateral and ipsilateral projections relative to each dye injection site was determined. Glutamate prevented the development of the structural asymmetry which is present in control chicks. The resultant symmetry of the thalamofugal projections in glutamate-treated chicks may cause deficits in visual discrimination.

Some Remarks on Special Conformal and Special Projective Symmetries in General Relativity

Those space-times admitting special conformal vector fields and those admitting special projective vector fields have recently been studied. In this paper these two classes of space times are shown to be very closely related to each other. Certain uniqueness features of (and necessary extra symmetries contained in) the associated Lie algebras are discussed and the dimensionality of each of the algebras is computed.

Three-Dimensional Reconstruction of a Collagen IV Analogue in the Dogfish Egg Case Wall

The wall of the egg case of the dogfish (Scyliorhinus canicula) contains an analogue of collagen Types IV and VIII organised into a regular 3-dimensional network. It presumably provides both a protective and a filtering role for the eggs contained within it. Electron micrographs of longitudinal and transverse sections, including systematically tilted sections, have been used both to define, for the first time, the space group symmetry of the lattice and to carry out 3-D reconstruction of the unit cell contents. This cell was found to be tetragonal, space group I422, with a = b = 11 ± 1 nm, and c = 74 ± 4 nm. Consistent with this, projection symmetries were c2mm for the [1,0,0] view, p2mm for the [1,1,0] view, and p4mm for the projection down the c axis (the [0,0,1] view), and all observed reflections in the computed Fourier transform obeyed the ruleh+k+l= 2n(ninteger) for body-centred lattices. The 3-D reconstruction, the first electron micrograph 3-D reconstruction of a collagen-containing material, is interpreted in terms of variations of previous molecular models for this structure. Type IV collagen is a constituent of the basal lamina, where it forms a network with both structural and filtering properties. The dogfish egg case structure may throw light on the (less regular) collagen IV structure of the basal lamina.

Symmetry of projection in the quantitative analysis of mammographic images.

Mammographic parenchymal patterns are among the strongest indicators of the risk of developing breast cancer. Risk evaluation through breast patterns may have an important role in studies of the aetiology of breast cancer and for monitoring changes in the breast in evaluating potential risk-modifying interventions. Typically, patterns are assessed by an experienced radiologist according to Wolfe grade, or on a coarse quantitative scale according to percent density. Parenchymal characterization methods, to overcome variability of classification by human observer, are under investigation. These include image segmentation using semi-automatic thresholding and automatic classification through textural and density measures. An important practical question relates to the extent to which information about mammographic pattern is carried by any one of the four views obtained in a typical examination. Specifically, variations of right-left breast symmetry and variations between the two standard views of each breast were tested. The mammograms of 30 premenopausal women, comprising 90 images [30 each of the right cranial-caudal (RCC), left cranial-caudal (LCC) and right medial-lateral oblique (RMLO)] were evaluated. Parameters included both subjective (radiologist classification and interactive image thresholding) and objective (fractal and skewness indices) quantitative measurements of parenchymal pattern. For the parameters tested, a high degree of correlations was observed for measurements on the RCC, LCC and RMLO views. Pearson correlation coefficients between 0.86-0.96 were found for the comparisons of quantitative parameters. The strong correlations suggest that, in the study and application of mammographic density classification, representative information is provided in a single view.

Structure of Al–Ni–Co decagonal quasicrystals

Al-Ni-Co stable decagonal quasicrystals had been believed to belong to P10 5 /mmc and to have decagonal symmetry of the basic atom-cluster column when projected along the decagonal axis. The space group and the projection symmetry of the cluster columns of Al 70 Ni 30x Cox (x = 15 or 20) quasicrystals were re-examined by convergent-beam electron diffraction and high-resolution electron microscopy (HREM). It has been revealed that the space group is P10 5 /mmc but the projection symmetry is pentagonal. The quasicrystals have been found to be isomorphic with Al-Cu-Co decagonal quasicrystals by Saitoh et al. in 1996 (Phil. Mag. A, 73, 381). The symmetries of the HREM image of the cluster column have been revealed to be pentagonal and decagonal at accelerating voltages of 200 and 300 kV respectively. The strong contribution of the 13420-type reflections at 300kV smears the polar (pentagonal) image of the cluster column because electron waves scattered from all the atoms are superposed nearly in phase for the reflections with a lattice spacing of about 0.2nm. It is demonstrated by computer simulations that a pentagonal atom-cluster-column model can produce pentagonal and decagonal HREM images at accelerating voltages of 200 and 300 kV respectively.

Surfactosomes: a novel approach to the reconstitution and 2-D crystallisation of membrane proteins

The formation of vesicle-like structures (termed surfactosomes) and lamellar sheets from solutions containing ammonium perfluoroocanoate (APFO) is illustrated using conventional and cryo-transmission electron microscopy. It is shown how this detergent can be used for the solubilisation, reconstitution, and 2-D crystallisation of membrane proteins as demonstrated for the major protein of the membrane sector of the V-type H +-ATPase (16-kDa protein). Electron microscopical analysis of 2-D crystals of the 16-kDa protein (a = b = 13.0 ± 0.2 nm with γ = 90° and p4 projection symmetry) revealed a unit cell comprising four dimeric complexes of the 16-kDa protein the significance of which is discussed.

Symmetry sensing through computer vision and a facial image recognition system

The symmetries exhibited by nearly front view photographs of a human face have been explored by using group theory. Thus we develop an algorithm for facial image recognition implemented through computerised symmetry sensing for dealing with problems of personal identification from face photographs. The questioned facial image is made superimposable over the control one by processing through a properly computed transformation of projective symmetry. A total superimposed image, as well as other composite images produced from the control and the processed questioned images, facilitate accurate decision-making in identity establishment. All such images can be stored as permanent records and their hard copies or photographs from the monitor screen are suitable for court demonstration. An example of personal identification in a case of suspected impersonation has been demonstrated as a practical application of this new method.

Projective transformations and symmetries of differential equation

The group properties of the equations of geodesics on a pseudo-Riemannian manifold are considered, in particular, when these are written as a system of second-order differential equations (resolved with respect to the second derivatives) with third-degree polynomials in the derivatives of the unknown function on the right-hand sides. Each point symmetry of such systems is proved to be a projective transformation. A connection between projective transformation in and symmetries of Hamiltonian systems and Lie-Bäcklund transformations of Hamilton-Jacobi equation with quadratic Hamiltonians is discovered. This provides tools for developing a systematic geometric approach to defining and investigating point and non-point symmetries of large classes of differential equations and partial differential equations and to obtaining their solutions. The dimension of the maximal symmetry group for system of second-order ordinary differential equations resolved with respect to the higher derivatives is found, and this group is proved to be the projective group. As a consequence, the dimension of the maximal symmetry group of the Newton equations is found. In case of three spatial dimensions this group (which is a 24-dimensional projective group) is proved to have as a subgroup the Poincaré group, which is fundamental for special relativity theory.

Three-dimensional membrane crystals in amphibian cone outer segments: 2. Crystal type associated with the saddle point regions of cone disks

In light-adapted, perfusion fixed retinas of the Congo eel salamander, Amphiuma, we have observed distinctive 3-D crystalline domains within the axial array of cone outer segment disks. These crystalline domains, each involving 2-12 disks, have been observed in the distal half of cone outer segment, and are associated with saddle point regions and immediately adjacent segments of the cone disk perimeter. In longitudinal sections, the crystals typically display an axially oriented array of cytoplasmic filaments with lateral spacings in the range of 12-13 nm. The width of the intradiskal compartment is expanded to 8-10 nm within the crystal, and approximates the width of the cytoplasmic compartment. In some sections, the cytoplasmic filaments are axially aligned with intradiskal filaments of similar length and width. In transverse sections, the projected lattice appears to be approximately rectangular, with unit cell dimensions of approximately 12 nm x 12.5 nm. In shape, orientation, dimension, location, projection symmetry and associated membrane spacing relationships, the cone outer segment crystal filaments share a strong resemblance with lattice filaments located along the perimeters and preincisures of rod disks. These similarities suggest that the cone crystal filaments may be related to the rim protein of rod disks. Lastly, the preferential association of these crystals with saddle point regions indirectly supports the hypothesis that reductions in COS disk area with apical displacement are accomplished by resorption of disk membrane components through the saddle points.

COMPACT U(1)×U(1) MODEL WITH MINIMAL INTERSPECIES INTERACTION

We introduce a minimally interacting pure gauge compact U(1)×U(1) model consistent with Abelian projection symmetries. This paradigm, whose interactions are entirely due to compactness, illustrates how compactness can contribute to interspecies interactions. Furthermore, it has a much richer phase structure than that obtained by naively tensoring together two compact U(1) copies.

The use of non-linear characteristics of a lattice fringe image for the characterisation of imaging parameters

A way of determining HREM experimental parameters from the analysis of non-linear interference is described. Thickness, defocus, beam divergence and isotropic specimen vibration can all be determined for relatively broad ranging characteristics of the material examined, the specific example treated here being [001] Al x Ga 1− x As. For structures of a certain two-dimensional symmetry in projection, quantitative information concerning the misalignments can also be obtained.

4-Å Projection Map of Bacteriophage T4 DNA Helix-Destabilizing Protein (gp32*I) Crystal by 400-kV Electron Cryomicroscopy

Ice-embedded crystals of bacteriophage T4 DNA helix-destabilizing protein gp32*I were imaged by computer-controlled spot-scanning on a 400-kV electron cryomicroscope. gp32*I crystals generally have different steps of thickness within a crystal; each step can have different symmetry. Multivariate statistical analysis enabled us to unambiguously select spot-scan images that consist entirely of one motif which were processed subsequently by crystallographic Fourier-averaging techniques. The computed phases of the resulting reflections were evaluated for symmetry in projection, and some of those images were correlated with independent thickness measurements of freeze-dried samples of the same crystals. The structure factors with pgg symmetry from nine spot-scan images were merged, and the mean figure of merit of merged phases was better than 0.9 for data at resolution up to 4 Å. A projection map was generated and showed multiple density peaks that corresponded to the high-resolution features of gp32*I.

Understanding geometrical phases in quantum mechanics: An elementary example

We discuss an exact solution to the simplest nontrivial example of a geometrical phase in quantum mechanics. By means of this example: (1) we elucidate the fundamental distinction between rays and vectors in describing quantum mechanical states; (2) we show that superposition of quantal states is invalid; only decomposition is allowed—which is adequate for the measurement process. Our example also shows that the origin of singularities in the analog vector potential is to be found in the unavoidable breaking of projective symmetry caused by using the Schrodinger equation.

A 9 Å two-dimensional projected structure of cholera toxin B-subunit-G M1 complexes determined by electron crystallography

Highly ordered two-dimensional crystals of cholera toxin B-subunit pentamers have been grown by specific interaction with planar lipid films containing monosialoganglioside G M1. Electron diffractograms of frozen-hydrated crystals show diffraction peaks extending to beyond 4 Å, while electron images diffract to 8 Å. A two-dimensional projected structure of cholera toxin B-subunit-G M1 complex has been calculated at 9 Å resolution by combining electron diffraction and image data. Crystals present an approximate pgg projection symmetry, with unit cell dimensions a = 119(±1) Å, b = 123(±1) Å, γ = 90 °. Each pentameric assembly presents two concentric rings of electron scattering density, separated by an area of lower density. The outer and inner rings are centered at 25 Å and 11 Å from the pentamer centre, respectively. The apparent projected density of the outer ring is larger than that of the inner ring. We propose that the outer and inner density rings correspond respectively to the peripheral β-sheet arrangement and the central α-helix barrel, recently identified in the crystal structure of the heat-labile enterotoxin from Escherichia coli.

Analysis of symmetry and three-dimensional reconstruction of thin gp32∗I crystals

Thin, multilayered crystals of gp32∗I were analyzed by negative stain electron microscopy and image processing. Images of untilted crystals exhibited different projection symmetries and structural motifs. Systematic analysis of these images categorized the projections into four types. Areas producing the type 1 projection were reconstructed in three-dimensions from four tilt series containing 111 images. The three-dimensional data has excellent p12 1 plane group symmetry and reveals that the gp32∗I molecule contains two large domains linked together by a small domain. Computer simulations utilizing projection data suggested that the type 2 and 3 projections arise from superposition of type 1 projections related by a 2 1 screw axis along the projection axis. The three-dimensional reconstruction was utilized in a final simulation that explained the occurrence of the fourth type of projection. This work provides a firm foundation for future high-resolution analysis of the crystal by electron cryomicroscopy.

Interhemispheric Integration: II. Symmetry and Convergence of the Corticostriatal Projections of the Left and the Right Principal Sulcus (PS) and the Left and the Right Supplementary Motor Area (SMA) of the Rhesus Monkey

The relationship between the termination zones of projections from paired homotopic areas of the frontal lobes was examined in the caudate nucleus and the putamen of the macaque monkey. Injections of WGA-HRP and tritiated amino acids were made in topographically matched regions of the principal sulcus (PS) or of the supplementary motor area (SMA) in each hemisphere, such that the projections from the same area on each side were differentially labeled in the same animal. Adjacent sections through the neostriatum were processed for the respective tracers, permitting the relationship between the converging projections to be defined. The topographic distribution and strength of ipsilateral corticostriatal projections observed for separately labeled left and right hemispheres were strikingly similar. Projections from the left and the right PS terminated preferentially in central parts of the left and right neostriata, respectively, while projections of the left and right SMAs terminated preferentially in dorsolateral parts of respective left and right neostriata. Therefore, little evidence for asymmetry of corticostriatal projections was found. The projections of the left and the right PS to the same neostriatum were also compared. Remarkably, whether in the left or right hemisphere, projections from the left and the right PS were in precise register in topographically specific territories of the caudate and putamen. Likewise, projections of the left and right SMAs converged in both the left and right neostriata. Such convergence allows for a remarkable degree of interhemispheric integration in the descending corticostriatal networks.

Structural and chemical characterization of the S layer of a Pseudomonas-like bacterium

Sections and freeze-fractured preparations showed an S layer on the surface of Pseudomonas-like strain EU2. Polyacrylamide gel electrophoresis of cell envelopes extracted with 1% sodium dodecyl sulfate (SDS) at room temperature showed three proteins (45K, 55K, and 110K). The 55K protein was identified as the S-layer protein. Incubation in 1.5 M guanidine hydrochloride removed the S layer from cell envelopes and dissociated the structure into subunits. The soluble 55K protein reassembled into planar sheets upon removal of the guanidine hydrochloride by dialysis. Electron microscopy and image processing indicated that these sheets had p4 symmetry in projection with a lattice constant of 13.2 +/- 0.1 nm (corresponding to 9.3 nm between adjacent fourfold axes). In some instances these reassemblies appeared to form small three-dimensional crystals which gave particularly clear views of the structure in projection because of the superimposition of information from a number of layers. A model is proposed with molecules having rounded lobes connected by a narrower linker region and joining at the lobes to form the fourfold axes of the array. The pattern superficially resembles those of other bacterial S layers, such as those of Aeromonas salmonicida, Aeromonas hydrophila, and Azotobacter vinelandii. Extraction of cell envelopes with 1% SDS at 50 degrees C released the 110K protein from the envelopes and removed an amorphous backing layer from the S layer. The 45K protein displayed heat-modifiable migration in SDS-polyacrylamide gel electrophoresis and was insoluble in SDS at 50 degrees C or in high concentrations of guanidine hydrochloride, suggesting that it was associated with the peptidoglycan.

Photographs of the human face and broken projective symmetry

The exact projective symmetry manifested in photographs is broken if the object is not flat. It has been found that front view perspectives of the human face, photographed with limited variations in camera angle (within 30 degrees) show a breaking projective symmetry sufficiently small to be considered for accurate comparison. As a practical implementation of this new approach, a computer-controlled video camera superimposition technique has been developed and demonstrated in the identification of a disguised human face.

Photographs of flat impressions and their projective symmetry

Evidence photographs of two-dimensional impressions, taken without aligning the camera focal plane parallel to the plane of the impression, should not be rejected because they suffer from perspective deformation. A consideration of projective symmetry offers a possible solution to the problem of identification with such photographs of questioned and control impressions which requires no knowledge of the exact size and shape of the original impressions.

Quadratic zeros of Racah 6j coefficients: A geometrical approach

It is shown that the projective symmetries of the polynomial Φ of quadratic 6j coefficients form the symmetrical group S6. Nonlinear rational symmetries of Φ are found. Partial parametrizations of the zeros of Φ are presented.