Octagonal Symmetry Research Articles

Assembly and Disassembly of Nuclear Pore Complex: a View from Structural Side

Nucleocytoplasmic exchange in the cell occurs through the nuclear pore complexes (NPCs). NPCs are large multiprotein complexes with octagonal symmetry about their axis and imperfect mirror symmetry about a plane parallel with the nuclear envelop (NE). NPC fuses the inner and outer nuclear membranes and opens up а channel between nucleus and cytoplasm. NPC is built of nucleoporins. Each nucleoporin occurs in at least eight copies per NPC. Inside the NPC forms a permeability barrier by which NPC can ensure fast and selectable transport of molecules from one side of nuclear membrane to another. NPC architecture is based on hierarchical principle of organization. Nucleoporins are integrated into complexes that oligomerizes into bigger octomeric high-order structures. These structures are the main components of NPC. In the first part of this work the main attention is paid to NPC structure and nucleoporins’ properties. The second part is dedicated to mechanisms of NPC assembly and disassembly at different stages of cell cycle.

Assembly and Disassembly of the Nuclear Pore Complex: A View from the Structural Side

Nucleocytoplasmic exchange in the cell occurs through the nuclear pore complexes (NPCs). NPCs are large multiprotein complexes with octagonal symmetry about their axis and imperfect mirror symmetry about a plane parallel with the nuclear envelop (NE). NPC fuses the inner and outer nuclear membranes and opens up a channel between nucleus and cytoplasm. NPC is built of nucleoporins. Each nucleoporin occurs in at least eight copies per NPC. Inside the NPC a permeability barrier forms by which NPCs can provide fast and selectable transport of molecules from one side of the nuclear membrane to the other. NPC architecture is based on hierarchical principle of organization. Nucleoporins are integrated into complexes that oligomerizes into bigger octomeric high-order structures. These structures are the main components of NPCs. In the first part of this work, the main attention is paid to NPC structure and nucleoporin properties. The second part is dedicated to mechanisms of NPC assembly and disassembly at different stages of the cell cycle.

Self-assembly of dodecagonal and octagonal quasicrystals in hard spheres on a plane.

Hard spheres are one of the most fundamental model systems in soft matter physics, and have been instrumental in shedding light on nearly every aspect of classical condensed matter. Here, we add one more important phase to the list that hard spheres form: quasicrystals. Specifically, we use simulations to show that an extremely simple, purely entropic model system, consisting of two sizes of hard spheres resting on a flat plane, can spontaneously self-assemble into two distinct random-tiling quasicrystal phases. The first quasicrystal is a dodecagonal square-triangle tiling, commonly observed in a large variety of colloidal systems. The second quasicrystal has, to our knowledge, never been observed in either experiments or simulations. It exhibits octagonal symmetry, and consists of three types of tiles: triangles, small squares, and large squares, whose relative concentration can be continuously varied by tuning the number of smaller spheres present in the system. The observed tile composition of the self-assembled quasicrystals agrees very well with the theoretical prediction we obtain by considering the four-dimensional (lifted) representation of the quasicrystal. Both quasicrystal phases form reliably and rapidly over a significant part of parameter space. Our results demonstrate that entropy combined with a set of geometrically compatible, densely packed tiles can be sufficient ingredients for the self-assembly of colloidal quasicrystals.

Localization in Two-Dimensional Quasicrystalline Lattices.

We investigate the emergence of localization in a weakly interacting Bose gas confined in quasicrystalline lattices with three different rotational symmetries: five, eight, and twelve. The analysis, performed at a mean field level and from which localization is detected, relies on the study of two observables: the inverse participation ratio (IPR) and the Shannon entropy in the coordinate space. Those physical quantities were determined from a robust statistical study for the stationary density profiles of the interacting condensate. Localization was identified for each lattice type as a function of the potential depth. Our analysis revealed a range of the potential depths for which the condensate density becomes localized, from partially at random lattice sites to fully in a single site. We found that localization in the case of five-fold rotational symmetry appears for (6ER,9ER), while it occurs in the interval (12ER,15ER) for octagonal and dodecagonal symmetries.

Flat bands and band-touching from real-space topology in hyperbolic lattices

Motivated by the recent experimental realizations of hyperbolic lattices in circuit quantum electrodynamics and in classical electric-circuit networks, we study flat bands and band-touching phenomena in such lattices. We analyze noninteracting nearest-neighbor hopping models on hyperbolic analogs of the kagome and dice lattices with heptagonal and octagonal symmetry. We show that two characteristic features of the energy spectrum of those models, namely the fraction of states in the flat band as well as the number of touching points between the flat band and the dispersive bands, can both be captured exactly by a combination of real-space topology arguments and a reciprocal-space description via the formalism of hyperbolic band theory. Furthermore, using real-space numerical diagonalization on finite lattices with periodic boundary conditions, we obtain new insights into higher-dimensional irreducible representations of the non-Euclidean (Fuchsian) translation group of hyperbolic lattices. First, we find that the fraction of states in the flat band is the same for Abelian and non-Abelian hyperbolic Bloch states. Second, we find that only Abelian states participate in the formation of touching points between the flat and dispersive bands.

Notes on Quasicrystals and the Origin of Life

We still do not understand the origins of life. Here I suggest that it may have arisen as a fractal quasicrystal of organic molecules on a metallic quasicrystalline template. It may have arisen at least twice, once with pentagonal symmetry and once with octagonal symmetry. In both instances these protolife forms are part of an imperfect fractal continuum from the subatomic to the galactic.

Analogy and Symmetry

This paper uses a heuristic approach to reflect on the generalization in geometry by analogy from the plane to the space. Focusing in the regular octagon, several criteria are presented to answer of the question of which polyhedra would be “analogous” to the octagon. The presented construction procedures generate polyhedra which are similar to the octagon, and the symmetry of the obtained solids is used as key indicator of the level of analogy. The procedures can be extended to other shapes and planar patterns with octagonal symmetry.

Chiral spiral cyclic twins.

A formula is presented for the generation of chiral m-fold multiply twinned two-dimensional point sets of even twin modulus m > 6 from an integer inclination sequence; in particular, it is discussed for the first three non-degenerate cases m = 8, 10, 12, which share a connection to the aperiodic crystallography of axial quasicrystals exhibiting octagonal, decagonal and dodecagonal long-range orientational order and symmetry.

Reconfigurable spin-wave dynamics in two-dimensional quasiperiodic magnonic crystals

Active control of spin-wave properties is at the heart of on-chip reconfigurable magnonics, which promise development of energy efficient and miniaturized on-chip processing and communication devices in the GHz and sub-THz frequency range. Here, we demonstrated active control of spin-wave spectra in two-dimensional nanoscale antidot lattices arranged in shifted honeycomb and octagonal symmetries by varying the antidot diameter using broadband ferromagnetic resonance. Both these lattices exhibited stark variation in the spin-wave spectra with antidot diameter, besides the variation of the strength and orientation of bias magnetic field. With increasing diameter number of spin-wave modes gradually increased, and the position of mode splitting shifted to higher magnetic field due to ensuing increase in the demagnetizing field governing the spin-wave extension or localization in the lattices. The bias-field angle variation revealed higher-order rotational symmetry characteristic of quasi-periodic structure. Micromagnetic simulations qualitatively reproduced the experimental observations, and simulated spin-wave mode profiles revealed extended and standing spin-wave modes and their mutual conversion with bias-field angle in both lattices. The contrasting variation in the amplitude of rotational symmetry of spin-wave frequency with antidot diameter was interpreted by the variation of spin-wave mode profiles as well as the simulated internal magnetic field variation. Such efficient tunability of spin-wave properties with internal structure and external control parameters and higher-order rotational anisotropy in these atypical magnonic crystals will promote them for applications in on-chip reconfigurable magnonic devices.

Description of Voronoi tiles in quasicrystals with octagonal symmetry

We describe families of Voronoi tiles which appear in tilings of certain standard quasicrystal-like point sets with octagonal symmetry built using the cut-and-project construction.

Stabilizing quasicrystals composed of patchy colloids by narrowing the patch width

We explore the behavior of two-dimensional patchy colloidal particles with 8 or 10 symmetrically arranged patches by employing Monte Carlo simulations. The particles interact according to an isotropic pair potential that possesses only one typical length. The patches lead to additional attractions that are anisotropic and depend on the relative orientation of two neighboring particles. We investigate the assembled structures with a special interest in quasicrystals. We found that the patch width is of great importance: Only in the case of narrow patch widths we are able to observe metastable octagonal and decagonal quasicrystals, while dodecagonal quasicrystals can also occur for broad patches. These results are important to understand the role of interactions with preferred binding angles in order to obtain quasicrystals. Our findings suggest that in the case of sharp binding angles, as they occur in metallic alloys, octagonal and decagonal symmetries might be observed more often than in systems with less sharp binding angles as it is the case in soft-matter systems where dodecagonal quasicrystals dominate.

Low-lying octupole isovector excitation in Nd144

The nature of low-lying 3(-) levels in Nd-144 was investigated in the Nd-143(n, gamma gamma) cold neutron-capture reaction. The combination of the high neutron flux from the research reactor at the Institut Laue-Langevin and the high gamma-ray detection efficiency of the EXILL setup allowed the recording of gamma gamma coincidences. From the coincidence data precise branching ratios were extracted. Furthermore, the octagonal symmetry of the setup allowed angular-distribution measurements to determine multipole-mixing ratios. Additionally, in a second measurement the ultra-high resolution spectrometer GAMS6 was employed to conduct lifetime measurements using the gamma-ray induced Doppler-shift technique (GRID). The confirmed strong M1 component in the 3(3)(-) -> 3(1)(-) decay strongly supports the assignment of the 3(3)(-) level at 2779 keV as low-lying isovector octupole excitation. Microscopic calculations within the quasiparticle phonon model confirm an isovector component in the wave function of the 3(3)(-) level, firmly establishing this fundamental mode of nuclear excitation in near-spherical nuclei.

Symmetry Incorporated Cost-Effective Architectures for Two-Dimensional Digital Filters

Professor Fettweis as far back as 1977 published a paper generalizing McClellan transformation to obtain circular symmetry in 2-D and spherical, hyper-spherical symmetries in multidimensional digital filters [1]. This survey paper presents stateof-the-art two-dimensional (2-D) VLSI digital filter architectures possessing various symmetries in the filter magnitude response. Preceding the symmetry structures, a generalized formulation is given that allows the derivation of various new 2-D VLSI filter structures of any order without global broadcast. Following this, two types (namely, Type 1 [20] and Type 3 [21], [25], [26]) of cost-effective 2-D magnitude symmetry filter architectures possessing diagonal, four-fold rotational, quadrantal, and octagonal symmetries with reduced number of multipliers are given. By combining the identities of the Types-1 and 3 symmetry filter structures, multimode 2-D symmetry filters which enable the above four symmetry modes are discussed. The Type-1 and Type-3 multimode filters can result in a 65.3% cost reduction in terms of number of multipliers compared with the sum of the multipliers of the four individual Type-1 symmetry filter structures studied in this paper. Furthermore, Type-3 has shorter critical path than Type-1 multimode filter. The paper is concluded with the presentation of a 2-D filter design example and a corresponding structure.

Multiple-scale structures: from Faraday waves to soft-matter quasicrystals.

For many years, quasicrystals were observed only as solid-state metallic alloys, yet current research is now actively exploring their formation in a variety of soft materials, including systems of macromolecules, nanoparticles and colloids. Much effort is being invested in understanding the thermodynamic properties of these soft-matter quasicrystals in order to predict and possibly control the structures that form, and hopefully to shed light on the broader yet unresolved general questions of quasicrystal formation and stability. Moreover, the ability to control the self-assembly of soft quasicrystals may contribute to the development of novel photonics or other applications based on self-assembled metamaterials. Here a path is followed, leading to quantitative stability predictions, that starts with a model developed two decades ago to treat the formation of multiple-scale quasiperiodic Faraday waves (standing wave patterns in vibrating fluid surfaces) and which was later mapped onto systems of soft particles, interacting via multiple-scale pair potentials. The article reviews, and substantially expands, the quantitative predictions of these models, while correcting a few discrepancies in earlier calculations, and presents new analytical methods for treating the models. In so doing, a number of new stable quasicrystalline structures are found with octagonal, octadecagonal and higher-order symmetries, some of which may, it is hoped, be observed in future experiments.

Influence of anisotropic dipolar interaction on the spin dynamics of Ni80Fe20 nanodot arrays arranged in honeycomb and octagonal lattices

Ultrafast spin dynamics in ferromagnetic nanodot arrays with dot diameter 100 nm and thickness 20 nm arranged in honeycomb and octagonal lattice symmetries are studied to explore the tunability of the collective magnetization dynamics. By varying the inter-dot separation between 30 nm and 300 nm drastic variation in the precessional dynamics from strongly collective to completely isolated regime has been observed by using all-optical time-resolved magneto-optical Kerr microscope. Micromagnetic simulation is exploited to gain insights about the resonant mode profiles and magnetic coupling between the nanodots. A significant spectral and spatial variation in the resonant mode with increasing dipolar interaction is demonstrated with increasing inter-dot separation. The spins driven by effective field inside single nanodots are prone to precess independently, generating two self-standing centre and edge modes in the array that are influenced by the relative orientation between the inter-dot coupling direction and bias magnetic field. The anisotropic behavior of dipolar field is rigorously investigated here. Splitting of the centre mode in case of octagonal lattice is experimentally observed here as a consequence of the anisotropic dipolar field between the nanodot pairs coupled horizontally and vertically, which is not found in the honeycomb lattice. In addition, proper understanding of the modification of dynamic mode profile by neighboring dipolar interaction built up here, is imperative for further control of the dynamic dipolar interaction and the corresponding collective excitation in magnonic crystals. The usage of nanodot lattices with complex basis structures can be advantageous for the designing of high density magnetic recording media, spin-wave filter and logic devices.

Tunable picosecond spin dynamics in two dimensional ferromagnetic nanodot arrays with varying lattice symmetry

Tunable configurational anisotropy in spin-waves with up to 8-fold symmetry in closely packed nanodot lattices with rectangular, honeycomb and octagonal symmetry. The extrinsic nature of the anisotropy is due to angular variation of the magnetostatic field distribution.

Modeling quasi-lattice with octagonal symmetry

We prove the possibility to use the method of modeling of a quasi-lattice with octagonal symmetry similar to that proposed earlier for the decagonal quasicrystal. The method is based on the multiplication of the groups of basis sites according to specified rules. This model is shown to be equivalent to the method of the periodic lattice projection, but is simpler because it considers merely two-dimensional site groups. The application of the proposed modeling procedure to the reciprocal lattice of octagonal quasicrystals shows a fairly good matching with the electron diffraction pattern. Similarly to the decagonal quasicrystals, the possibility of three-index labeling of the diffraction reflections is exhibited in this case. Moreover, the ascertained ratio of indices provides information on the intensity of diffraction reflections.

Octagonal symmetry in low-discrepancy β-manganese.

A low-discrepancy cubic variant of β-Mn is presented exhibiting local octagonal symmetry upon projection along any of the three mutually perpendicular 〈100〉 axes. Ideal structural parameters are derived to be x(8c) = (2-\sqrt{2})\big/16 and y(12d) = 1\big/(4 \sqrt{2}) for the P4132 enantiomorph. A comparison of the actual and ideal structure models of β-Mn is made in terms of the newly devised concept of geometrical discrepancy maps. Two-dimensional maps of both the geometrical star discrepancy D(*) and the minimal interatomic distance dmin are calculated over the combined structural parameter range 0 \leq x(8c) \,\lt\, 1/8 and 1/8 \leq y(12d)\, \lt\, 1/4 of generalized β-Mn type structures, showing that the `octagonal' variant of β-Mn is almost optimal in terms of globally minimizing D(*) while at the same time globally maximizing dmin. Geometrical discrepancy maps combine predictive and discriminatory powers to appear useful within a wide range of structural chemistry studies.

Peptide Ordering within Nuclear Pores in Living Cells

It is a profound belief of biologists, based upon decades of structural biochemistry, that understanding the three-dimensional structure of individual proteins and protein complexes illuminates the basis of protein function. X-ray crystallography and other structural techniques have revealed remarkable high-resolution maps of even large, multimolecular structures. However, the approaches which give such spectacular information in vitro are unable to probe structures of proteins that are functioning in their natural milieu in living cells. This frontier between structural biology and cellular biology is only beginning to be addressed. An example of the progress is work from the laboratory of Sanford Simon (Rockefeller University, New York). Earlier works by Mattheyses et al. (1) and Kampman et al. (2), and a new article by Atkinson et al. (3) (this issue) describe the elegant application of fluorescence polarization to probe the orientation of specific protein components in functioning nuclear pore complexes in living cells. Nuclear pores are an essential component of eukaryotic cells that are responsible for the highly regulated, bidirectional transport of proteins and RNA between the cytoplasm and nucleus. They are very large structures that contain a large aqueous channel that spans the double membrane of the nuclear envelope. Pores are composed of ∼30 different proteins (nucleoporins) present in multiple copies, with several thousand proteins comprising a single nuclear pore. Crystallography has provided structural information about isolated nucleoporins and their interaction sites (4). Electron microscopy of intact nuclear pore complexes has revealed a structure with octagonal symmetry (5,6). Simon and colleagues have taken advantage of the random distribution of nuclear pores around the approximately circular cross section of the nuclear envelope. Nuclear pore proteins labeled with green fluorescent protein were introduced into cells by molecular biology techniques. By illumination of the sample with polarized light and collection of the emission in two orthogonally polarized channels, the fluorescence anisotropies of nuclear pores with their varying orientations around the nuclear envelope cross section were determined. Thus, the intrinsic geometry of the positions of the nuclear pores in the nuclear envelope enabled investigation of the anisotropy at various angles. Channel nucleoporins have a predicted α-helical region and an unfolded domain containing phenylalaine-glycine (FG) repeats. The ensemble of FG domains from multiple nucleoporins in a channel is thought to function as a diffusion barrier to regulate transport of large macromolecules (Fig. 1 (7)). This article addresses the question of whether FG regions in different nucleoporins are oriented relative to the central axis of functioning nuclear pores in living cells. The findings were significant: some of the FG domains of specific nucleoporins were ordered. Ordering was more likely for nucleoporins lining the center of the nuclear pore complex than for those at the cytoplasmic or nuclear entrances. Ordering was not altered by active transport or attachment of cargo. Although one might have predicted (wished?) a change with function, the time resolution of anisotropy measurements (2 s) was likely insufficient to capture fluctuations due to individual transport events. Figure 1 Schematic of nuclear pore. Adapted from Fig. 12-10 in Alberts et al. (7). These studies are a hopeful beginning, demonstrating the possibility of asking structural questions within living cells. We can look forward to improved techniques, including faster time resolution approaches and the application of single-molecule fluorescence imaging, that will enable the detection of function-related structural changes in proteins in their normal, intracellular milieu. These types of biophysical techniques should be applicable to numerous other dynamic protein structures within cells.

Tunable Magnonic Spectra in Two‐Dimensional Magnonic Crystals with Variable Lattice Symmetry

AbstractTunable magnonic properties are demonstrated in two‐dimensional magnonic crystals in the form of artificial ferromagnetic nanodot lattices with variable lattice symmetry. An all‐optical time‐domain excitation and detection of the collective precessional dynamics is performed in the strongly magnetostatically coupled Ni80Fe20 (Py) circular dot lattices arranged in different lattice symmetry such as square, rectangular, hexagonal, honeycomb, and octagonal symmetry. As the symmetry changes from square to octagonal through rectangular, hexagonal and honeycomb, a significant variation in the spin wave spectra is observed. The single uniform collective mode in the square lattice splits in two distinct modes in the rectangular lattice and in three distinct modes in the hexagonal and octagonal lattices. However, in the honeycomb lattice a broad band of modes are observed. Micromagnetic simulations qualitatively reproduce the experimentally observed modes, and the simulated mode profiles reveal collective modes with different spatial distributions with the variation in the lattice symmetry determined by the magnetostatic field profiles. For the hexagonal lattice, the most intense peak shows a six‐fold anisotropy with the variation in the azimuthal angle of the external bias magnetic field. Analysis shows that this is due to the angular variation of the dynamical component of magnetization for this mode, which is directly influenced by the variation of the magnetostatic field on the elements in the hexagonal lattice. The observations are important for tunable and anisotropic propagation of spin waves in magnonic crystal based devices.