Axis Of Rotational Symmetry Research Articles

Beyond bilateral symmetry: geometric morphometric methods for any type of symmetry

BackgroundStudies of symmetric structures have made important contributions to evolutionary biology, for example, by using fluctuating asymmetry as a measure of developmental instability or for investigating the mechanisms of morphological integration. Most analyses of symmetry and asymmetry have focused on organisms or parts with bilateral symmetry. This is not the only type of symmetry in biological shapes, however, because a multitude of other types of symmetry exists in plants and animals. For instance, some organisms have two axes of reflection symmetry (biradial symmetry; e.g. many algae, corals and flowers) or rotational symmetry (e.g. sea urchins and many flowers). So far, there is no general method for the shape analysis of these types of symmetry.ResultsWe generalize the morphometric methods currently used for the shape analysis of bilaterally symmetric objects so that they can be used for analyzing any type of symmetry. Our framework uses a mathematical definition of symmetry based on the theory of symmetry groups. This approach can be used to divide shape variation into a component of symmetric variation among individuals and one or more components of asymmetry. We illustrate this approach with data from a colonial coral that has ambiguous symmetry and thus can be analyzed in multiple ways. Our results demonstrate that asymmetric variation predominates in this dataset and that its amount depends on the type of symmetry considered in the analysis.ConclusionsThe framework for analyzing symmetry and asymmetry is suitable for studying structures with any type of symmetry in two or three dimensions. Studies of complex symmetries are promising for many contexts in evolutionary biology, such as fluctuating asymmetry, because these structures can potentially provide more information than structures with bilateral symmetry.

Generic role of the anisotropic surface free energy on the morphological evolution in a strained-heteroepitaxial solid droplet on a rigid substrate

A systematic study based on self-consistent dynamical simulations is presented for the spontaneous evolution of an isolated thin solid droplet on a rigid substrate, which is driven by the surface drift diffusion induced by the anisotropic capillary forces (surface stiffness) and mismatch stresses. In this work, we studied the effect of surface free energy anisotropies [weak and strong (anomalous)] on the development kinetics of the “Stranski–Krastanow” island type morphologies. The anisotropic surface free energy and the surface stiffness were treated with well accepted trigonometric functions. Although, various tilt angles and anisotropy constants were considered during simulations, the main emphasis was given on the effect of rotational symmetries associated with the surface Helmholtz free energy topography in two-dimensional space. Our computer simulations revealed the formation of an extremely thin wetting layer during the development of the bell-shaped Stranski-Krastanow island through the mass accumulation at the central region of the droplet via surface drift-diffusion. For weak anisotropy constant levels, instead of singlet islanding, we observed formation of doublet islanding, separated by a shallow wetting layer, for a set of specific tilt angles, ϕ=90° and ϕ=45°, respectively, for the twofold and fourfold rotational symmetry axis. No such formation has been detected for the sixfold symmetry. In the strong (anomalous) anisotropy constant domain, we demonstrated the existence of two distinct morphological modes: (i) the complete stability of the initial Cosine-shaped droplet just above a certain anisotropy constant threshold level by spontaneous slight readjustments of the base and the height of the cluster; (ii) the Frank-van der Merwe mode of thin film formation for very large values of the anisotropy constant by the spreading and coalescence of the droplets over the substrate surface. During the course of the simulations, we continuously tracked both the morphology (i.e., the peak height, the extension of the wetting layer beyond the domain boundaries, and the triple junction contact angle) and the energetics (the global Helmholtz free energy changes associated with the total strain and surface energy variations) of the system.

The Cowin–Mehrabadi theorem for an axis of symmetry

The well-known Cowin–Mehrabadi Theorem deals with necessary and sufficient conditions for a normal n to a symmetry plane. Necessary conditions require that n be a common eigenvector of C ijkk , C ikjk and C ijkl n j n l . It is shown that a vector parallel to an axis of symmetry must also satisfy these conditions. An axis of rotational symmetry is also a normal to a plane of symmetry except in the case of a trigonal material. Being a common eigenvector of C ijkk and C ikjk belonging to a nondegenerate eigenvalue guarantees it to be an axis of symmetry.

Symmetry, winding number, and topological charge of vortex solitons in discrete-symmetry media

We determine the functional behavior near the discrete rotational symmetry axis of discrete vortices of the nonlinear Schr\"odinger equation. We show that these solutions present a central phase singularity whose charge is restricted by symmetry arguments. Consequently, we demonstrate that the existence of high-charged discrete vortices is related to the presence of other off-axis phase singularities, whose positions and charges are also restricted by symmetry arguments. To illustrate our theoretical results, we offer two numerical examples of high-charged discrete vortices in photonic crystal fibers showing hexagonal discrete rotational invariance.

Design of Peptide Nanoparticles Using Simple Protein Oligomerization Domains

Viruses are naturally formed bionanoparticles ranging in size from about 20 to 150 nm. Remarkably, small vi- ruses are composed of one single protein chain folding into a capsid structure with icosahedral symmetry. The icosahe- dron is built from 60 asymmetric units and is the largest closed shell in which every subunit is in an identical environ- ment. It is characterized by 2-fold, 3-fold and 5-fold rotational symmetry axes. By superposition of different protein oli- gomerization domains onto the symmetry axes of an icosahedron, a nanoparticle with icosahedral symmetry can be de- signed. We have recently described such a design of peptide nanoparticles using coiled-coil protein oligomerization do- mains. Here we show that oligomerization motifs other than coiled-coils can be used to form nanoparticles by incorporat- ing the globular foldon domain from fibritin with a trimeric  -sheet conformation into the design. We expressed and puri- fied 8 different peptides and performed refolding studies and biophysical characterization with analytical ultra centrifuga- tion (AUC) and electron microscopy (EM). In the first design version we joined the foldon domain to the pentameric coiled-coil domain of COMP and varied the lengths of the linker sequences between the two domains. In this design we observed only smaller nanoparticles. When in the second design the foldon domain was extended with an additional trimeric coiled-coil domain as a combined trimerization domain that is linked to the COMP pentamer, we observed nanoparticles of sizes and molecular weights as would be expected for icosahedral symmetry. Viruses and virus-like parti- cles (VLPs) are known for their ability to induce a strong humoral and hence antibody mediated immune response due to their repetitive antigen display. Peptide based nanoparticles have similar properties to VLPs, which are in clinical trials as a carrier in vaccination. Therefore, these peptide nanoparticles represent an alternative platform for subunit vaccine using the concept of repetitive antigen display.

A digital simulation study of steady-state voltammograms for the ion transfer across the liquid–liquid interface formed at the orifice of a micropipette

Digital simulation of cyclic voltammograms for the ion transfer across the micro liquid–liquid interface formed at the orifice of a glass micropipette shows that the electrochemically driven flow of ions from the inside of the micropipette to the infinitely large outer phase gives steady-state current–potential characteristics, when the shape of the tip is not cylindrical but truncated conical as usual micropipettes. Such steady-state behavior well explains experimentally observed voltammograms for ion transfer across the hydrophobic ionic liquid–water interface formed at the orifice of a micropipette with a small radius. Both the limiting current and the half-wave potential of the voltammogram vary with the angle of the inner wall of the micropipette with respect to the axis of rotational symmetry of the pipette, θ 1 , and both reach the limits predicted by the radial diffusion of the ions to the inlaid disk, that is, θ 1 = 90 ° in an asymptotic manner. The conditions for obtaining such steady-state voltammograms depend on the size of the interface and the scan rate of the applied voltage, aside from θ 1 .

Supramolecular interaction patterns in the zinc(II) dichloride and tin(IV) tetrachloride complexes with dipyrido[f,h]quinoxaline-6,7-dicarbonitrile

This study characterizes the supramolecular synthons that dominate the intermolecular organization of the title compounds, namely dichloridobis(dipyrido[f,h]quinoxaline-6,7-dicarbonitrile)zinc(II), [ZnCl(2)(C(16)H(6)N(6))(2)], (I), and tetrachlorido(dipyrido[f,h]quinoxaline-6,7-dicarbonitrile)tin(IV), [SnCl(4)(C(16)H(6)N(6))], (II), in their respective crystal structures. Molecules of (I) are located on 2(b) axes of rotational symmetry. Their crystal packing is stabilized mostly by pi-pi stacking and dipole-dipole attractions between the organic ligand fragments of inversion-related neighbouring species, as well as by weak intermolecular C-H...N hydrogen bonds. On the other hand, Cl...pi and N...pi interactions seem to direct the crystal packing in (II), which is unusual in the sense that its aromatic fragments are not involved in pi-pi stacking. Molecules of (II) are located on m(b) planes of mirror symmetry. This study confirms the diverse structural chemistry of this organic ligand, which can be involved in a wide range of supramolecular interactions. These include effective coordination to various metal ions via the phenathroline N-atom sites, pi-pi stacking and pi...halogen contacts through its extended pi-system, and hydrogen bonding and N...pi interactions through its nitrile groups. The competing natures of the latter make it difficult to predict a priori the preferred supramolecular motif that may form in a given structure.

Shear compliance of two-dimensional pores possessing N -fold axis of rotational symmetry

We use the complex variable method and conformal mapping to derive a closed-form expression for the shear compliance parameters of some two-dimensional pores in an elastic material. The pores have an N -fold axis of rotational symmetry and can be represented by at most three terms in the mapping function that conformally maps the exterior of the pore into the interior of the unit circle. We validate our results against the solutions of some special cases available in the literature, and against boundary-element calculations. By extrapolation of the results for pores obtained from two and three terms of the Schwarz–Christoffel mapping function for regular polygons, we find the shear compliance of a triangle, square, pentagon and hexagon. We explicitly verify the fact that the shear compliance of a symmetric pore is independent of the orientation of the pore relative to the applied shear, for all cases except pores of fourfold symmetry. We also show that pores having fourfold symmetry, or no symmetry, will have shear compliances that vary with cos 4 θ . An approximate scaling law for the shear compliance parameter, in terms of the ratio of perimeter squared to area, is proposed and tested.

Wave reflection at an anelastic transversely isotropic ocean bottom

We study the reflection of waves at the ocean bottom, which is modeled as a plane interface separating a viscoacoustic medium (water) and a viscoelastic transversely isotropic solid whose axis of rotational symmetry is perpendicular to the bottom. We compute the plane-wave reflection coefficient (including the phenomenon known as the Rayleigh window) both numerically — by amplitude variation with offset (AVO) analysis of synthetic seismograms generated using a domain decomposition method and analytically. A first simulation considers the water-steel interface, for which experimental data is available. Then, we consider soft sediments and stiff crustal rocks for various values of the anellipticity parameter [Formula: see text]. The domain-decomposition technique relies on one grid for the fluid and another grid for the solid and uses Fourier and Chebyshev differential operators. The ane-lastic and anisotropic stress-strain relation is described by the Zener model. Special attention is given to modeling the boundary conditions at the ocean bottom. For this purpose, we further develop the technique for wave propagation at fluid/anelastic-anisotropic-solid interfaces. AVO slowness-frequency-domain analysis is used to compute the reflection coefficient and phase angle from the synthetic seismograms. This allows us to verify the domain-decomposition algorithm, which is shown to model with high accuracy the Rayleigh window for varying [Formula: see text]. The comparison also verifies the calculation of the analytical plane-wave reflection coefficient because a wrong choice of the sign of the vertical slowness of the reflected wave may cause nonphysical discontinuities in the coefficient. Moreover, the pseudospectral modeling code allows a general material variability and a complete and accurate characterization of the seismic response of an anisotropic ocean bottom.

Rule based system for archaeological pottery classification

Motivated by the requirements of the present archaeology, we are developing an automated system for archaeological classification of ceramics. Classification and reconstruction of archaeological fragments is based on the profile, which is the cross-section of the fragment in the direction of the rotational axis of symmetry. In order to segment the profile into primitives like rim, wall, and base, rules based on human expert knowledge are created. The input data for the estimation of the profile is a set of points produced by the acquisition system. A function fitting this set is constructed and later on processed to find the characteristic points necessary to classify the original fragment. We demonstrate the method and give results on real archaeological data.

Simulating nanoscale functional motions of biomolecules

We are describing efficient dynamics simulation methods for the characterization of functional motion of biomolecules on the nanometer scale. Multivariate statistical methods are widely used to extract and enhance functional collective motions from molecular dynamics (MD) simulations. A dimension reduction in MD is often realized through a principal component analysis (PCA) or a singular value decomposition (SVD) of the trajectory. Normal mode analysis (NMA) is a related collective coordinate space approach, which involves the decomposition of the motion into vibration modes based on an elastic model. Using the myosin motor protein as an example we describe a hybrid technique termed amplified collective motions (ACM) that enhances sampling of conformational space through a combination of normal modes with atomic level MD. Unfortunately, the forced orthogonalization of modes in collective coordinate space leads to complex dependencies that are not necessarily consistent with the symmetry of biological macromolecules and assemblies. In many biological molecules, such as HIV-1 protease, reflective or rotational symmetries are present that are broken using standard orthogonal basis functions. We present a method to compute the plane of reflective symmetry or the axis of rotational symmetry from the trajectory frames. Moreover, we develop an SVD that best approximates the given trajectory while respecting the symmetry. Finally, we describe a local feature analysis (LFA) to construct a topographic representation of functional dynamics in terms of local features. The LFA representations are low-dimensional, and provide a reduced basis set for collective motions, but unlike global collective modes they are sparsely distributed and spatially localized. This yields a more reliable assignment of essential dynamics modes across different MD time windows.

Structuring by multi-beam interference using symmetric pyramids

A method for producing optical structures using rotationally symmetric pyramids is proposed. Two-dimensional structures can be achieved using acute prisms. They form by multi-beam interference of plane waves that impinge from directions distributed symmetrically around the axis of rotational symmetry. Flat-topped pyramids provide an additional beam along the axis thus generating three-dimensional structures. Experimental results are consistent with the results of numerical simulations. The advantages of the method are simplicity of operation, low cost, ease of integration, good stability, and high transmittance. Possible applications are the fabrication of photonic micro-structures such as photonic crystals or array waveguides as well as multi-beam optical tweezers.

Symmetry Types, Systems and Their Multiplicity in the Structure of Adenovirus Capsid

The icosahedral adenovirus capsid has three rotational symmetry axes of different types. The six five-fold, ten three-fold and the fifteen two-fold axes have two superficial points each, altogether 62. The axes determine the number and location of the identical rotational facet groups and that during the different rotational phases which other regular facets and with what multiplicity shall be covered by them. The number of rotational facets of the five-, three- and two-fold rotational symmetry axes is 4, 6.66 and 10, respectively. In all the three cases, there are two kinds of possible arrangements of the facets. During the rotation--when the facets of the facet group placed on one by one to the neighbouring identical facet groups--at the five-fold axes, the facets of the rotational facet group get into cover position 12 times with all the 20 regular capsid facets, 20 times at the three-fold axes, and 30 times at the two-fold axes in a way that a different facet combination (facet hit) falls to every facet, and the original symmetry is not disturbed. After all, this means 240, 400 and 600 facet combinations, i.e. multiplicity in case of five-, three- and two-fold symmetry axes respectively, and these numbers correspond with that of the theoretically possible variations. The same results can be calculated by multiplying the number of real rotations of the capsid bringing the body into itself, i.e. the number 60 with the number of facets contributing to the five-, three- and two-fold rotational phases. The other way of the determination of multiplicity takes into account that all the facet groups of the capsid rotate simultaneously during all the rotational phases, and this multiplies the number of multiplicity with the number of the rotational types five-, three- and two-fold which result in one and the same multiplicity number in the case of five-, three- and two-fold symmetry, alike 1200. Perpendicular to the five-fold symmetry axes with the line of intersection drawn horizontally in the middle along the 6 geodetic ribbon like motifs a regular decagonal intersection forms and the capsid can be cut into two equal parts, in which the polypeptides show a 72 degree rotation from each other, but with a proper rotation the polypeptides get into a congruent position, which means 300 or 600 specific facet combinations. The capsid similar to the icosahedron has also 15 virtual mirror planes which divide the capsid into two, identically arranged halves, forming six right angle triangles on each facet, altogether 120 smaller rectangular so-called Mobius-triangles on the surface. In the three-fold symmetry axis of the facets, these triangles in two separate groups of three can be rotated symmetrically with 120 degrees according to the orientation of the polypeptide subunits in a way that the hexon and other polypeptides too nearly cover each other. Consequently, the adenovirus capsid is a symmetrically arranged body in which several various symmetry types and symmetry systems can be found and their structural symmetry elements exist simultaneously and covering each other. The icosahedral symmetry types and systems are valid and functional simultaneously and in parallel with great multiplicity, but the existence of more than 1500 structural elements in several depth levels, their order of location and distribution make the symmetry of the capsid richer and more complex.

Three-dimensional superconducting wire network in a magnetic field

Vortex states of icosahedral superconducting wire network are investigated by the de Gennes–Alexander theory. We obtained the magnetic field dependence of the transition temperature and fluxon arrangements in the network. Fluxon arrangements of icosahedral network have some different symmetry related to directions of the applied fields. In a magnetic field parallel to the fivefold rotational symmetry axis, we found that the order parameter vanishes at the central nodal point along the field axis. In a magnetic field parallel to the threefold rotational symmetry axis, anti-fluxon can appear at the central triangular loop.

Evolution and propagation of magnetic vortices in chains of Permalloy nanospheres

Magnetization reversal in a chain of Permalloy Fe0.2Ni0.8 spheres, whose diameters of 40–60nm are large enough to support vortex structures, is investigated using micromagnetic modeling based on the Landau-Lifshitz-Gilbert equations. The emphasis is on a chain of two spheres with fields applied along and perpendicular to the axis of rotational symmetry. Magnetization processes and critical fields are given for (1) inversion symmetry, with opposite senses of vortex rotation in the two spheres, and (2) a more uniform curling mode, with the sense of rotation the same in both spheres. Symmetry breaking perturbations are shown to be important in the nucleation of changes from one magnetic configuration to the next. As the geometry is approximated by cubic grid cells, whose centers lie within the boundaries of the ideally smooth surfaces, the critical fields are influenced by the grid size. The results for two spheres are generalized in the description of a chain of n spheres, in which at least 2n symmetry states can be selected by the application of inhomogeneous fields.

A Symmetry Preserving Singular Value Decomposition

A reduced order representation of a large data set is often realized through a principal component analysis based upon a singular value decomposition (SVD) of the data. The left singular vectors of a truncated SVD provide the reduced basis. In several applications such as facial analysis and protein dynamics, structural symmetry is inherent in the data. Typically, reflective or rotational symmetry is expected to be present in these applications. In protein dynamics, determining this symmetry allows one to provide SVD major modes of motion that best describe the symmetric movements of the protein. In face detection, symmetry in the SVD allows for more efficient compression algorithms. Here we present a method to compute the plane of reflective symmetry or the axis of rotational symmetry of a large set of points. Moreover, we develop a symmetry preserving singular value decomposition (SPSVD) that best approximates the given set while respecting the symmetry. Interesting subproblems arise in the presence of noisy data or in situations where most, but not all, of the structure is symmetric. An important part of the determination of the axis of rotational symmetry or the plane of reflective symmetry is an iterative reweighting scheme. This scheme is rapidly convergent in practice and seems to be very effective in ignoring outliers (points that do not respect the symmetry).

Wideband dielectric resonator antenna

AbstractThis paper presents a new wideband dielectric resonator (DR) antenna. To enhance the bandwidth, two layers of stacked resonators having the same rotational axis of symmetry are used. At the bottom layer, a smaller cylindrical DR is placed concentrically inside a large annular‐ring DR, whereas at the top layer, there is only one cylindrical DR. A coaxial probe is placed off the antenna axis in the DR antenna in order to produce broadside far‐field radiation patterns. The effects of the antenna parameters such as dielectric constant, probe length, and excitation position as well as the air gap between the bottom layer of the stacked DRA and the ground plane are investigated numerically. A bandwidth up to 67% for a −10‐dB return loss is achieved for the proposed antenna. © 2005 Wiley Periodicals, Inc. Microwave Opt Technol Lett 48: 222–226, 2006; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.21312

Functional Role and Affinity of Inorganic Cations in Stabilizing the Tetrameric Structure of the KcsA K+ Channel

Crystal structures of the tetrameric KcsA K+ channel reveal seven distinct binding sites for K+ ions within the central pore formed at the fourfold rotational symmetry axis. Coordination of an individual K+ ion by eight protein oxygen atoms within the selectivity filter suggests that ion-subunit bridging by cation–oxygen interactions contributes to structural stability of the tetramer. To test this hypothesis, we examined the effect of inorganic cations on the temperature dependence of the KcsA tetramer as monitored by SDS-PAGE. Inorganic cations known to permeate or strongly block K+ channels (K+, Rb+, Cs+, Tl+, NH4 +, Ba2+, and Sr2+) confer tetramer stability at higher temperatures (T0.5 range = 87°C to >99°C) than impermeant cations and weak blockers (Li+, Na+, Tris+, choline+; T0.5 range = 59°C to 77°C). Titration of K+, Ba2+, and other stabilizing cations protects against rapid loss of KcsA tetramer observed in 100 mM choline Cl at 90°C. Tetramer protection titrations of K+, Rb+, Cs+, Tl+, and NH4 + at 85°C or 90°C exhibit apparent Hill coefficients (N) ranging from 1.7 to 3.3 and affinity constants (K0.5) ranging from 1.1 to 9.6 mM. Ba2+ and Sr2+ titrations exhibit apparent one-site behavior (N ≅ 1) with K0.5 values of 210 nM and 11 μM, respectively. At 95°C in the presence of 5 mM K+, titration of Li+ or Na+ destabilizes the tetramer with K0.5 values of 57 mM and 109 mM, respectively. We conclude that specific binding interactions of inorganic cations with the selectivity filter are an important determinant of tetramer stability of KscA.

Unsteady hydrodynamic effect of rotation on steady rigid-body motion

Owing to the inertial effect of the flow, an unsteady hydrodynamic force will act on a particle of arbitrary shape undergoing a steady rigid-body motion with small but finite Reynolds number if the axis of rotation of the particle is not its axis of rotational symmetry. Unsteady flow field is generated owing to such rotation of the body and as a result the particle experiences a time-dependent translational resistance. In this paper, we analyse this time-dependent hydrodynamic force and obtain its higher-order correction by systematically expanding the Navier–Stokes equation in small Reynolds number.

Effect of a Hydrodynamic Flow on the Orientation Profiles of a Nematic Liquid Crystal Around a Spherical Particle

ABSTRACT We study the effect of a hydrodynamic flow on the orientational configuration of a nematic liquid crystal around a spherical particle imposing rigid normal anchoring. We discuss the cases of large Ericksen numbers (characterizing the ratio of viscous force to the elastic force), where strong elastic deformation by the flow is expected. We assume the Stokes flow profile and deal with a simplified dynamic equation of the orientational order parameter of a second-rank tensor Q αβ. We consider the cases where a particle accompanying a hyperbolic hedgehog or a Saturn ring is subjected to a flow along the axis of rotational symmetry. In both cases the orientation profiles exhibit considerable deformation by a flow.