Informational Measure of Symmetry vs. Voronoi Entropy and Continuous Measure of Entropy of the Penrose Tiling. Part II of the “Voronoi Entropy vs. Continuous Measure of Symmetry of the Penrose Tiling”

The notion of the informational measure of symmetry is introduced according to: Hsym(G)=−∑i=1kP(Gi)lnP(Gi), where P(Gi) is the probability of appearance of the symmetry operation Gi within the given 2D pattern. Hsym(G) is interpreted as an averaged uncertainty in the presence of symmetry elements from the group G in the given pattern. The informational measure of symmetry of the “ideal” pattern built of identical equilateral triangles is established as Hsym(D3)= 1.792. The informational measure of symmetry of the random, completely disordered pattern is zero, Hsym=0. The informational measure of symmetry is calculated for the patterns generated by the P3 Penrose tessellation. The informational measure of symmetry does not correlate with either the Voronoi entropy of the studied patterns nor with the continuous measure of symmetry of the patterns. Quantification of the “ordering” in 2D patterns performed solely with the Voronoi entropy is misleading and erroneous.

The Role of Symmetry in Non-Hermitian Scattering1

We review recent work on asymmetric scattering by Non-Hermitian (NH) Hamiltonians. Quantum devices with an asymmetric scattering response to particles incident from right or left in effective ID waveguides will be important to develop quantum technologies. They act as microscopic equivalents of familiar macroscopic devices such as diodes, rectifiers, or valves. The symmetry of the underlying NH Hamiltonian leads to selection rules which restrict or allow asymmetric response. NH-symmetry operations may be organized into group structures that determine equivalences among operations once a symmetry is satisfied. The NH Hamiltonian posseses a particular symmetry if it is invariant with respect to the corresponding symmetry operation, which can be conveniently expressed by a unitary or antiunitary superoperator. A simple group is formed by eight symmetry operations, which include the ones for Parity-Time symmetry and Hermiticity as specific cases. The symmetries also determine the structure of poles and zeros of the S matrix. The ground-state potentials for two-level atoms crossing properly designed laser beams realize different NH symmetries to achieve transmission or reflection asymmetries.

Informational Measure of Symmetry vs. Voronoi Entropy and Continuous Measure of Entropy of the Penrose Tiling. Part II of the “Voronoi Entropy vs. Continuous Measure of Symmetry of the Penrose Tiling”.

The notion of the informational measure of symmetry is introduced according to: HsymG=-i=1kPGilnPGi, where PGi is the probability of appearance of the symmetry operation Gi within the given 2D pattern. HsymG is interpreted as an averaged uncertainty in the presence of symmetry elements from the group G in the given pattern. The informational measure of symmetry of the “ideal” pattern built of identical equilateral triangles is established as HsymD3=1.792. The informational measure of symmetry of the random, completely disordered pattern is zero, Hsym=0. Informational measure of symmetry is calculated for the patterns generated by the P3 Penrose tessellation. Informational measure of symmetry does not correlate neither with the Voronoi entropy of the studied patterns nor with the continuous measure of symmetry of the patterns.

Synthesis, Structural Studies, and Anticancer Properties of [CuBr(PPh3)2(4,6-Dimethyl-2-Thiopyrimidine-κS]

CuBr(PPh3)2(4,6-dimethylpyrimidine-2-thione) (Cu-L) was synthesized by stirring CuBr(PPh3)3 and 4,6-dimethylpyrimidine-2-thione in dichloromethane. The crystal structure of Cu-L was obtained, and indicated that the complex adopts a distorted tetrahedral structure with several intramolecular hydrogen bonds. Moreover, a centrosymmetric dimer is formed by the intermolecular hydrogen bonding of the bromine acceptor created by symmetry operation 1−x, 1−y, 1−z to the methyl group (D3 = C42) of the pyrimidine–thione ligand. HSA-binding of Cu-L and its ligand were evaluated, revealing that Cu-L binds to HSA differently than its ligand. The HSA-bindings were modeled by molecular docking, which suggested that Cu-L binds to the II A domain while L binds between the I B and II A domains. Anticancer activities toward OVCAR-3 and HeLa cell lines were tested and indicated the significance of the copper center in enhancing the cytotoxic effect; negligible toxicities for L and Cu-L were observed towards a non-cancer cell line. The current study highlights the potential of copper(I)-phosphine complexes containing thione ligands as therapeutic agents.

Removing the C-terminal protecting group enlarges the crystal size: Z–(Gly–Aib)2–OH·H2O

The achiral tetrapeptide monohydrate N-(benzyloxycarbonyl)glycyl-α-aminoisobutyrylglycyl-α-aminoisobutyric acid monohydrate, Z–Gly–Aib–Gly–Aib–OH·H2O (Z is benzyloxycarbonyl, Aib is α-aminoisobutyric acid and Gly is glycine) or C20H28N4O7·H2O, exhibits two conformations related by the symmetry operation of an inversion centre. It adopts only one of two possible intramolecular hydrogen bonds in a type I (and I′) β-turn and forms a maximum of intermolecular hydrogen bonds partly mediated by water. The space group, the molecular structure and the crystal packing differ from two already described (Gly–Aib)2 peptides which vary only in the protecting groups. This structure confirms the high structural flexibility of Gly–Aib peptides and points to a strong relationship between intermolecular hydrogen bonding and crystal quality and size.

A symmetry-derived mechanism for atomic resolution imaging

We introduce an image-contrast mechanism for scanning transmission electron microscopy (STEM) that derives from the local symmetry within the specimen. For a given position of the electron probe on the specimen, the image intensity is determined by the degree of similarity between the exit electron-intensity distribution and a chosen symmetry operation applied to that distribution. The contrast mechanism detects both light and heavy atomic columns and is robust with respect to specimen thickness, electron-probe energy, and defocus. Atomic columns appear as sharp peaks that can be significantly narrower than for STEM images using conventional disk and annular detectors. This fundamentally different contrast mechanism complements conventional imaging modes and can be acquired simultaneously with them, expanding the power of STEM for materials characterization.

Norpsilocin: freebase and fumarate salt

The solid-state structures of the naturally occurring psychoactive tryptamine norpsilocin {4-hydroxy-N-methyltryptamine (4-HO-NMT); systematic name: 3-[2-(methylamino)ethyl]-1H-indol-4-ol}, C11H14N2O, and its fumarate salt (4-hydroxy-N-methyltryptammonium fumarate; systematic name: bis{[2-(4-hydroxy-1H-indol-3-yl)ethyl]methylazanium} but-2-enedioate), C11H15N2O+·0.5C4H2O4 2−, are reported. The freebase of 4-HO-NMT has a single molecule in the asymmetric unit joined together by N—H...O and O—H...O hydrogen bonds in a two-dimensional network parallel to the (100) plane. The ethylamine arm of the tryptamine is modeled as a two-component disorder with a 0.895 (3) to 0.105 (3) occupancy ratio. The fumarate salt of 4-HO-NMT crystallizes with a tryptammonium cation and one half of a fumarate dianion in the asymmetric unit. The ions are joined together by N—H...O and O—H...O hydrogen bonds to form a three-dimensional framework, as well as π–π stacking between the six-membered rings of inversion-related indoles (symmetry operation: 2 − x, 1 − y, 2 – z).

Flavor Mixing and the Permutation Symmetry among Generations

In the standard model, the permutation symmetry among the three generations of fundamental fermions is usually regarded to be broken by the Higgs couplings. It is found that the symmetry is restored if we include the mass matrix parameters as physical variables which transform appropriately under the symmetry operation. Known relations between these variables, such as the renormalization group equations, as well as formulas for neutrino oscillations (in vacuum and in matter), are shown to be covariant tensor equations under the permutation symmetry group.

Generating a High Valency Biotin Binder by Selecting Uniform Protein Assemblies via Crystallization

Crystallization is a common practice in the purification process in small molecule synthesis while selecting the wanted product. For proteins it is rarely applied due to the methodological predicaments in obtaining crystals. Our observation of the stabilized octamers in the crystal structure of hoefavidin, a novel dimeric member of the avidin family, led to the notion of developing a novel biotechnological tool via covalent crosslinking. The avidin–biotin system has been exploited for decades utilizing the ultra-high affinity between avidin and biotin as a basis for numerous applications. Optimizing the system led to the discovery of a novel group of dimeric avidins including hoefavidin. Hoefavidin has a dynamic quaternary structure, where a dimer is the basis for generating the octamer via crystallographic symmetry operation. Upon biotin binding in solution hoefavidin dissociates solely into dimers. In order to stabilize the octamer, we designed the P61C mutant to form a disulfide bridge stabilizing the octamer and preventing dissociation upon biotin binding. The process of selecting P61C hoefavidin uniform octamers includes crystallization followed by dissolving the crystals. The P61C modified hoefavidin octamer can have a substantial added value to the various biotechnological applications and advances of the biotin based high affinity systems.

The chromatic symmetry of twins and allotwins

The symmetry of twins is described by chromatic point groups obtained from the intersection group {\cal H}^* of the oriented point groups of the individuals {\cal H}_i extended by the operations mapping different individuals. This article presents a revised list of twin point groups through the analysis of their groupoid structure, followed by the generalization to the case of allotwins. Allotwins of polytypes with the same type of point group can be described by a chromatic point group like twins. If the individuals are all differently oriented, the chromatic point group is obtained in the same way as in the case of twins; if they are mapped by symmetry operation of the individuals, the chromatic point group is neutral. If the same holds true for some but not all individuals, then the allotwin can be seen as composed of twinned regions described by a twin point group, that are then allotwinned and described by a colour identification group; the allotwin is then described by a chromatic group obtained as an extension of the former by the latter, and requires the use of extended symbols reminiscent of the extended Hermann–Mauguin symbols of space groups. In the case of allotwins of polytypes with different types of point groups, as well as incomplete (allo)twins, a chromatic point group does not reveal the full symmetry: the groupoid has to be specified instead.