Broken symmetry and strangeness of the semiconductor impurity band metal-insulator transition

The filamentary model of the metal-insulator transition in randomly doped semiconductor impurity bands is geometrically equivalent to similar models for continuous transitions in dilute antiferromagnets and even to the λ transition in liquid He, but the critical behaviors are different. The origin of these differences lies in two factors: quantum statistics and the presence of long range Coulomb forces on both sides of the transition in the electrical case. In the latter case, in addition to the main transition, there are two satellite transitions associated with disappearance of the filamentary structure in both insulating and metallic phases. These two satellite transitions were first identified by Fritzsche in 1958, and their physical origin is explained here in geometrical and topological terms that facilitate calculation of critical exponents.

Broken symmetry and critical transport properties of random metals

Recent experimental data on the conductivity σ+(T), T → 0, on the metallic side of the metal–insulator transition in ideally random (neutron transmutation-doped) 70Ge:Ga have shown that σ+(0) ∝ (N − Nc)μ with μ = ½, confirming earlier ultra-low-temperature results for Si:P. This value is inconsistent with theoretical predictions based on diffusive classical scaling models, but it can be understood by a quantum-directed percolative filamentary amplitude model in which electronic basis states exist which have a well-defined momentum parallel but not normal to the applied electric field. The model, which is based on a new kind of broken symmetry, also explains the anomalous sign reversal of the derivative of the temperature dependence in the critical regime.

Electrostatic interaction between helical macromolecules in dense aggregates: An impetus for DNA poly- and meso-morphism

DNA exhibits a surprising multiplicity of structures when it is packed into dense aggregates. It undergoes various polymorphous transitions (e.g., from the B to A form) and mesomorphous transformations (from hexagonal to orthorhombic or monoclinic packing, changes in the mutual alignment of nearest neighbors, etc). In this report we show that such phenomena may have their origin in the specific helical symmetry of the charge distribution on DNA surface. Electrostatic interaction between neighboring DNA molecules exhibits strong dependence on the patterns of molecular surface groups and adsorbed counter-ions. As a result, it is affected by such structural parameters as the helical pitch, groove width, the number of base pairs per helical turn, etc. We derive expressions which relate the energy of electrostatic interaction with these parameters and with the packing variables characterizing the axial and azimuthal alignment between neighboring macromolecules. We show, in particular, that the structural changes upon the B-to-A transition reduce the electrostatic energy by ≈kcal/mol per base pair, at a random adsorption of counter ions. Ion binding into the narrow groove weakens or inverts this effect, stabilizing B-DNA, as it is presumably the case in Li+-DNA assemblies. The packing symmetry and molecular alignment in DNA aggregates are shown to be affected by the patterns of ion binding.

Solid and liquid phase 59Co NMR studies of cobalamins and their derivatives

We describe the application of 59Co NMR to the study of naturally occurring cobalamins. Targets of these investigations included vitamin B12, the B12 coenzyme, methylcobalamin, and dicyanocobyrinic acid heptamethylester. These measurements were carried out on solutions and powders of different origins, and repeated at a variety of magnetic field strengths. Particularly informative were the solid-state central transition NMR spectra, which when combined with numerical line shape analyses provided a clear description of the cobalt coupling parameters. These parameters showed a high sensitivity to the type of ligands attached to the metal and to the crystallization history of the sample. 59Co NMR determinations also were carried out on synthetic cobaloximes possessing alkyl, cyanide, aquo, and nitrogenated axial groups, substituents that paralleled the coordination of the natural compounds. These analogs displayed coupling anisotropies comparable to those of the cobalamins, as well as systematic up-field shifts that can be rationalized in terms of their stronger binding affinity to the cobalt atom. Cobaloximes also displayed a higher regularity in the relative orientations of their quadrupole and shielding coupling tensors, reflecting a higher symmetry in their in-plane coordination. For the cobalamines, poor correlations were observed between the values measured for the quadrupole couplings in the solid and the line widths observed in the corresponding solution 59Co NMR resonances.

Twist fields, the elliptic genus, and hidden symmetry

We combine infinite dimensional analysis (in particular a priori estimates and twist positivity) with classical geometric structures, supersymmetry, and noncommutative geometry. We establish the existence of a family of examples of two-dimensional, twist quantum fields. We evaluate the elliptic genus in these examples. We demonstrate a hidden SL(2,ℤ) symmetry of the elliptic genus, as suggested by Witten.

Metabolism to a response pathway selective retinoid ligand during axial pattern formation

We report identification of 9-cis-4-oxo-retinoic acid (9-cis-4-oxo-RA) as an in vivo retinoid metabolite in Xenopus embryos. 9-Cis-4-oxo-RA bound receptors (RARs) α, β, and γ as well as retinoid X receptors (RXRs) α, β, and γ in vitro. However, this retinoid displayed differential RXR activation depending on the response pathway used. Although it failed to activate RXRs in RXR homodimers, it activated RXRs and RARs synergistically in RAR-RXR heterodimers. 9-Cis-4-oxo-RA thus acted as a dimer-specific agonist. Considering that RAR-RXR heterodimers are major functional units involved in transducing retinoid signals during embryogenesis and that 9-cis-4-oxo-RA displayed high potency for modulating axial pattern formation in Xenopus, metabolism to 9-cis-4-oxo-RA may provide a mechanism to target retinoid action to this and other RAR-RXR heterodimer-mediated processes.

Nanohedra: Using symmetry to design self assembling protein cages, layers, crystals, and filaments

A general strategy is described for designing proteins that self assemble into large symmetrical nanomaterials, including molecular cages, filaments, layers, and porous materials. In this strategy, one molecule of protein A, which naturally forms a self-assembling oligomer, An, is fused rigidly to one molecule of protein B, which forms another self-assembling oligomer, Bm. The result is a fusion protein, A-B, which self assembles with other identical copies of itself into a designed nanohedral particle or material, (A-B)p. The strategy is demonstrated through the design, production, and characterization of two fusion proteins: a 49-kDa protein designed to assemble into a cage approximately 15 nm across, and a 44-kDa protein designed to assemble into long filaments approximately 4 nm wide. The strategy opens a way to create a wide variety of potentially useful protein-based materials, some of which share similar features with natural biological assemblies.

A brief survey of symmetry in mathematics

This paper presents a brief survey of the idea of symmetry in mathematics, as exemplified by some particular developments in algebra, differential equations, topology, and number theory.

Symmetry, stability, and dynamics of multidomain and multicomponent protein systems

Symmetry is commonly observed in many biological systems. Here we discuss representative examples of the role of symmetry in structural molecular biology. Point group symmetries are observed in many protein oligomers whose three-dimensional atomic structures have been elucidated by x-ray crystallography. Approximate symmetry also occurs in multidomain proteins. Symmetry often confers stability on the molecular system and results in economical usage of basic components to build the macromolecular structure. Symmetry is also associated with cooperativity. Mild perturbation from perfect symmetry may be essential in some systems for dynamic functions.

Symmetry and the energy landscapes of biomolecules

The role of symmetry in the folding of proteins is discussed using energy landscape theory. An analytical argument shows it is much easier to find sequences with funneled energy landscape capable of fast folding if the structure is symmetric. The analogy with phase transitions of small clusters with magic numbers is discussed.

The role of symmetry in fundamental physics

The role of symmetry in fundamental physics is reviewed.

Symmetry arguments in chemistry

The use (and misuse) of symmetry arguments in constructing molecular models and in the interpretation of experimental observations bearing on molecular structure (spectroscopy, diffraction, etc.) is discussed. Examples include the development of point groups and space groups for describing the external and internal symmetry of crystals, the derivation of molecular symmetry by counting isomers (the benzene structure), molecular chirality, the connection between macroscopic and molecular chirality, pseudorotation, the symmetry group of nonrigid molecules, and the use of orbital symmetry arguments in discussing aspects of chemical reactivity.

Five-fold symmetry in crystalline quasicrystal lattices

To demonstrate that crystallographic methods can be applied to index and interpret diffraction patterns from well-ordered quasicrystals that display non-crystallographic 5-fold symmetry, we have characterized the properties of a series of periodic two-dimensional lattices built from pentagons, called Fibonacci pentilings, which resemble aperiodic Penrose tilings. The computed diffraction patterns from periodic pentilings with moderate size unit cells show decagonal symmetry and are virtually indistinguishable from that of the infinite aperiodic pentiling. We identify the vertices and centers of the pentagons forming the pentiling with the positions of transition metal atoms projected on the plane perpendicular to the decagonal axis of quasicrystals whose structure is related to crystalline η phase alloys. The characteristic length scale of the pentiling lattices, evident from the Patterson (autocorrelation) function, is ∼τ2 times the pentagon edge length, where τ is the golden ratio. Within this distance there are a finite number of local atomic motifs whose structure can be crystallographically refined against the experimentally measured diffraction data.

From symmetry to asymmetry: Phylogenetic patterns of asymmetry variation in animals and their evolutionary significance

Phylogenetic analyses of asymmetry variation offer a powerful tool for exploring the interplay between ontogeny and evolution because (i) conspicuous asymmetries exist in many higher metazoans with widely varying modes of development, (ii) patterns of bilateral variation within species may identify genetically and environmentally triggered asymmetries, and (iii) asymmetries arising at different times during development may be more sensitive to internal cytoplasmic inhomogeneities compared to external environmental stimuli. Using four broadly comparable asymmetry states (symmetry, antisymmetry, dextral, and sinistral), and two stages at which asymmetry appears developmentally (larval and postlarval), I evaluated relations between ontogenetic and phylogenetic patterns of asymmetry variation. Among 140 inferred phylogenetic transitions between asymmetry states, recorded from 11 classes in five phyla, directional asymmetry (dextral or sinistral) evolved directly from symmetrical ancestors proportionally more frequently among larval asymmetries. In contrast, antisymmetry, either as an end state or as a transitional stage preceding directional asymmetry, was confined primarily to postlarval asymmetries. The ontogenetic origin of asymmetry thus significantly influences its subsequent evolution. Furthermore, because antisymmetry typically signals an environmentally triggered asymmetry, the phylogenetic transition from antisymmetry to directional asymmetry suggests that many cases of laterally fixed asymmetries evolved via genetic assimilation.

Symmetry in chemistry from the hydrogen atom to proteins

The last 2 decades have seen discoveries in highly excited states of atoms and molecules of phenomena that are qualitatively different from the “planetary” model of the atom, and the near-rigid model of molecules, characteristic of these systems in their low-energy states. A unified view is emerging in terms of approximate dynamical symmetry principles. Highly excited states of two-electron atoms display “molecular” behavior of a nonrigid linear structure undergoing collective rotation and vibration. Highly excited states of molecules described in the “standard molecular model” display normal mode couplings, which induce bifurcations on the route to molecular chaos. New approaches such as rigid–nonrigid correlation, vibrons, and quantum groups suggest a unified view of collective electronic motion in atoms and nuclear motion in molecules.