A note on Hadwiger’s conjecture for W5-free graphs with independence number two

The Hadwiger number of a graph $G$, denoted $h\left(G\right)$, is the largest integer $t$ such that $G$ contains $K_t$ as a minor. A famous conjecture due to Hadwiger in 1943 states that for every graph $G$, $h\left(G\right)\ge \chi \left(G\right)$, where $\chi \left(G\right)$ denotes the chromatic number of $G$. Let $\alpha \left(G\right)$ denote the independence number of $G$. A graph is $H$-free if it does not contain the graph $H$ as an induced subgraph. In 2003, Plummer, Stiebitz and Toft proved that $h\left(G\right)\ge \chi \left(G\right)$ for all $H$-free graphs $G$ with $\alpha \left(G\right)\le 2$, where $H$ is any graph on four vertices with $\alpha \left(H\right)\le 2$, $H=C_5$, or $H$ is a particular graph on seven vertices. In 2010, Kriesell subsequently generalized the statement to include all forbidden subgraphs $H$ on five vertices with $\alpha \left(H\right)\le 2$. In this note, we prove that $h\left(G\right)\ge \chi \left(G\right)$ for all $W_5$-free graphs $G$ with $\alpha \left(G\right)\le 2$, where $W_5$ denotes the wheel on six vertices.     

Linear synthesis of the tumor-associated carbohydrate antigens Globo-H,SSEA-3, and Gb3

The tumor-associated carbohydrate antigens Globo-H, SSEA-3, and Gb3 were synthesized in a linear fashion using glycosyl phosphate monosaccharide building blocks. All of the building blocks were prepared from readily available common precursors. The difficult alpha-(1-->4-cis)-galactosidic linkage was installed using a galactosyl phosphate donor with high selectivity. Introduction of the beta-galactosamine unit required the screening a variety of amine protecting groups to ensure good donor reactivity and protecting group compatibility. An N-trichloroacetyl-protected galactosamine donor performed best for the installation of the beta-glycosidic linkage. Conversion of the trichloroacetyl group to the N-acetyl group was achieved under mild conditions, fully compatible with the presence of multiple glycosidic bonds. This synthetic strategy is expected to be amenable to the synthesis of the globo-series of tumor antigens on solid-support.     

The Dynamics of the Neuropeptide Y Receptor Type 1 Investigated by Solid-State NMR and Molecular Dynamics Simulation

We report data on the structural dynamics of the neuropeptide Y (NPY) G-protein-coupled receptor (GPCR) type 1 (Y1R), a typical representative of class A peptide ligand GPCRs, using a combination of solid-state NMR and molecular dynamics (MD) simulation. First, the equilibrium dynamics of Y1R were studied using ¹⁵N-NMR and quantitative determination of ¹H-¹³C order parameters through the measurement of dipolar couplings in separated-local-field NMR experiments. Order parameters reporting the amplitudes of the molecular motions of the C-H bond vectors of Y1R in DMPC membranes are 0.57 for the Cα sites and lower in the side chains (0.37 for the CH₂ and 0.18 for the CH₃ groups). Different NMR excitation schemes identify relatively rigid and also dynamic segments of the molecule. In monounsaturated membranes composed of longer lipid chains, Y1R is more rigid, attributed to a higher hydrophobic thickness of the lipid membrane. The presence of an antagonist or NPY has little influence on the amplitude of motions, whereas the addition of agonist and arrestin led to a pronounced rigidization. To investigate Y1R dynamics with site resolution, we conducted extensive all-atom MD simulations of the apo and antagonist-bound state. In each state, three replicas with a length of 20 μs (with one exception, where the trajectory length was 10 μs) were conducted. In these simulations, order parameters of each residue were determined and showed high values in the transmembrane helices, whereas the loops and termini exhibit much lower order. The extracellular helix segments undergo larger amplitude motions than their intracellular counterparts, whereas the opposite is observed for the loops, Helix 8, and termini. Only minor differences in order were observed between the apo and antagonist-bound state, whereas the time scale of the motions is shorter for the apo state. Although these relatively fast motions occurring with correlation times of ns up to a few µs have no direct relevance for receptor activation, it is believed that they represent the prerequisite for larger conformational transitions in proteins.     

Halophilic reactions of a stable silylene with chloro and bromocarbons

A number of disilanes have been synthesized from a stable silylene, 1 (N,N'-di-tert-butyl-1,3-diaza-2-silacyclopent-4-en-2-ylidene), and a variety of halocarbons. It is proposed that disilane formation is a result of an initial halophilic interaction between the silylene and halocarbon. Formation of disilanes from 1 and CCl4, 2a, CHCl3, 2b, CH2Cl2, 2c, benzyl chloride, 2d, and bromobenzene, 5, are described here. An X-ray crystal structure of 2b was determined.     

Perturbed angular distributions in the presence of isotropic paramagnetic relaxation

The effect of paramagnetic relaxation on perturbed angular distributions is treated for nuclei interacting with their electronic shells via isotropic hyperfine interaction. The conditions are given under which Blume's analytical stochastic-model result for the nuclear perturbation factorsG _k (t) can be derived quantum mechanically. Systems with arbitrary nuclear spin, but electronic spinS=1/2 may be calculated without resorting to the assumption necessary forS>1/2. Explicit closed expressions forG _k (t) can be found for this particular case.     

Mode-coupling approach to the dynamics of simple liquids

Summary A new derivation of the mode-coupling theory of liquid dynamics is presented which is applicable to both quantum and classical systems. A clear distinction between the basic underlying hyphotheses and the more technical approximations has been attempted. Asymptotic corrections to the dynamics of modes due to mode-coupling effects and selfconsistent mode-coupling theories emerge from the original idea as two markedly different branches of application. Paper presented at the workshop ?Highlights on Simple Liquids?, held in Turin at ISI on 1–3 May, 1989.     

Reaction Kinetics and Oxidation Mechanisms of the Conversion of Pyrite to Ferrous Sulphate: A Mössbauer Spectroscopy Study

Pyrite undergoes a series of exothermic reactions during mine roasting to porous hematite. At low temperatures, the first non-refractive phase to form is ferrous sulphate and could be a cheaper alternative to hematite roasting for the mining industry. In this study, pyrite powder is heated in air at temperatures between 200 and 370 °C for 1 to 256 h in a temperature and time series. The rate of oxidation of pyrite to ferrous sulphate is modelled by combining the Arrhenius equation with the Weibull function to extract reliable thermodynamic data, including the energy of activation, the frequency factor and the overall order of reaction. From the thermodynamic data obtained, two possible oxidation mechanisms are recognized, depending on the bond dissociation energies of the S–S and Fe–S bonds in pyrite.