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.     

Formation of tetra(ethylene oxide) terminated Si-C linked monolayers and their derivatization with glycine: an example of a generic strategy for the immobilization of biomolecules on silicon

Surface modification with oligo(ethylene oxide) functionalized monolayers terminated with reactive headgroups constitutes a powerful strategy to provide specific coupling of biomolecules with simultaneous protection from nonspecific adsorption on surfaces for the preparation of biorecognition interfaces. To date, oligo(ethylene oxide) functionalized monolayer-forming molecules which can be activated for attachment of biomolecules but which can selectively form monolayers onto hydrogen terminated silicon have yet to be developed. Here, self-assembled monolayers (SAMs) containing tetra(ethylene oxide) moieties protected with tert-butyl dimethylsilyl groups were formed by thermal hydrosilylation of alkenes with single-crystal Si(111)-H. The protection group was used to avoid side reactions with the hydride terminated silicon surface. Monolayer formation was carried out using solutions of the alkene in the high-boiling-point solvent 1,3,5-triethylbenzene. The protecting group was removed under very mild acidic conditions to yield a free hydroxyl functionality, a convenient surface moiety for coupling of biological entities via carbamate bond formation. The chemical composition and structure of the monolayers before and after deprotection were characterized by X-ray photoelectron spectroscopy (XPS) and X-ray reflectometry. To demonstrate the utility of this surface for covalent modification, two reagents were compared and contrasted for their ability to activate the surface hydroxyl groups for coupling of free amines, carbonyl diimidazole (CDI), and disuccinimidyl carbonate (DSC). Analysis of XP spectra before and after activation by CDI or DSC, and after subsequent reaction with glycine, provided quantitative information on the extent of activation and overall coupling efficiencies. CDI activated surfaces gave poor coupling yields under various conditions, whereas DSC mediated activation followed by aminolysis at neutral pH was found to be an efficient method for the immobilization of amines on tetra(ethylene oxide) modified surfaces.     

Colloidal nano-MOFs nucleate and stabilize ultra-small quantum dots of lead bromide perovskites

The development of synthetic routes to access stable, ultra-small (i.e. <5 nm) lead halide perovskite (LHP) quantum dots (QDs) is of fundamental and technological interest. The considerable challenges include the high solubility of the ionic LHPs in polar solvents and aggregation to form larger particles. Here, we demonstrate a simple and effective host–guest strategy for preparing ultra-small lead bromide perovskite QDs through the use of nano-sized MOFs that function as nucleating and host sites. Cr₃O(OH)(H₂O)₂(terephthalate)₃ (Cr-MIL-101), made of large mesopore-sized pseudo-spherical cages, allows fast and efficient diffusion of perovskite precursors within its pores, and promotes the formation of stable, ∼3 nm-wide lead bromide perovskite QDs. CsPbBr₃, MAPbBr₃ (MA⁺ = methylammonium), and (FA)PbBr₃ (FA⁺ = formamidinium) QDs exhibit significantly blue-shifted emission maxima at 440 nm, 446 nm, and 450 nm, respectively, as expected for strongly confined perovskite QDs. Optical characterization and composite modelling confirm that the APbBr₃ (A = Cs, MA, FA) QDs owe their stability within the MIL-101 nanocrystals to both short- and long-range interfacial interactions with the MOF pore walls.

We demonstrate a simple and effective host–guest strategy for preparing ultra-small lead bromide perovskite QDs through the use of nano-sized MOFs that function as nucleating and host sites.     

Synthesis of (chlorovinyl)cobaloxime complexes,model complexes of proposed intermediates in the B12-catalyzed dehalogenation of chlorinated ethylenes

(Chlorovinyl)cobaloxime complexes, (dmgH)2(py)CoR (R = cis-dichlorovinyl, trans-2-chlorovinyl or 1-chlorovinyl, dmgH = dimethylglyoximate) were synthesized from the reaction of chlorinated ethylenes with in situ generated (dmgH)2Co(I) in the presence of pyridine.     

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.     

Ribosomal synthesis of unnatural peptides

Combinatorial libraries of non-biological polymers and drug-like peptides could in principle be synthesized from unnatural amino acids by exploiting the broad substrate specificity of the ribosome. The ribosomal synthesis of such libraries would allow rare functional molecules to be identified using technologies developed for the in vitro selection of peptides and proteins. Here, we use a reconstituted E. coli translation system to simultaneously re-assign 35 of the 61 sense codons to 12 unnatural amino acid analogues. This reprogrammed genetic code was used to direct the synthesis of a single peptide containing 10 different unnatural amino acids. This system is compatible with mRNA-display, enabling the synthesis of unnatural peptide libraries of 10(14) unique members for the in vitro selection of functional unnatural molecules. We also show that the chemical space sampled by these libraries can be expanded using mutant aminoacyl-tRNA synthetases for the incorporation of additional unnatural amino acids or by the specific posttranslational chemical derivitization of reactive groups with small molecules. This system represents a first step toward a platform for the synthesis by enzymatic tRNA aminoacylation and ribosomal translation of cyclic peptides comprised of unnatural amino acids that are similar to the nonribosomal peptides.     

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.     

Kinetics and mechanistic details of bulk ZnO dissolution using a thiol–imidazole system

Oxide dissolution is important for metal extraction from ores and has become an attractive route for the preparation of inks for thin film solution deposition; however, oxide dissolution is often kinetically challenging. While binary “alkahest” systems comprised of thiols and N-donor species, such as amines, are known to dissolve a wide range of oxides, the mechanism of dissolution and identity of the resulting solute(s) remain unstudied. Here, we demonstrate facile dissolution of both bulk synthetic and natural mineral ZnO samples using an “alkahest” that operates via reaction with thiophenol and 1-methylimidazole (MeIm) to give a single, pseudotetrahedral Zn(SPh)₂(MeIm)₂ molecular solute identified by X-ray crystallography. The kinetics of ZnO dissolution were measured using solution ¹H NMR, and the reaction was found to be zero-order in the presence of excess ligands, with more electron withdrawing para-substituted thiophenols resulting in faster dissolution. A negative entropy of activation was measured by Eyring analysis, indicating associative ligand binding in, or prior to, the rate determining step. Combined experimental and computational surface binding studies on ZnO reveal stronger, irreversible thiophenol binding compared to MeIm, leading to a proposed dissolution mechanism initiated by thiol binding to the ZnO surface with the liberation of water, followed by alternating MeIm and thiolate ligand additions, and ultimately cleavage of the ligated zinc complex from the ZnO surface. Design rules garnered from the mechanistic insight provided by this study should inform the dissolution of other bulk oxides into inks for solution processed thin films.

Oxide dissolution is important for metal extraction from ores and has become an attractive route for the preparation of inks for thin film solution deposition; however, oxide dissolution is often kinetically challenging.