Crossing numbers of sequences of graphs II: Planar tiles

We describe a method of creating an infinite family of crossing‐critical graphs from a single small planar map, the tile, by gluing together many copies of the tile together in a circular fashion. This method yields all known infinite families of k‐crossing‐critical graphs. Furthermore, the method yields new infinite families, which extend from (4,6) to (3.5,6) the interval of rationals r for which there is, for some k, an infinite sequence of k‐crossing‐critical graphs all having average degree r. © 2003 Wiley Periodicals, Inc. J Graph Theory 42: 332–341, 2003     

Gas permeability and n‐butane solubility of poly(1‐trimethylgermyl‐1‐propyne)

The gas permeability and n‐butane solubility in glassy poly(1‐trimethylgermyl‐1‐propyne) (PTMGP) are reported. As synthesized, the PTMGP product contains two fractions: (1) one that is insoluble in toluene and soluble only in carbon disulfide (the toluene‐insoluble polymer) and (2) one that is soluble in both toluene and carbon disulfide (the toluene‐soluble polymer). In as‐cast films, the gas permeability and n‐butane solubility are higher in films prepared from the toluene‐soluble polymer (particularly in those films cast from toluene) than in films prepared from the toluene‐insoluble polymer and increase to a maximum in both fractions after methanol conditioning. For example, in as‐cast films prepared from carbon disulfide, the oxygen permeability at 35 °C is 330 × 10^?10 cm³ (STP) cm/(cm² s cmHg) for the toluene‐soluble polymer and 73 × 10^?10 cm³ (STP) cm/(cm² s cmHg) for the toluene‐insoluble polymer. After these films are conditioned in methanol, the oxygen permeability increases to 5200 × 10^?10 cm³ (STP) cm/(cm² s cmHg) for the toluene‐soluble polymer and 6200 × 10^?10 cm³ (STP) cm/(cm² s cmHg) for the toluene‐insoluble polymer. The rankings of the fractional free volume and nonequilibrium excess free volume in the various PTMGP films are consistent with the measured gas permeability and n‐butane solubility values. Methanol conditioning increases gas permeability and n‐butane solubility of as‐cast PTMGP films, regardless of the polymer fraction type and casting solvent used, and minimizes the permeability and solubility differences between the various films (i.e., the permeability and solubility values of all conditioned PTMGP films are similar). © 2002 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 40: 2228–2236, 2002     

Crystallographic analysis of metal-ion binding to human ubiquitin

The metal-binding ability of human ubiquitin (hUb) towards a selection of biologically relevant metal ions and complexes has been probed. Different techniques have been used to obtain crystals suitable for crystallographic analysis. In the first type of experiments, crystals of hUb have been soaked in solutions containing copper(II) acetate and two metallodrugs, Zeise salt (K[PtCl(3)(η(2)-C(2)H(4))]·H(2)O) and cisplatin (cis-[PtCl(2)(NH(3))(2)]). The Zeise salt is used in a test for hepatitis, whereas cisplatin is one of the most powerful anticancer drugs in clinical use. The Zeise salt readily reacts with hUb crystals to afford an adduct with three platinum residues per protein molecule, Pt(3)-hUb. In contrast, copper(II) acetate and cisplatin were found to be unreactive for contact times up to one hour and to cause degradation of the hUb crystals for longer times. In the second type of experiments, hUb was cocrystallized with a solution of copper(II) or zinc(II) acetate or cisplatin. Zinc(II) acetate gives, at low metal-to-protein molar ratios (8:1), crystals containing one metal ion per three molecules of protein, Zn-hUb(3) (already reported in previous work), whereas at high metal-to-protein ratios (70:1) gives crystals containing three Zn(II) ions per protein molecule, Zn(3)-hUb. In contrast, once again, copper(II) acetate and cisplatin, even at low metal-to-protein ratios, do not give crystalline material. In the soaking experiment, the Zeise anion leads to simultaneous platination of His68, Met1, and Lys6. Present and previous results of cocrystallization experiments performed with Zn(II) and other Group 12 metal ions allow a comprehensive understanding of the metal-ion binding properties of hUb with His68 as the main anchoring site, followed by Met1 and carboxylic groups of Glu16, Glu18, Glu64, Asp21, and Asp32, to be reached. In the case of platinum, Lys6 can also be a binding site. The amount of bound metal ion, with respect to that of the protein, appears to be a relevant parameter influencing crystal packing.     

Evaluation of extraction methods for quantification of aqueous fullerenes in urine

There is a growing concern about the human and environmental health effects of fullerenes (e.g., C(60)) due to their increasing application in research, medicine, and industry. Toxicological and pharmacokinetic research requires standard methods for extraction and detection of fullerenes from biological matrices such as urine. The present study validates the use of liquid-liquid extraction (LLE) and solid-phase extraction (SPE) methods in conjunction with liquid chromatography-mass spectrometry (LC-MS) for the quantitative determination of C(60) in human and synthetic urine as compared with ultrapure water. Glacial acetic acid, which is necessary to prevent emulsions during LLE, inhibited C(60) detection by LC-MS, but this could be mitigated with evaporation. Aqueous C(60) aggregates (nC(60)) were spiked at 180 μg/L into the components of a synthetic urine recipe to determine their individual impacts on extraction and detection. Urea, creatinine, and a complex protein (i.e., gelatin) were found to impair SPE, leading to a low recovery rate of 43 ± 4% for C(60) spiked into human urine. In contrast, C(60) was consistently recovered from synthetic matrices using LLE, and recovery in human urine was 80 ± 6%. These results suggest that LLE combined with LC-MS is suitable for studying the clearance of fullerenes from the body. LLE is a robust technique that holds promise for extracting C(60) from other complex biological matrices (e.g., blood, sweat, amniotic fluid) in toxicological studies, enabling a better understanding of the behavior of fullerenes in human and animal systems and facilitating a more comprehensive risk evaluation of fullerenes.     

Conformational transitions of glycine induced by vibrational excitation of the O-H stretch

Vibrational energy flow and conformational transitions following excitation of the OH stretching mode of the most stable conformer of glycine are studied by classical trajectories. "On the fly" simulations with the PM3 semiempirical electronic structure method for the potential surface are used. Initial conditions are selected to correspond to the ν=1 excitation of the OH stretch. The main findings are: (1) An an equilibrium-like ratio is established between the populations of the 3 lowest-lying conformers after about 10 picoseconds. (2) There is a high probability throughout the 150 ps of the simulations for finding the molecule in geometries far from the equilibrium structures of the lowest-energy conformers. (3) Energy from the initial excited OH (ν=1) stretch flows preferentially to 5 other vibrational modes, including the bending motion of the H atom. (4) RRK theory yields conformational transition rates that deviate substantially from the classical trajectory results. Possible implication of these results for vibrational energy flow and conformational transitions in small biological molecules are discussed.     

A conjecture of Erd?s on graph Ramsey numbers

The Ramsey number r(G) of a graph G is the minimum N such that every red–blue coloring of the edges of the complete graph on N vertices contains a monochromatic copy of G. Determining or estimating these numbers is one of the central problems in combinatorics.One of the oldest results in Ramsey Theory, proved by Erd?s and Szekeres in 1935, asserts that the Ramsey number of the complete graph with m edges is at most . Motivated by this estimate Erd?s conjectured, more than a quarter century ago, that there is an absolute constant c such that for any graph G with m edges and no isolated vertices. In this short note we prove this conjecture.     

Small subgraphs of random regular graphs

The main aim of this short paper is to answer the following question. Given a fixed graph H, for which values of the degree d does a random d-regular graph on n vertices contain a copy of H with probability close to one?     

A New Lower Bound For A Ramsey-Type Problem

Let 3 ≤ r < s be fixed integers and let G be a graph on n vertices not containing a complete graph on s vertices. The main aim of this paper is to provide a new lower bound on the size of the maximum subset of G without a copy of complete graph K_r. Our results substantially improve previous bounds of Krivelevich and Bollobás and Hind. * Research supported in part by NSF grants DMS-0106589, CCR-9987845 and by the State of New Jersey. Part of this research was done while visiting Microsoft Research.