Note on a conjecture of toft

A conjecture of Toft [17] asserts that any 4-critical graph (or equivalently, every 4-chromatic graph) contains a fully odd subdivision ofK ₄. We show that if a graphG has a degree three nodev such thatG-v is 3-colourable, then eitherG is 3-colourable or it contains a fully oddK ₄. This resolves Toft's conjecture in the special case where a 4-critical graph has a degree three node, which is in turn used to prove the conjecture for line-graphs. The proof is constructive and yields a polynomial algorithm which given a 3-degenerate graph either finds a 3-colouring or exhibits a subgraph that is a fully odd subdivision ofK ₄. (A graph is 3-degenerate if every subgraph has some node of degree at most three.)     

A density functional theory computational study of the role of ligand on the stability of CuI and CuII species associated with ATRP and SET‐LRP

Atom transfer radical polymerization (ATRP) and single electron‐transfer living radical polymerization (SET‐LRP) both utilize copper complexes of various oxidation states with N‐ligands to perform their respective activation and deactivation steps. Herein, we utilize DFT (B3YLP) methods to determine the preferred ligand‐binding geometries for Cu/N‐ligand complexes related to ATRP and SET‐LRP. We find that those ligands capable of achieving tetrahedral complexes with Cu^I and trigonal bipyramidal with axial halide complexes with [Cu^IIX]⁺ have higher energies of stabilization. We were able to correlate calculated preferential stabilization of [Cu^IIX]⁺ with those ligands that perform best in SET‐LRP. A crude calculation of energy of disproportionation revealed that the same preferential binding of [Cu^IIX]⁺ results in increased propensity for disproportionation. Finally, by examining the relative energies of the basic steps of ATRP and SET‐LRP, we were able to rationalize the transition from the ATRP mechanism to the SET‐LRP mechanism as we transition from typical nonpolar ATRP solvents to polar SET‐LRP solvents. © 2007 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 45: 4950–4964, 2007     

Practical syntheses of [1,3,5]‐triazine dendritic molecules on solid supports

A practical and divergent synthesis of supported [1,3,5]‐triazine dendritic molecules on Wang resin, PEGA resin, SynPhase? Lanterns, and silica gel is described. The alkylamine linkers used allow derivatization with functionality for both synthetic (e.g., supported reagent and scavenger activity) and chemical biology applications. The use of supported intermediates allows differentiation of symmetric linkers without the need for protecting group chemistry. The synthetic route uses inexpensive, readily available starting materials in a straightforward and scaleable strategy. © 2006 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 44: 2248–2259, 2006     

Low temperature specific heat calculations for amorphous polystyrene

Numerous attempts have previously been made to explain low temperature specific heat anomalies in glasses. The potential value of low frequency Raman data in interpreting such results is demonstrated here: the density of states for the 60 cm^?1 side-group mode in amorphous polystyrene is calculated from Raman measurements. Using the harmonic approximation, the contribution to the specific heat from this side-group motion is calculated and added to the Debye term. The form of C/T³ between 1.5 and 4 K is shown to be reproduced, although the magnitude of C/T³ is only about 70% of experimental values. This discrepancy and the anomalous behaviour below 1.5 K is believed to the due to lower-lying vibrational states of the system, perhaps not accessible through Raman scattering experiments.     

Stereoselective reductase-catalysed deoxygenation of sulfoxides in aerobic and anaerobic bacteria

Direct and indirect evidence, of unexpected stereoselective reductase-catalysed deoxygenations of sulfoxides, was found. The deoxygenations proceeded simultaneously, with the expected dioxygenase-catalysed asymmetric sulfoxidation of sulfides, during some biotransformations with the aerobic bacterium Pseudomonas putida UV4. Stereoselective reductase-catalysed asymmetric deoxygenation of racemic alkylaryl, dialkyl and phenolic sulfoxides was observed, without evidence of the reverse sulfoxidation reaction, using anaerobic bacterial strains. A purified dimethyl sulfoxide reductase, obtained from the intact cells of the anaerobic bacterium Citrobacter braakii DMSO 11, yielded, from the corresponding racemates, enantiopure alkylaryl sulfoxide and thiosulfinate samples.     

Mechanistic implications for the formation of the diiron cluster in ribonucleotide reductase provided by quantitative EPR spectroscopy

The small subunit of Escherichia coli ribonucleotide reductase (R2) is a homodimeric (betabeta) protein, in which each beta-peptide contains a diiron cluster composed of two inequivalent iron sites. R2 is capable of reductively activating O(2) to produce a stable tyrosine radical (Y122*), which is essential for production of deoxyribonucleotides on the larger R1 subunit. In this work, the paramagnetic Mn(II) ion is used as a spectroscopic probe to characterize the assembly of the R2 site with EPR spectroscopy. Upon titration of Mn(II) into samples of apoR2, we have been able to quantitatively follow three species (aquaMn(II), mononuclear Mn(II)R2, and dinuclear Mn(2)(II)R2) and fit each to a sequential two binding site model. As previously observed for Fe(II) binding within apoR2, one of the sites has a greater binding affinity relative to the other, K(1) = (5.5 +/- 1.1) x 10(5) M(-)(1) and K(2) = (3.9 +/- 0.6) x 10(4) M(-)(1), which are assigned to the B and A sites, respectively. In multiple titrations, only one dinuclear Mn(2)(II)R2 site was created per homodimer of R2, indicating that only one of the two beta-peptides of R2 is capable of binding Mn(II) following addition of Mn(II) to apoR2. Under anaerobic conditions, addition of only 2 equiv of Fe(II) to R2 (Fe(2)(II)R2) completely prevented the formation of any bound MnR2 species. Upon reaction of this sample with O(2) in the presence of Mn(II), both Y122* and Mn(2)(II)R2 were produced in equal amounts. Previous stopped-flow absorption spectroscopy studies have indicated that apoR2 undergoes a protein conformational change upon binding of metal (Tong et al. J. Am. Chem. Soc. 1996, 118, 2107-2108). On the basis of these observations, we propose a model for R2 metal incorporation that invokes an allosteric interaction between the two beta-peptides of R2. Upon binding the first equiv of metal to a beta-peptide (beta(I)), the aforementioned protein conformational change prevents metal binding in the adjacent beta-peptide (beta(II)) approximately 25 A away. Furthermore, we show that metal incorporation into beta(II) occurs only during the O(2) activation chemistry of the beta(I)-peptide. This is the first direct evidence of an allosteric interaction between the two beta-peptides of R2. Furthermore, this model can explain the generally observed low Fe occupancy of R2. We also demonstrate that metal uptake and this newly observed allosteric effect are buffer dependent. Higher levels of glycerol cause loss of the allosteric effect. Reductive cycling of samples in the presence of Mn(II) produced a novel mixed metal Fe(III)Mn(III)R2 species within the active site of R2. The magnitude of the exchange coupling (J) determined for both the Mn(2)(II)R2 and Fe(III)Mn(III)R2 species was determined to be -1.8 +/- 0.3 and -18 +/- 3 cm(-)(1), respectively. Quantitative spectral simulations for the Fe(III)Mn(III)R2 and mononuclear Mn(II)R2 species are provided. This work represents the first instance where both X- and Q-band simulations of perpendicular and parallel mode spectra were used to quantitatively predict the concentration of a protein bound mononuclear Mn(II) species.     

An investigation of mononuclear and binuclear palladium(II) complexes of egta (egta = glycine,N,N′-(1,2-ethanediylbis(oxy-2,1-ethanediyl)bis[N-carboxymethyl]) by 1H-, 13C- and 15N-n.m.r. methods

1:1 and 2:1 palladium(II) complexes of egta^4– (egta^4– = glycine, N,N-(1,2-ethanediylbis)(oxy-2,1-ethanediyl)bis[N-carboxymethyl]) were prepared by 1:1 and 2:1 addition of K₂PdCl₄ to K₄egta, and examined by ¹H-, ¹³C- and ¹⁵N-n.m.r. methods. The 1:1 complex, [Pd(egta)]^2– in solution, utilizes a square-planar coordination comprised of two nitrogen and two glycinato carboxylate donors of egta^4–, leaving two glycinato carboxylates pendant. The complex has a cis-(R,S) stereochemistry which places both pendant carboxylates below the PdN₂O₂ square plane and the tether backbone of egta^4– in the up, up sense above the same plane. The cis-(R,S) assignment was assisted by computer simulations of the ¹³C-n.m.r. spectrum for four possible isomers. Only cis-(R,S) and trans-(R,R) calculated ¹³C-spectra were compatible with the observed ¹³C-n.m.r. pattern. The HH NOESY spectrum of [Pd(egta)]^2– detects long range coupling of the backbone –OCH₂CH₂O– linkage with both coordinated and pendant glycinato CH₂ moieties. The cis-(R,S) isomer's tortional movements allow such contacts whereas a trans-(R,R) isomer does not. The 2:1 complex, [Pd₂(egta)(H₂O)₂] in solution has an extended-chain structure with each palladium(II) center coordinated in the mer-iminodiacetate-like coordination with two bound glycinato-functionalities.     

Near-perfect matrices

A 0, 1 matrixA isnear-perfect if the integer hull of the polyhedron {x0: Ax } can be obtained by adding one extra (rank) constraint. We show that in general, such matrices arise as the cliquenode incidence matrices of graphs. We give a colouring-like characterization of the corresponding class of near-perfect graphs which shows that one need only check integrality of a certain linear program for each 0, 1, 2-valued objective function. This in contrast with perfect matrices where it is sufficient to check 0, 1-valued objective functions. We also make the following conjecture: a graph is near-perfect if and only if sequentially lifting any rank inequality associated with a minimally imperfect graph results in the rank inequality for the whole graph. We show that the conjecture is implied by the Strong Perfect Graph Conjecture. (It is also shown to hold for graphs with no stable set of size eleven.) Our results are used to strengthen (and give a new proof of) a theorem of Padberg. This results in a new characterization of minimally imperfect graphs: a graph is minimally imperfect if and only if both the graph and its complement are near-perfect.The research has partially been done when the author visited Mathematic Centrum, CWI, Amsterdam, The Netherlands.