The vertex-cover polynomial of a graph

In this paper we define the vertex-cover polynomial Ψ(G,τ) for a graph G. The coefficient of τ^r in this polynomial is the number of vertex covers V′ of G with |V′|=r. We develop a method to calculate Ψ(G,τ). Motivated by a problem in biological systematics, we also consider the mappings f from {1, 2,…,m} into the vertex set V(G) of a graph G, subject to f⁻¹(x)f⁻¹(y)≠ for every edge xy in G. Let F(G,m) be the number of such mappings f. We show that F(G,m) can be determined from Ψ(G,τ).     

Correction to: On Webs of Maximum Rank (Geom. Dedicata 31 (1989), 19–35)

We correct an error in Example (3.4) in Geom. Dedicata 31 (1989), 19–35.     

The effect of solvent upon molecularly thin rotaxane film formation

We have investigated variations in molecularly thin rotaxane films deposited by solvent evaporation, using atomic force microscopy (AFM). Small changes in rotaxane structure result in significant differences in film morphology. The addition of exo-pyridyl moietes to the rotaxane macrocycle results in uniform domains having orientations corresponding to the underlying substrate lattice, while a larger, less symmetric molecule results in a greater lattice mismatch and smaller domain sizes. We have measured differences in film heights both as a function of the solvent of deposition and as a function of surface coverage of rotaxanes. Based on these observations we describe how the use of solvents with higher hydrogen-bond basicity results in films which are more likely to favour sub-molecular motion.     

Site-to-site peptide transport on a molecular platform using a small-molecule robotic arm

Peptides attached to a cysteine hydrazide ‘transporter module’ are transported selectively in either direction between two chemically similar sites on a molecular platform, enabled by the discovery of new operating methods for a molecular transporter that functions through ratcheting. Substrate repositioning is achieved using a small-molecule robotic arm controlled by a protonation-mediated rotary switch and attachment/release dynamic covalent chemistry. A polar solvent mixtures were found to favour Z to E isomerization of the doubly-protonated switch, transporting cargo in one direction (arbitrarily defined as ‘forward’) in up to 85% yield, while polar solvent mixtures were unexpectedly found to favour E to Z isomerization enabling transport in the reverse (‘backward’) direction in >98% yield. Transport of the substrates proceeded in a matter of hours (compared to 6 days even for simple cargoes with the original system) without the peptides at any time dissociating from the machine nor exchanging with others in the bulk. Under the new operating conditions, key intermediates of the switch are sufficiently stabilized within the macrocycle formed between switch, arm, substrate and platform that they can be identified and structurally characterized by ¹H NMR. The size of the peptide cargo has no significant effect on the rate or efficiency of transport in either direction. The new operating conditions allow detailed physical organic chemistry of the ratcheted transport mechanism to be uncovered, improve efficiency, and enable the transport of more complex cargoes than was previously possible.

Peptides are transported in either direction between chemically similar sites on a molecular platform, substrate repositioning is achieved using a cysteine hydrazide transporter module and a small-molecule robotic arm controlled by a rotary switch.     

The mixed-aldehyde synthesis of difunctional tetraarylporphyrins

The synthesis is reported of nine unsymmetrical, meso-substituted porphyrins. Among the compounds prepared are the following 5-(R)-10,15,20-tri-p-tolylporphyrins; R = 2,6-dinitrophenyl, 4-hydroxy-3-ethoxy-phenyl, 4-hydroxy-3-methoxy-5-nitrophenyl, 5-hydroxy-2-nitrophenyl and 4-hydroxy-3-nitrophenyl. Other porphyrins reported include 5-(2-(1-butoxy)phenyl)-15-(2-nitrophenyl)-10-15-di-p-tolylporphyrin and the two 5-(R)-10-15,20-tripropylporphyrins in which R = 2-nitrophenyl and 2-hydroxyphenyl. The disubstituted porphyrins offer a rational route to the synthesis of difunctional “tailed-porphyrins”.     

Mechanistic Aspects of Carbon Nanotube Nucleation and Growth

The discovery, synthesis, characterization, and applicability of carbon nanotubes have produced tremendous excitement and interest among scientists and engineers. In particular, the use of these unique tubular nanostructures for new strong lightweight materials, nanoelectronics, fuel storage and cells, electron emitters and bio, scanning probe microscopy, and chemical sensing devices has created an intense effort to advance the synthesis so as to mass produce carbon nanotubes with control over diameter and helicity. The massive and controlled synthesis of this heralded nanostructure has been a great challenge. Although significant progress has advanced the preparation, more synthetic development is required. The syntheses have so far involved three main approaches: arc discharge vaporization, laser vaporization, and catalytic chemical vapor deposition. The synthetic trend has progressed to a point where further advancement with these techniques will require a better understanding of the mechanism of nucleation and growth. The mechanics of carbon nanotube nucleation and growth involve very complex and diverse phenomena occurring under extreme conditions and on the mesoscopic scale. As yet the detail mechanism is unknown. Difficulties with experimental probing and computational simulation have increased the mystery of this mechanism. This review presents an account of research on the synthesis of carbon nanotubes and the mechanism of formation. This overview includes all three mentioned synthetic approaches and hybrids thereof. On the basis of this broad account a comprehensive mechanism for carbon nanotube nucleation and growth naturally arises. This mechanism is qualitative and it hopes to inspire more quantitative exploration and synthetic advancement.     

Forward and backward pericyclic photochemical reactions have intermediates in common, yet cyclobutenes break the rules.

Photochemical pericyclic reactions are believed to proceed via a so-called pericyclic minimum on the lowest excited potential surface (S(1)), which is common to both the forward and backward reactions. Such a common intermediate has never been directly detected. The photointerconversion of 1,3-butadiene and cyclobutene is the prevailing prototype for such reactions, yet only diene ring closure proceeds with the stereospecificity that the Woodward-Hoffmann rules predict. This contrast seems to exclude a common intermediate. Using ultrafast spectroscopy, we show that the excited states of two cyclobutene/diene isomeric pairs are linked by not one, but by two common minima, p* and ct*. Starting from the diene side (cyclohepta-1,3-diene and cycloocta-1,3-diene), electrocyclic ring closure passes via the pericyclic minimum p*, whereas ct* is mainly responsible for cis-trans isomerization. Starting from the corresponding cyclobutenes (bicyclo[3.2.0]heptene-6 and bicyclo[4.2.0]octene-7), the forbidden isomer is formed from ct*. The path branches at the first (S(2)/S(1)) conical intersection towards p* and ct*. The fact that the energetically unfavorable ct* path can compete is ascribed to a dynamic effect: the momentum in C=C twist direction, acquired--such as in other olefins--in the Franck-Condon region of the cyclobutenes.     

A biological process for the reclamation of flue gas desulfurization gypsum using mixed sulfate-reducing bacteria with inexpensive carbon sources

A combined chemical and biological process for the recycling of flue gas desulfurization (FGD) gypsum into calcium carbonate and elemental sulfur is demonstrated. In this process, a mixed culture of sulfate-reducing bacteria (SRB) utilizes inexpensive carbon sources, such as sewage digest or synthesis gas, to reduce FGD gypsum to hydrogen sulfide. The sulfide is then oxidized to elemental sulfur via reaction with ferric sulfate, and accumulating calcium ions are precipitated as calcium carbonate using carbon dioxide. Employing anaerobically digested municipal sewage sludge (AD-MSS) medium as a carbon source, SRBs in serum bottles demonstrated an FGD gypsum reduction rate of 8 mg/L/h (10⁹ cells)^-1. A chemostat with continuous addition of both AD-MSS media and gypsum exhibited sulfate reduction rates as high as 1.3 kg FGD gypsum/m³d. The increased biocatalyst density afforded by cell immobilization in a columnar reactor allowed a productivity of 152 mg SO₄ ^-2/Lh or 6.6 kg FGD gypsum/m³d. Both reactors demonstrated 100% conversion of sulfate, with 75–100% recovery of elemental sulfur and chemical oxygen demand utilization as high as 70%. Calcium carbonate was recovered from the reactor effluent on precipitation using carbon dioxide. It was demonstrated that SRBs may also use synthesis gas (CO, H₂, and CO₂ in the reduction of gypsum, further decreasing process costs. The formation of two marketable products—elemental sulfur and calcium carbonate—from FGD gypsum sludge, combined with the use of a low-cost carbon source and further improvements in reactor design, promises to offer an attractive alternative to the landfilling of FGD gypsum.