Families of Line-Graphs and Their Quantization

Any directed graph G with N vertices and J edges has an associated line-graph L(G) where the J edges form the vertices of L(G). We show that the non-zero eigenvalues of the adjacency matrices are the same for all graphs of such a family L ⁿ (G). We give necessary and sufficient conditions for a line-graph to be quantisable and demonstrate that the spectra of associated quantum propagators follow the predictions of random matrices under very general conditions. Line-graphs may therefore serve as models to study the semiclassical limit (of large matrix size) of a quantum dynamics on graphs with fixed classical behaviour.     

Cationic poly(methacryl oxyethyl trimethylammonium) and its poly(ethylene glycol)‐grafted analogue as capillary coating materials in electrophoresis

Cationic polyelectrolytes were synthesized and used as semipermanent coating materials for capillaries in electrophoresis. The polyelectrolytes used were a homopolymer of poly(methacryl oxyethyl trimethylammonium chloride) (PMOTAC) and its poly(ethylene glycol) (PEG)‐grafted analogue. Two PMOTAC polyelectrolytes, with molar masses of 85,000 and 300,000 g/mol, and PEG‐grafted PMOTAC with a molar mass of 280,000 g/mol were synthesized and then characterized by size exclusion chromatography (SEC) and nuclear magnetic resonance (NMR) spectroscopy. Attachment of the polyelectrolytes to the wall of the fused silica capillary for electrophoresis caused the electroosmotic flow (EOF) to reverse. The polyelectrolyte coatings were tested over the pH range 2–11 at different buffer ionic strengths, and the most stable and strongest anodic EOFs were obtained at acidic pH values with low ionic strength buffers. Between runs the capillary is merely rinsed for 2 or 3 min with the background electrolyte solution. With the PMOTAC coatings at pH values ≤5, the RSDs of the EOFs were less than 2.9% after 60 injections. The effects of the molar mass of the polycation and of PEGylation of PMOTAC on the interactions between the polycations and basic proteins were studied at acidic pH values. The differences in the effective electrophoretic mobilities, resolution values, and plate numbers of the proteins with the different coatings were due to the EOF, as demonstrated through calculations of reduced mobilities, relative resolution values, and relative plate numbers. © 2007 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 45: 2655–2663, 2007     

The effect of ph and gas composition on the bubble fractionation of proteins

Studies were conducted to establish the effect of the variation of environmental factors on the separation occurring in protein systems, resulting from bubble fractionation in a bioreactor. The measure of separation was selected to be the separation ratio. This is defined to be the ratio of either the top or the middle position concentration in the vessel to the bottom concentration of the vessel. Invertase and α-amylase were the two “model” enzymes considered. It was observed that, under certain conditions, i.e., a combination of the nature of the sparging gas and the medium pH, varying degrees of protein separation were achieved. The pH of the system dramatically influenced the separation. It was found that the best separation occurred at a certain pH, assumed to be at or close to thepI of the protein in question. Furthermore, it was observed that systems sparged with CO₂ exhibited greater separation than systems sparged with air. In fact, in the case of invertase, almost threefold separation was observed at the top port when the solution was sparged with CO₂.     

Enantio- and Diastereoselective Aldol-Reaction of 2,6-Dimethylphenyl Propionate Using Titanium-Carbohydrate Complexes

Chloro(cyclopentadienyl)bis(1,2:5,6-di-O-isopropylidene-α-D -glucofuranos-3-O-yl)titanium ( 1 ) is used for the transmetallation of Li-enolates obtained from propionyl derivatives. While such Ti-enolates of ketones and hydrazones appear to be unreactive, the (E)enolate 13 of 2,6-dimethylphenyl propionate ( 11 ) adds to the re-side of aldehydes, affording various syn-aldols 14 with high dia- and enantioselectivity (92–97% ds, 91–97% ee, cf. Scheme 2 and Table 1). Racemic syn-aldols (±)- 14 are obtained analogously from the achiral bis(2-propyloxy)-Ti-enolate 15 (Scheme 2 and Table 2). In contrast to the unstable Li-enolate 10 , the Ti-enolates 13 and 15 isomerize at ?30°, presumably to the thermodynamically more stable (Z)-enolates (Scheme 4), While the diastereoselectivity of the achiral enolate 15 is lost upon this equilibration, the chiral (Z)-enolate 27 quite unexpectedly affords anti-aldols 12 of high optical purity (94–98% ec) and, in most cases, with acceptable-to-good diastereoselectivity (82–90% ds). Notable exceptions are branched unsaturated and aromatic aldehydes which form a greater proportion of synepimers of moderate optical purity (Scheme 5 and Table 3). Consistent with these findings, re-facial-and ami -selective aldol-addition is also exhibited by the (Z)-configurated Ti-enolate 22 of N-propionyl-oxazolidi-none 19 (Scheme 3).