Some examples of random walks on free products of discrete groups

Summary We consider the random walk (X_n) associated with a probability p on a free product of discrete groups. Knowledge of the resolvent (or Green's function) of p yields theorems about the asymptotic behaviour of the n-step transition probabilities p^*n(x)=P(X_n= x¦ X₀=e) as n. Woess [15], Cartwright and Soardi [3] and others have shown that under quite general conditions there is behaviour of the type p^*n(x)C_x^– n n^– 3/2. Here we show on the other hand that if G is a free product of m copies ofZ ^r and if (X_n) is the « average » of the classical nearest neighbour random walk on each of the factorsZ ^r, then while it satisfies an « n^–3/2 — law » for r small relative to m, it switches to an n^– r/2 -law for large r. Using the same techniques, we give examples of irreducible probabilities (of infinite support) on the free groupZ ^*m which satisfyn ^– for .     

Characterization of a Nudix hydrolase from <Emphasis Type="Italic">Deinococcus radiodurans</Emphasis> with a marked specificity for (deoxy)ribonucleoside 5'-diphosphates

Background  

Nudix hydrolases form a protein family whose function is to hydrolyse intracellular nucleotides and so regulate their levels and eliminate potentially toxic derivatives. The genome of the radioresistant bacterium Deinococcus radiodurans encodes 25 nudix hydrolases, an unexpectedly large number. These may contribute to radioresistance by removing mutagenic oxidised and otherwise damaged nucleotides. Characterisation of these hydrolases is necessary to understand the reason for their presence. Here, we report the cloning and characterisation of the DR0975 gene product, a nudix hydrolase that appears to be unique to this organism.     

Calculations of the electronic structures of cage molecules using free-electron orbitals as a basis

A non-empirical molecular orbital method, particularly suitable for calculations on cage-like molecules, is described. The method uses as basis functions the set of free-electron functions which are the solutions of Schrödinger's equation for an electron confined between two concentric, spherical potential energy barriers. Application of the theory to the SCF calculation of the energies of the delocalized electrons in benzene and tetrasulphur tetranitride shows that the model is capable of interpreting the properties of such systems. However, it does highlight a difficulty in the calculation of excited state energies with one-centre models which appears to be largely unrecognized.Extension of the method to a consideration of all the valence electrons, using P₄ as an example, reveals problems the origin of which is an inadequate treatment of the core electrons. It is suggested that these problems may best be dealt with by use of a suitable pseudo potential.     

Organophotoredox/palladium dual catalytic decarboxylative Csp3–Csp3 coupling of carboxylic acids and π-electrophiles

A dual catalytic decarboxylative allylation and benzylation method for the construction of new C(sp³)–C(sp³) bonds between readily available carboxylic acids and functionally diverse carbonate electrophiles has been developed. The new process is mild, operationally simple, and has greatly improved upon the efficiency and generality of previous methodology. In addition, new insights into the reaction mechanism have been realized and provide further understanding of the harnessed reactivity.

A dual catalytic decarboxylative allylation and benzylation method for the construction of new C(sp³)–C(sp³) bonds between readily available carboxylic acids and functionally diverse carbonate electrophiles has been developed.     

Imidazopyridine and pyrimidinopyridine systems from perfluorinated pyridine derivatives

Annelation of pentafluoropyridine via an intramolecular nucleophilic aromatic substitution process with benzamidine gave an imidazopyridine system in high yield in a two step process whilst alkyl amidines gave 4-aminotetrafluoropyridine by a competing elimination reaction. 4-Phenylsulfonyl tetrafluoropyridine reacts with amidines to give the corresponding imidazo[4,5-b]pyridine systems. In contrast, 4-cyanotetrafluoropyridine gave a [6,6]-fused pyrimidinopyridine system arising from initial nucleophilic substitution at the C-3 position of the pyridine ring followed by intramolecular cyclization onto the pendant cyano group. The systems prepared by this annelation methodology further demonstrate the utility of perfluorinated heterocyclic substrates for the synthesis of heterocyclic scaffolds that possess multiple functionality and have potential applications in the drug discovery arena.     

Peptide microarrays for the determination of protease substrate specificity

A method is described for the preparation of substrate microarrays that allow for the rapid determination of protease substrate specificity. Peptidyl coumarin substrates, synthesized on solid support using standard techniques, are printed onto glass slides using DNA microarraying equipment. The linkage from the peptide to the slide is formed through a chemoselective reaction, resulting in an array of uniformly displayed fluorogenic substrates. The arrays can be treated with proteases to yield substrate specificity profiles. Standard instrumentation for visualization of microarrays can be used to obtain comparisons of the specificity constants for all of the prepared substrates. The utility of these arrays is demonstrated by the selective cleavage of preferred substrates with trypsin, thrombin, and granzyme B, and by assessing the extended substrate specificity of thrombin using a microarray of 361 different peptidyl coumarin substrates.     

The heat of formation of sulphuryl fluoride and the stability of fluorosulphates

A standard enthalphy of -767.2 kJ mol^-1 was determined for sulphuryl fluoride from the heat of alkaline hydrolysis, and a standard free energy of -720.8 kJ mol^-1 derived. The thermodynamically preferred product of ionic fluorosulphate decompositions is always sulphuryl fluoride. Kinetic control must predominate when sulphur trioxide is the main product.     

Self-assembled nanoparticle probes for recognition and detection of biomolecules

Colloidal gold nanocrystals have been used to develop a new class of nanobiosensors that is able to recognize and detect specific DNA sequences and single-base mutations in a homogeneous format. At the core of this biosensor is a 2.5-nm gold nanoparticle that functions as both a nano-scaffold and a nano-quencher (efficient energy acceptor). Attached to this core are oligonucleotide molecules labeled with a thiol group at one end and a fluorophore at the other. This hybrid bio/inorganic construct is found to spontaneously assemble into a constrained arch-like conformation on the particle surface. Binding of target molecules results in a conformational change, which restores the fluorescence of the quenched fluorophore. Unlike conventional molecular beacons with a stem-and-loop structure, the nanoparticle probes do not require a stem, and their background fluorescence increases little with temperature. In comparison with the organic quencher Dabcyl (4,4'-dimethylaminophenyl azo benzoic acid), metal nanoparticles have unique structural and optical properties for new applications in biosensing and molecular engineering.     

Flexible and stretchable electrodes for next generation polymer electronics: a review

Transparent conductive electrodes play a significant role in the fabrication and development of optoelectronic devices. As next generation optoelectronic devices tend towards mobile and wearable devices, the added attribute of flexibility or stretchability for these electrodes becomes increasingly important. However, mechanical requirements aside, transparent conductive electrodes must still retain high transparency and conductivity, with the metrics for these parameters being compared to the standard, indium tin oxide. In the search to replace indium tin oxide, two materials that have risen to the forefront are carbon nanotubes and silver nanowires due to their high transparency, conductivity, mechanical compliance, and ease of fabrication. This review highlights recent innovations made by our group in electrodes utilizing carbon nanotubes and silver nanowires, in addition to the use of these electrodes in discrete devices and integrated systems.     

From Ion‐Like Ethylzinc Aluminates to [EtZn(arene)2]+[Al(ORF)4]− Salts

Attempts to prepare previously unknown simple and very Lewis acidic [RZn]⁺[Al(OR^F)₄]^? salts from ZnR₂, AlR₃, and HO?R^F delivered the ion‐like RZn(Al(OR^F)₄) (R=Me, Et; R^F=C(CF₃)₃) with a coordinated counterion, but never the ionic compound. Increasing the steric bulk in RZn⁺ to R=CH₂CMe₃, CH₂SiMe₃, or Cp*, thus attempting to induce ionization, failed and led only to reaction mixtures including anion decomposition. However, ionization of the ion‐like EtZn(Al(OR^F)₄) compound with arenes yielded the [EtZn(arene)₂]⁺[Al(OR^F)₄]^? salts with arene=toluene, mesitylene, or o‐difluorobenzene (o‐DFB)/toluene. In contrast to the ion‐like EtZn(η³‐C₆H₆)(CHB₁₁Cl₁₁), which co‐crystallizes with one benzene molecule, the less coordinating nature of the [Al(OR^F)₄]^? anion allowed the ionization and preparation of the purely organometallic [EtZn(arene)₂]⁺ cation. These stable materials have further applications as, for example, initiators of isobutene polymerization. DFT calculations to compare the Lewis acidities of the zinc cations to those of a large number of organometallic cations were performed on the basis of fluoride ion affinity. The complexation energetics of EtZn⁺ with arenes and THF was assessed and related to the experiments.