Toll number of the Cartesian and the lexicographic product of graphs

Toll convexity is a variation of the so-called interval convexity. A tolled walk $T$ between two non-adjacent vertices $u$ and $v$ in a graph $G$ is a walk, in which $u$ is adjacent only to the second vertex of $T$ and $v$ is adjacent only to the second-to-last vertex of $T$. A toll interval between $u,v\in V\left(G\right)$ is a set $T_G(u,v)=\{x\in V\left(G\right):x\text{ lies on a tolled walk between }u\text{ and }v\}$. A set $S?V\left(G\right)$ is toll convex, if $T_G(u,v)?S$ for all $u,v\in S$. A toll closure of a set $S?V\left(G\right)$ is the union of toll intervals between all pairs of vertices from $S$. The size of a smallest set $S$ whose toll closure is the whole vertex set is called a toll number of a graph $G$, $tn\left(G\right)$. The first part of the paper reinvestigates the characterization of convex sets in the Cartesian product of two graphs. It is proved that the toll number of the Cartesian product of two graphs equals 2. In the second part, the toll number of the lexicographic product of two graphs is studied. It is shown that if $H$ is not isomorphic to a complete graph, $tn(G°H)\le 3?tn\left(G\right)$. We give some necessary and sufficient conditions for $tn(G°H)=3?tn\left(G\right)$. Moreover, if $G$ has at least two extreme vertices, a complete characterization is given. Furthermore, graphs with $tn(G°H)=2$ are characterized. Finally, the formula for $tn(G°H)$ is given — it is described in terms of the so-called toll-dominating triples or, if $H$ is complete, toll-dominating pairs.     

Comparative analysis of CE‐SSCP to standard RFLP‐CE‐FLA method in quantification of known viral variants within an RNA virus quasispecies

RNA viruses display the highest replication error rate in our biosphere, leading to highly diverse viral populations termed quasispecies. The gold standard method for detection and quantification of variants in a quasispecies is cloning and sequencing, but it is expensive, laborious and time consuming. Therefore, other mutation detection approaches, including SSCP, are often used. In this study, we demonstrate development and the usage of a CE‐SSCP method for quantification of two nearly identical viral variants in heterogenic population of a mumps virus strain and its comparison to RFLP‐CE‐fragment length analysis (RFLP‐CE‐FLA). Analyzed PCR fragments were of the same size (245 bp) with one difference in their nucleotide sequence. The limit of detection of both methods was at 5% of the minor variant. When PCR amplicons of the two variants were pooled, methods' results were very similar. On the contrary, the quantification results of samples in which variants were mixed prior to PCR showed substantial difference between the two methods. Our results indicate that although both methods can be used for detection and monitoring of a specific mutation within a viral population, caution should be taken when quantitative analysis of complex samples is based solely on results of one method.     

Strategies for "wiring" redox-active proteins to electrodes and applications in biosensors, biofuel cells, and nanotechnology

In this tutorial review the basic approaches to establish electrochemical communication between redox-active proteins and electrodes are elucidated and examples for applications in electrochemical biosensors, biofuel cells and nanotechnology are presented. The early stage of protein electrochemistry is described giving a short overview over electron transfer (ET) between electrodes and proteins, followed by a brief introduction into experimental procedures for studying proteins at electrodes and possible applications arising thereof. The article starts with discussing the electrochemistry of cytochrome c, the first redox-active protein, for which direct reversible ET was obtained, under diffusion controlled conditions and after adsorption to electrodes. Next, examples for the electrochemical study of redox enzymes adsorbed on electrodes and modes of immobilization are discussed. Shortly the experimental approach for investigating redox-active proteins adsorbed on electrodes is outlined. Possible applications of redox enzymes in electrochemical biosensors and biofuel cells working by direct ET (DET) and mediated ET (MET) are presented. Furthermore, the reconstitution of redox active proteins at electrodes using molecular wire-like units in order to "wire" the proteins to the electrode surface and possible applications in nanotechnology are discussed.     

Simple rules for the understanding of Heusler compounds

Heusler compounds are a remarkable class of intermetallic materials with 1:1:1 (often called Half-Heusler) or 2:1:1 composition comprising more than 1500 members. Today, more than a century after their discovery by Fritz Heusler, they are still a field of active research. New properties and potential fields of applications emerge constantly; the prediction of topological insulators is the most recent example. Surprisingly, the properties of many Heusler compounds can easily be predicted by the valence electron count. Their extremely flexible electronic structure offers a toolbox which allows the realization of demanded but apparently contradictory functionalities within one ternary compound. Devices based on multifunctional properties, i.e. the combination of two or more functions such as superconductivity and topological edge states will revolutionize technological applications. The subgroup of more than 250 semiconductors is of high relevance for the development of novel materials for energy technologies. Their band gaps can readily be tuned from zero to ≈4 eV by changing the chemical composition. Thus, great interest has been attracted in the fields of thermoelectrics and solar cell research. The wide range of their multifunctional properties is also reflected in extraordinary magneto-optical, magnetoelectronic, and magnetocaloric properties. The most prominent example is the combination of magnetism and exceptional transport properties in spintronic devices. To take advantage of the extremely high potential of Heusler compounds simple rules for the understanding of the structure, the electronic structure and the relation to the properties are reviewed.     

Bioactive unnatural somatostatin analogues through bioorthogonal iodo- and ethynyl-disulfide intercalators

Iodo- and ethynyl-containing bisalkylating bioconjugation agents 5 and 8 were achieved and allow the introduction of reactive unnatural substituents into proteins and peptides whilst the bioactive 3D structure is retained. Derivatives of the peptide hormone somatostatin bearing a single iodo or ethynyl group were prepared through intercalation into the disulfide bridge. For the first time, the exact reaction mechanism of the intercalation was elucidated by applying 2D NMR experiments and it was shown that, during the reaction, somatostatin diastereomers were formed. Site-directed modification of the ethynyl-modified peptide with a coumarin chromophore was achieved through a [1,3] dipolar Huisgen cycloaddition reaction; this suggests that such a derivative could serve as an attractive platform to prepare artificial somatostatin compound libraries. The biological activity and specificity of a representative modified somatostatin derivative was demonstrated and efficient receptor-mediated cell uptake occurred in a dose-dependent manner into receptor positive cells only. The iodo and ethynyl bioconjugation reagents presented herein could be applied for introducing such substituents into alternative peptides and proteins and, in principle, could facilitate the efficient design of a broad variety of artificial protein and peptide analogues with previously unknown bioactivities.     

1,1-carboboration of 1-alkynes: a conceptual alternative to the hydroboration reaction

Strongly electrophilic boranes R-B(C(6)F(5))(2) react readily with a variety of 1-alkynes by means of a 1,1-carboboration reaction to yield alkenylborane products, which can subsequently be used as reagents in metal catalyzed cross-coupling reactions.     

A complete set of addition laws for incomplete Edwards curves

Edwards curves were the first curves shown to have a complete addition law. However, the completeness of the addition law depends on the curve parameters and even a complete Edwards curve becomes incomplete over a quadratic field extension. This paper covers arbitrary Edwards curves and gives a set of two addition laws that for any pair of input points P₁, P₂ produce the sum P₁+P₂.     

Quantitative detection of C-deuterated drugs by CARS microscopy and Raman microspectroscopy

The introduction of carbon-deuterium (C-D) bonds into drug compounds by organic synthesis is a non-invasive labelling approach, which does not alter the chemical and physiological properties of the drug itself. C-deuterated drugs exhibit characteristic vibrational signatures in the C-D stretching region around 2100-2300 cm(-1), which avoids spectral interference with contributions from a complex biological environment. In this paper, the quantitative detection of C-deuterated drugs by Raman microspectroscopy and single-band CARS microscopy is examined. Concentration-dependent studies on drugs with aliphatic and aromatic C-D moieties were performed in a two-channel microfluidic chip, using the corresponding non-deuterated (C-H) isotopologues as an internal reference.     

Primitive symmetric designs with up to 2500 points

By this article we conclude the construction of all primitive ( v, k,λ ) symmetric designs with v < 2500 , up to a few unsolved cases. Complementary to the designs with prime power number of points published previously, here we give 55 primitive symmetric designs with v ≠ p ^m , p prime and m positive integer, together with the analysis of their full automorphism groups. The research involves programming and wide‐range computations. We make use of the software package GAP and the library of primitive groups which it contains. © 2011 Wiley Periodicals, Inc. J Combin Designs 19:463‐474, 2011