Balancing signed graphs

A signed graph based on F is an ordinary graph F with each edge marked as positive or negative. Such a graph is called balanced if each of its cycles includes an even number of negative edges. Psychologists are sometimes interested in the smallest number d=d(G) such that a signed graph G may be converted into a balanced graph by changing the signs of d edges. We investigate the number D(F) defined as the largest d(G) such that G is a signed graph based on F. We prove that $\text{1}\text{2}\text{m?}\text{nm}\text{≤D(F)≤}\text{1}\text{2}\text{m}$ for every graph F with n vertices and m edges. If F is the complete bipartite graph with t vertices in each part, then $\text{D(F)≤}\text{1}\text{2}\text{t}{}^2\text{?ct}{}^{\mathrm{<mtext>3</mtext><mtext>2</mtext>}}$ for some positive constant c.     

Finite Sholander trees, trees, and their betweenness

We provide a proof of Sholander’s claim [M. Sholander, Trees, lattices, order, and betweenness, Proceedings of the American Mathematical Society 3 (1952) 369–381] concerning the representability of collections of so-called segments by trees, which yields a characterization of the interval function of a tree. Furthermore, we streamline Burigana’s characterization [L. Burigana, Tree representations of betweenness relations defined by intersection and inclusion, Mathematics and Social Sciences 185 (2009) 5–36] of tree betweenness and provide a relatively short proof.     

A novel fused-silica capillary dropping mercury electrode with long drop-time and its application on determination of critical micelle concentration

A novel long drop time mercury electrode has been constructed from common fused-silica capillary (50 μm I.D., 360 μm E.D.). Proposed device provides reproducible mercury drops with typical lifetime 40-120 s. The electrode was used for a set of electrocapillary measurements aimed at determination of critical micelle concentration of anionic surfactants by a convection controlled drop-time technique. A critical micelle concentration of sodium dodecyl sulfate 5.6 ± 0.4 mmol L⁻¹ and 4.3 ± 0.4 mmol L⁻¹ were obtained in 1 mmol L⁻¹ and 5 mmol L⁻¹ phosphate buffer (pH 7.0), respectively. The values were comparable to those obtained from conductometric measurements under the same conditions (7.0 ± 0.1 mmol L⁻¹ and 5.2 ± 0.1 mmol L⁻¹, respectively) and the difference was explained in accordance with theory of hemi-micelle formation.     

Like-charge guanidinium pairing from molecular dynamics and ab initio calculations

Pairing of guanidinium moieties in water is explored by molecular dynamics simulations of short arginine-rich peptides and ab initio calculations of a pair of guanidinium ions in water clusters of increasing size. Molecular dynamics simulations show that, in an aqueous environment, the diarginine guanidinium like-charged ion pairing is sterically hindered, whereas in the Arg-Ala-Arg tripeptide, this pairing is significant. This result is supported by the survey of protein structure databases, where it is found that stacked arginine pairs in dipeptide fragments exist solely as being imposed by the protein structure. In contrast, when two arginines are separated by a single amino acid, their guanidinium groups can freely approach each other and they frequently form stacked pairs. Molecular dynamics simulations results are also supported by ab initio calculations, which show stabilization of stacked guanidinium pairs in sufficiently large water clusters.     

Lines in hypergraphs

One of the De Bruijn-Erd?s theorems deals with finite hypergraphs where every two vertices belong to precisely one hyperedge. It asserts that, except in the perverse case where a single hyperedge equals the whole vertex set, the number of hyperedges is at least the number of vertices and the two numbers are equal if and only if the hypergraph belongs to one of simply described families, near-pencils and finite projective planes. Chen and Chvátal proposed to define the line uv in a 3-uniform hypergraph as the set of vertices that consists of u, v, and all w such that {u;v;w} is a hyperedge. With this definition, the De Bruijn-Erd?s theorem is easily seen to be equivalent to the following statement: If no four vertices in a 3-uniform hypergraph carry two or three hyperedges, then, except in the perverse case where one of the lines equals the whole vertex set, the number of lines is at least the number of vertices and the two numbers are equal if and only if the hypergraph belongs to one of two simply described families. Our main result generalizes this statement by allowing any four vertices to carry three hyperedges (but keeping two forbidden): the conclusion remains the same except that a third simply described family, complements of Steiner triple systems, appears in the extremal case.     

A de Bruijn-Erdős theorem for 1–2 metric spaces

A special case of a combinatorial theorem of De Bruijn and Erd?s asserts that every noncollinear set of n points in the plane determines at least n distinct lines. Chen and Chvátal suggested a possible generalization of this assertion in metric spaces with appropriately defined lines. We prove this generalization in all metric spaces where each nonzero distance equals 1 or 2.     

Proximity effect in the 13C NMR spectra of alkylpyridines: An aid in structure elucidation

The structures of seven polysubstituted alkylpyridines, recently isolated from coal tar, were elucidated using their ¹³C NMR spectra. For tri- and tetra-substituted derivatives the correct isomeric structure was found from their conformity with the proximity effect, i.e. the deviations from additivity when calculating chemical shifts from substituent increments (SCS) for sterically crowded molecules. This effect manifests itself both on the ring and on the alkyl carbon signals, and in most cases shifts their position upfield.     

Gyration- and inertia-tensor-based collective coordinates for metadynamics. Application on the conformational behavior of polyalanine peptides and Trp-cage folding

Effective simulations of proteins, their complexes, and other amino-acid polymers such as peptides or peptoids are critically dependent on the performance of the simulation methods and their ability to map the conformational space of the molecule in question. The most important step in this process is the choice of the coordinates in which the conformational sampling will be executed and their uniqueness regarding the capability to unambiguously determine the energy minimum on the free-energy hypersurface. In the presented study, we show that metadynamics and chosen collective coordinates-the principal moments of the tensors of gyration and inertia, the principal radii of gyration around the principal axes, asphericity, acylindricity, and anisotropy-can be used as a powerful combination to map the conformational space of peptides and proteins. We show that the combination of these coordinates with metadynamics produces a powerful tool for the study of biologically relevant molecules.