Edge-coloured cube decompositions

An edge-colored graph is a graph H together with a function f:E(H) → C where C is a set of colors. Given an edge-colored graph H, the graph induced by the edges of color c C is denoted by H(c). Let G, H, and J be graphs and let μ be a positive integer. A (J, H, G, μ) edge-colored graph decomposition is a set S = {H 1,H 2,...,H t} of edge-colored graphs with color set C = {c 1, c 2,..., c k} such that Hi ≅ H for 1 ≤ i ≤ t; Hi (cj) ≅ G for 1 ≤ i ≤ t and ≤ j ≤ k; and for j = 1, 2,..., k, each edge of J occurs in exactly μ of the graphs H 1(c j ), H 2(c j ),..., H t (c j ). Let Q 3 denote the 3-dimensional cube. In this paper, we find necessary and sufficient conditions on n, μ and G for the existence of a (K n ,Q 3,G, μ) edge-colored graph decomposition. © Birkhäuser Verlag, Basel 2007.

The chemistry and biology of human relaxin-3

A novel member of the human relaxin subclass of the insulin superfamily was recently discovered during a genomics database search and named relaxin-3. Like human relaxin-1 and relaxin-2, relaxin-3 is predicted to consist of a two-chain structure and three disulfide bonds in a disposition identical to that of insulin. To undertake detailed biophysical and biological characterization of the peptide, its chemical synthesis was undertaken. In contrast to human relaxin-1 and relaxin-2, however, relaxin-3 could not be successfully prepared by simple combination of the individual chains, thus necessitating recourse to the use of a regioselective disulfide bond formation strategy. Solid phase synthesis of the separate, selectively S-protected A and B chains followed by their purification and the subsequent stepwise formation of each of the three disulfides led to the successful acquisition of human relaxin-3. Comprehensive chemical characterization confirmed both the correct chain orientation and the integrity of the synthetic product. Relaxin-3 was found to bind to and activate native relaxin receptors in vitro and stimulate water drinking through central relaxin receptors in vivo. Recent studies have demonstrated that relaxin-3 will bind to and activate human LGR7, but not LGR8, in vitro. Secondary structural analysis showed it to adopt a less ordered confirmation than either relaxin-1 or relaxin-2, reflecting the presence in the former of a greater percentage of nonhelical forming amino acids. NMR spectroscopy and simulated annealing calculations were used to determine the three-dimensional structure of relaxin-3 and to identify key structural differences between the human relaxins.

Erratum: Two-Dimensional Failure Modeling with Minimal Repair which appeared in this journal of April 2004, Volume 51:3 on pages 345–362

In this Erratum, we point out the reason for an error in the derivation of a result in our earlier paper, “Two-Dimensional Failure Modeling with Minimal Repair” [1], which appeared in the April 2004 issue of this journal, 51:3, on pages 345–362, and give the correct derivation.