Acid promoted cinnamyl ion mobility within peptide derived macrocycles

We are developing methods that restrict the conformational mobility of peptides and related heteropolymers while simultaneously altering their properties. Our experiments occur as processes wherein a conserved, lipophilic reagent is activated in stages to form composite products with unprotected polyamides in parallel. For each starting oligomer, the goal is to create not one, but rather a collection of products. The intent is for those materials to retain molecular recognition elements of the biopolymer, yet display that functionality as part of stable, cyclic structures having defined shapes and enhanced membrane solubility/permeability. Here we describe reagent 2 and its two-step integration into peptides to afford macrocyclic ethers (e.g., 4 when starting with W-W-Y). When those materials are treated with protic acid in anhydrous solvent, the cinnamyl unit migrates from the oxygen of tyrosine to distribute throughout the structure, forming new products via carbon/carbon bonding. These changes occur concomitantly with acid-promoted rearrangements/cyclizations of the dienyne appendage to generate mixtures containing unique macrocycles such as 15. Similar amalgamations of 2 with more diverse peptides is a means to begin accessing complex peptidomimetics systematically. From a library of screening fractions generated in this way, we have identified a small molecule that selectively promotes hippocampal neurogenesis in the adult mouse brain.     

Wiener Index of Hexagonal Systems

The Wiener index W is the sum of distances between all pairs of vertices of a (connected) graph. Hexagonal systems (HS's) are a special type of plane graphs in which all faces are bounded by hexagons. These provide a graph representation of benzenoid hydrocarbons and thus find applications in chemistry. The paper outlines the results known for W of the HS: method for computation of W, expressions relating W with the structure of the respective HS, results on HS's extremal w.r.t. W, and on integers that cannot be the W-values of HS's. A few open problems are mentioned. The chemical applications of the results presented are explained in detail.     

The degree sequence of Fibonacci and Lucas cubes

The Fibonacci cube Γ_n is the subgraph of the n-cube induced by the binary strings that contain no two consecutive 1’s. The Lucas cube Λ_n is obtained from Γ_n by removing vertices that start and end with 1. It is proved that the number of vertices of degree k in Γ_n and Λ_n is and , respectively. Both results are obtained in two ways, since each of the approaches yields additional results on the degree sequences of these cubes. In particular, the number of vertices of high resp. low degree in Γ_n is expressed as a sum of few terms, and the generating functions are given from which the moments of the degree sequences of Γ_n and Λ_n are easily computed.     

The median function on graphs with bounded profiles

The median of a profile π=(u₁,…,u_k) of vertices of a graph G is the set of vertices x that minimize the sum of distances from x to the vertices of π. It is shown that for profiles π with diameter θ the median set can be computed within an isometric subgraph of G that contains a vertex x of π and the r-ball around x, where r>2θ−1−2θ/|π|. The median index of a graph and r-joins of graphs are introduced and it is shown that r-joins preserve the property of having a large median index. Consensus strategies are also briefly discussed on a graph with bounded profiles.     

Cover-Incomparability Graphs of Posets

Cover-incomparability graphs (C-I graphs, for short) are introduced, whose edge-set is the union of edge-sets of the incomparability and the cover graph of a poset. Posets whose C-I graphs are chordal (resp. distance-hereditary, Ptolemaic) are characterized in terms of forbidden isometric subposets, and a general approach for studying C-I graphs is proposed. Several open problems are also stated.     

Transitive,Locally Finite Median Graphs with Finite Blocks

The subject of this paper are infinite, locally finite, vertex-transitive median graphs. It is shown that the finiteness of the Θ-classes of such graphs does not guarantee finite blocks. Blocks become finite if, in addition, no finite sequence of Θ-contractions produces new cut-vertices. It is proved that there are only finitely many vertex-transitive median graphs of given finite degree with finite blocks. An infinite family of vertex-transitive median graphs with finite intransitive blocks is also constructed and the list of vertex-transitive median graphs of degree four is presented. Sandi Klavžar: Supported by the Ministry of Science of Slovenia under the grant P1-0297. The author is also with the Faculty of Mathematics and Natural Sciences, University of Maribor, Slovenia and the Institute of Mathematics, Physics and Mechanics, Ljubljana.     

Asymptotic number of isometric generalized Fibonacci cubes

For a binary word f, let Q_d(f) be the subgraph of the d-dimensional cube Q_d induced on the set of all words that do not contain f as a factor. Let G_n be the set of words f of length n that are good in the sense that Q_d(f) is isometric in Q_d for all d. It is proved that lim_n→∞|G_n|/2ⁿ exists. Estimates show that the limit is close to 0.08, that is, about eight percent of all words are good.     

Extending partial representations of circle graphs

The partial representation extension problem is a recently introduced generalization of the recognition problem. A circle graph is an intersection graph of chords of a circle. We study the partial representation extension problem for circle graphs, where the input consists of a graph and a partial representation giving some predrawn chords that represent an induced subgraph of . The question is whether one can extend to a representation of the entire graph , that is, whether one can draw the remaining chords into a partially predrawn representation to obtain a representation of . Our main result is an time algorithm for partial representation extension of circle graphs, where is the number of vertices. To show this, we describe the structure of all representations of a circle graph using split decomposition. This can be of independent interest.