Trades and Graphs

The trade spectrum of a simple graph G is defined to be the set of all t for which it is possible to assemble together t copies of G into a simple graph H, and then disassemble H into t entirely different copies of G. Trade spectra of graphs have applications to intersection problems, and defining sets, of G-designs. In this investigation, we give several constructions, both for specific families of graphs, and for graphs in general.

Packing complete multipartite graphs with 4-cycles

In this paper we completely solve the problem of finding a maximum packing of any complete multipartite graph with edge-disjoint 4-cycles, and the minimum leaves are explicitly given.

Decomposition of complete tripartite graphs into gregarious 4-cycles

A 4-cycle in a tripartite graph with vertex partition {V-1, V-2, V-3} is said to be gregarious if it has at least one vertex in each V-i, 1 less than or equal to i less than or equal to 3. In this paper, necessary and sufficient conditions are given for the existence of an edge-disjoint decomposition of any complete tripartite graph into gregarious 4-cycles.

Common multiples of complete graphs

A graph H is said to divide a graph G if there exists a set S of subgraphs of G, all isomorphic to H, such that the edge set of G is partitioned by the edge sets of the subgraphs in S. Thus, a graph G is a common multiple of two graphs if each of the two graphs divides G.

The metal directed assembly of a trinuclear macrocyclic copper(II) complex

A trinuclear macrocyclic complex is reported from the metal directed condensation between melamine, formaldehyde and the Cu-II complex of a linear tetraamine.

Decomposing complete tripartite graphs into cycles of lengths 3 and 4

Necessary and sufficient conditions are given for the edge-disjoint decomposition of a complete tripartite graph K-r,K-s,K-t into exactly alpha 3-cycles and beta 4-cycles. (C) 1999 Elsevier Science B.V. All rights reserved.

Site-directed mutants of RTP of Bacillus subtilis and the mechanism of replication fork arrest

DNA replication fork arrest during the termination phase of chromosome replication in Bacillus subtilis is brought about by the replication terminator protein (RTP) bound to specific DNA terminator sequences (Tev sites) distributed throughout the terminus region. An attractive suggestion by others was that crucial to the functioning of the RTP-Ter complex is a specific interaction between RTP positioned on the DNA and the helicase associated with the approaching replication fork. Ln support of this was the behaviour of two site-directed mutants of RTP. They appeared to bind Ter DNA normally but were ineffective in fork arrest as ascertained by in vitro Escherichia coli DnaB helicase and replication assays. We describe here a system for assessing the fork-arrest behaviour of RTP mutants in a bona fide in vivo assay in B. subtilis. One of the previously studied mutants, RTP.Y33N, was non-functional in fork arrest in vivo, as predicted. But through extensive analyses, this RTP mutant was shown to be severely defective in binding to Ter DNA, contrary to expectation. Taken in conjunction with recent findings on the other mutant (RTP.E30K), it is concluded that there is as yet no substantive evidence from the behaviour of RTP mutants to support the Rm-helicase interaction model for fork arrest. In an extension of the present work on RTP.Y33N, we determined the dissociation rates of complexes formed by wild-type (wt) RTP and another RTP mutant with various terminator sequences. The functional wtRTP-TerI complex was quite stable (half-life of 182 minutes), reminiscent of the great stability of the E. coli Tus-Ter complex. More significant were the exceptional stabilities of complexes comprising wtRTP and an RTP double-mutant (E39K.R42Q) bound to some particular terminator sequences. From the measurement of in vivo fork-arrest activities of the various complexes, it is concluded that the stability (half-life) of the whole RTP-Ter complex is not the overriding determinant of arrest, and that the RTP-Ter complex must be actively disrupted, or RTP removed, by the action of the approaching replication fork. (C) 1999 Academic Press.

Analysis of the substrate specificity of human sulfotransferases SULT1A1 and SULT1A3: Site-directed mutagenesis and kinetic studies

Sulfonation is an important metabolic process involved in the excretion and in some cases activation of various endogenous compounds and xenobiotics. This reaction is catalyzed by a family of enzymes named sulfotransferases. The cytosolic human sulfotransferases SULT1A1 and SULT1A3 have overlapping yet distinct substrate specificities. SULT1A1 favors simple phenolic substrates such as p-nitrophenol, whereas SULT1A3 prefers monoamine substrates such as dopamine. In this study we have used a variety of phenolic substrates to functionally characterize the role of the amino acid at position 146 in SULT1A1 and SULT1A3. First, the mutation A146E in SULT1A1 yielded a SULT1A3-like protein with respect to the Michaelis constant for simple phenols. The mutation E146A in SULT1A3 resulted in a SULT1A1-like protein with respect to the Michaelis constant for both simple phenols and monoamine compounds. When comparing the specificity of SULT1A3 toward tyramine with that for p-ethylphenol (which differs from tyramine in having no amine group on the carbon side chain), we saw a 200-fold preference for tyramine. The kinetic data obtained with the E146A mutant of SULT1A3 for these two substrates clearly showed that this protein preferred substrates without an amine group attached. Second, changing the glutamic acid at position 146 of SULT1A3 to a glutamine, thereby neutralizing the negative charge at this position, resulted in a 360-fold decrease in the specificity constant for dopamine. The results provide strong evidence that residue 146 is crucial in determining the substrate specificity of both SULT1A1 and SULT1A3 and suggest that there is a direct interaction between glutamic acid 146 in SULT1A3 and monoamine substrates.

Active site-directed protein regulation

Regulation of protein function is vital for the control of cellular processes. Proteins are often regulated by allosteric mechanisms, in which effecters bind to regulatory sites distinct from the active sites and alter protein function. Intrasteric regulation, directed at the active site and thus the counterpart of allosteric control, is now emerging as an important regulatory mechanism.

Acyclic permutants of naturally occurring cyclic proteins - Characterization of cystine knot and beta-sheet formation in the macrocyclic polypeptide kalata B1

Kalata B1 is a prototypic member of the unique cyclotide family of macrocyclic polypeptides in which the major structural features are a circular peptide backbone, a triple stranded beta-sheet, and a cystine knot arrangement of three disulfide bonds. The cyclotides are the only naturally occurring family of circular proteins and have prompted us to explore the concept of acyclic permutation, i.e. opening the backbone of a cross-linked circular protein in topologically permuted ways. We have synthesized the complete suite of acyclic permutants of kalata B1 and examined the effect of acyclic permutation on structure and activity. Only two of six topologically distinct backbone loops are critical for folding into the native conformation, and these involve disruption of the embedded ring in the cystine knot. Surprisingly, it is possible to disrupt regions of the p-sheet and still allow folding into native-like structure, provided the cystine knot is intact. Kalata B1 has mild hemolytic activity, but despite the overall structure of the native peptide being retained in all but two cases, none of the acyclic permutants displayed hemolytic activity. This loss of activity is not localized to one particular region and suggests that cyclization is critical for hemolytic activity.

Metal-directed synthesis routes to a 16-membered tetraazamacrocycle with two pendant primary amine groups

The reaction of the bis(propane-1,3-diamine)copper(II) ion with paraformaldehyde and nitroethane in dry methanol under basic conditions produces a macrocyclic product, (cis-3,11-dimethyl-3,11-dinitro-1,5,9,13-tetraazacyclohexadecane)copper(II) perchlorate, in low yield, compared with the good yield obtained in the parallel chemistry possible even under aqueous conditions using palladium(II) as a template. The palladium complex was reduced with zinc amalgam in dilute aqueous acid to yield the metal-free 16-membered macrocyclic hexaamine, in this case re-complexed and characterised by an X-ray crystal structure as the (cis-3,11-dimethyl-1,5,9,13-tetraazacyclohexadecane-3,11-diamine)copper(II) perchlorate. The copper ion is found in a tetragonally elongated and trigonally-distorted octahedral environment, with all six of the ligand nitrogens coordinated, the two primary amine pendant groups occupying cis sites. (C) 2000 Elsevier Science S.A. All rights reserved.

The fine structure of (v, 3) directed triple systems: v equivalent to 2 (mod 3)

The fine structure of a directed triple system of index lambda is the vector (c(1), c(2),...,C-lambda), where c(i) is the number of directed triples appearing precisely i times in the system. We determine necessary and sufficient conditions for a vector to be the fine structure of a directed triple system of index 3 for upsilon = 2 (mod 3).

Decompositions of complete multipartite graphs into cycles of even length

Necessary and sufficient conditions for the existence of an edge-disjoint decomposition of any complete multipartite graph into even length cycles are investigated. Necessary conditions are listed and sufficiency is shown for the cases when the cycle length is 4, 6 or 8. Further results concerning sufficiency, provided certain small decompositions exist, are also given for arbitrary even cycle lengths.

A family of perfect factorisations of complete bipartite graphs

A 1-factorisation of a graph is perfect if the union of any two of its 1-factors is a Hamiltonian cycle. Let n = p(2) for an odd prime p. We construct a family of (p-1)/2 non-isomorphic perfect 1-factorisations of K-n,K-n. Equivalently, we construct pan-Hamiltonian Latin squares of order n. A Latin square is pan-Hamiltoilian if the permutation defined by any row relative to any other row is a single Cycle. (C) 2002 Elsevier Science (USA).

C4-saturated bipartite graphs

Let H be a graph. A graph G is said to be H-free if it contains no subgraph isomorphic to H. A graph G is said to be an H-saturated subgraph of a graph K if G is an H-free subgraph of K with the property that for any edge e is an element of E(K)\E(G), G boolean OR {e} is not H-free. We present some general results on K-s,K-t-saturated subgraphs of the complete bipartite graph K-m,K-n and study the problem of finding, for all possible values of q, a C-4-saturated subgraph of K., having precisely q edges. (C) 2002 Elsevier Science B.V. All rights reserved.