The ‘Butterfly effect’ in Cayley graphs with applications to genomics

Suppose a finite set X is repeatedly transformed by a sequence of permutations of a certain type acting on an initial element x to produce a final state y. For example, in genomics applications, X could be a set of genomes and the permutations certain genome ‘rearrangements’ or, in group theory, X could be the set of configurations of a Rubik’s cube and the permutations certain specified moves. We investigate how ‘different’ the resulting state y′ to y can be if a slight change is made to the sequence, either by deleting one permutation, or replacing it with another. Here the ‘difference’ between y and y′ might be measured by the minimum number of permutations of the permitted type required to transform y to y′, or by some other metric. We discuss this first in the general setting of sensitivity to perturbation of walks in Cayley graphs of groups with a specified set of generators. We then investigate some permutation groups and generators arising in computational genomics, and the statistical implications of the findings.     

Tree-Based Unrooted Phylogenetic Networks

Phylogenetic networks are a generalization of phylogenetic trees that are used to represent non-tree-like evolutionary histories that arise in organisms such as plants and bacteria, or uncertainty in evolutionary histories. An unrooted phylogenetic network on a non-empty, finite set X of taxa, or network, is a connected, simple graph in which every vertex has degree 1 or 3 and whose leaf set is X. It is called a phylogenetic tree if the underlying graph is a tree. In this paper we consider properties of tree-based networks, that is, networks that can be constructed by adding edges into a phylogenetic tree. We show that although they have some properties in common with their rooted analogues which have recently drawn much attention in the literature, they have some striking differences in terms of both their structural and computational properties. We expect that our results could eventually have applications to, for example, detecting horizontal gene transfer or hybridization which are important factors in the evolution of many organisms.     

Chemotaxis by Pseudomonas aeruginosa.

Chemotaxis by Pseudomonas aeruginosa RM46 has been studied, and conditions required for chemotaxis have been defined, by using the Adler capillary assay technique. Several amino acids, organic acids, and glucose were shown to be attractants of varying effectiveness for this organism. Ethylenediaminetetraacetic acid was absolutely required for chemotaxis, and magnesium was also necessary for a maximum response. Serine taxis was greatest when the chemotaxis medium contained 1.5 X 10(-5) M ethylenediaminetetraacetic acid and 0.005 M magnesium chloride. It was not necessary to include methionine in the chemotaxis medium. The strength of the chemotactic responses to glucose and to citrate was dependent on prior growth of the bacteria on glucose and citrate, respectively. Accumulation in response to serine was inhibited by the addition of succinate, citrate, malate, glucose, pyruvate, or methionine to the chemotaxis medium. Inhibition by succinate was not dependent on the concentration of attractant in the capillary. However, the degree to which glucose and citrate inhibited serine taxis was dependent on the carbon source utilized for growth. Further investigation of this inhibition may provide information about the mechanisms of chemotaxis in P. aeruginosa.