A matroid associated with a phylogenetic tree

A (pseudo-)metric D on a finite set X is said to be a `tree metric' if there is a finite tree with leaf set X and non-negative edge weights so that, for all x,y ∈X, D(x,y) is the path distance in the tree between x and y. It is well known that not every metric is a tree metric. However, when some such tree exists, one can always find one whose interior edges have strictly positive edge weights and that has no vertices of degree 2, any such tree is 13; up to canonical isomorphism 13; uniquely determined by D, and one does not even need all of the distances in order to fully (re-)construct the tree's edge weights in this case. Thus, it seems of some interest to investigate which subsets of X, 2 suffice to determine (`lasso') these edge weights. In this paper, we use the results of a previous paper to discuss the structure of a matroid that can be associated with an (unweighted) X-tree T defined by the requirement that its bases are exactly the `tight edge-weight lassos' for T, i.e, the minimal subsets of X, 2 that lasso the edge weights of T

On Describing Multivariate Skewness: A Directional Approach

Most multivariate measures of skewness in the literature measure the overall skewness of a distribution. While these measures are perfectly adequate for testing the hypothesis of distributional symmetry, their relevance for describing skewed distributions is less obvious. In this article, we consider the problem of characterising the skewness of multivariate distributions. We define directional skewness as the skewness along a direction and analyse parametric classes of skewed distributions using measures based on directional skewness. The analysis brings further insight into the classes, allowing for a more informed selection of particular classes for particular applications. In the context of Bayesian linear regression under skewed error we use the concept of directional skewness twice. First in the elicitation of a prior on the parameters of the error distribution, and then in the analysis of the skewness of the posterior distribution of the regression residuals.Bayesian methods, Multivariate distribution, Multivariate regression, Prior elicitation, Skewness.

Reconstructing pedigrees: some identifiability questions for a recombination-mutation model

Pedigrees are directed acyclic graphs that represent ancestral relationships between individuals in a population. Based on a schematic recombination process, we describe two simple Markov models for sequences evolving on pedigrees - Model R (recombinations without mutations) and Model RM (recombinations with mutations). For these models, we ask an identifiability question: is it possible to construct a pedigree from the joint probability distribution of extant sequences? We present partial identifiability results for general pedigrees: we show that when the crossover probabilities are sufficiently small, certain spanning subgraph sequences can be counted from the joint distribution of extant sequences. We demonstrate how pedigrees that earlier seemed difficult to distinguish are distinguished by counting their spanning subgraph sequences.Comment: 40 pages, 9 figure

Circular Networks from Distorted Metrics

Trees have long been used as a graphical representation of species relationships. However complex evolutionary events, such as genetic reassortments or hybrid speciations which occur commonly in viruses, bacteria and plants, do not fit into this elementary framework. Alternatively, various network representations have been developed. Circular networks are a natural generalization of leaf-labeled trees interpreted as split systems, that is, collections of bipartitions over leaf labels corresponding to current species. Although such networks do not explicitly model specific evolutionary events of interest, their straightforward visualization and fast reconstruction have made them a popular exploratory tool to detect network-like evolution in genetic datasets. Standard reconstruction methods for circular networks, such as Neighbor-Net, rely on an associated metric on the species set. Such a metric is first estimated from DNA sequences, which leads to a key difficulty: distantly related sequences produce statistically unreliable estimates. This is problematic for Neighbor-Net as it is based on the popular tree reconstruction method Neighbor-Joining, whose sensitivity to distance estimation errors is well established theoretically. In the tree case, more robust reconstruction methods have been developed using the notion of a distorted metric, which captures the dependence of the error in the distance through a radius of accuracy. Here we design the first circular network reconstruction method based on distorted metrics. Our method is computationally efficient. Moreover, the analysis of its radius of accuracy highlights the important role played by the maximum incompatibility, a measure of the extent to which the network differs from a tree.Comment: Submitte

Response of two mouse tumours to hyperthermia with CCNU or melphalan.

The in vivo response of B16 melanoma and Lewis lung carcinoma to combinations of hyperthermia and graded doses of CCNU or Melphalan was studied. To obtain dose-response curves and quantitative comparisons of different treatments, an agar-colony assay was used to measure survival of cells from excised tumours. For heating experiments, the use of 2 tumours per animal, one heated and one not, allowed all other factors to be kept constant. When tumours were immersed in a water-bath at 43 degrees C for 1 h, Thermal Enhancement Ratios (TER) measured from the slopes of the dose-response curves were up to 1.6 for CCNU and 2.4 for Melphalan. Direct heat killing of about 1 decade was seen for 1 h at 43 degrees C. The anaesthetic Saffan also enhanced drug cell kill; the largest Dose Modifying Factor (2.7) was measured for Melphalan in the Lewis lung tumour. The duration of heating, and waterbath temperature, both influenced the enhancement of cell killing by CCNU, as did the time of excision of tumours between 0 and 3 1/2 h after treatment. There was no difference in effect between 3 1/2 and 24 h. The interaction between heat and CCNU varied if the interval between them was altered. The maximum effect was found if the heat and drug were given in close sequence

The Josephson plasmon as a Bogoliubov quasiparticle

We study the Josephson effect in alkali atomic gases within the two-mode approximation and show that there is a correspondence between the Bogoliubov description and the harmonic limit of the phase representation. We demonstrate that the quanta of the Josephson plasmon can be identified with the Bogoliubov excitations of the two-site Bose fluid. We thus establish a mapping between the Bogoliubov approximation for the many-body theory and the linearized pendulum Hamiltonian.Comment: 9 pages, LaTeX, submitted to J. Phys.

Economic inequalities in burden of illness, diagnosis and treatment of five long-term conditions in England: panel study

We compared the distribution by wealth of self-reported illness burden (estimated from validated scales, biomarker and reported symptoms) for angina, cataract, depression, diabetes and osteoarthritis, with the distribution of self-reported medical diagnosis and treatment. We aimed to determine if the greater illness burden borne by poorer participants was matched by appropriately higher levels of diagnosis and treatment