Reconstructing fully-resolved trees from triplet cover distances

It is a classical result that any finite tree with positively weighted edges, and without vertices of degree 2, is uniquely determined by the weighted path distance between each pair of leaves. Moreover, it is possible for a (small) strict subset L of leaf pairs to suffice for reconstructing the tree and its edge weights, given just the distances between the leaf pairs in L. It is known that any set L with this property for a tree in which all interior vertices have degree 3 must form a cover for T {that is, for each interior vertex v of T, L must contain a pair of leaves from each pair of the three components of T ̶ v. Here we provide a partial converse of this result by showing that if a set L of leaf pairs forms a cover of a certain type for such a tree T then T and its edge weights can be uniquely determined from the distances between the pairs of leaves in L. Moreover, there is a polynomial-time algorithm for achieving this reconstruction. The result establishes a special case of a recent question concerning `triplet covers', and is relevant to a problem arising in evolutionary genomics

Twisted trees and inconsistency of tree estimation when gaps are treated as missing data -- the impact of model mis-specification in distance corrections

Statistically consistent estimation of phylogenetic trees or gene trees is possible if pairwise sequence dissimilarities can be converted to a set of distances that are proportional to the true evolutionary distances. Susko et al. (2004) reported some strikingly broad results about the forms of inconsistency in tree estimation that can arise if corrected distances are not proportional to the true distances. They showed that if the corrected distance is a concave function of the true distance, then inconsistency due to long branch attraction will occur. If these functions are convex, then two "long branch repulsion" trees will be preferred over the true tree -- though these two incorrect trees are expected to be tied as the preferred true. Here we extend their results, and demonstrate the existence of a tree shape (which we refer to as a "twisted Farris-zone" tree) for which a single incorrect tree topology will be guaranteed to be preferred if the corrected distance function is convex. We also report that the standard practice of treating gaps in sequence alignments as missing data is sufficient to produce non-linear corrected distance functions if the substitution process is not independent of the insertion/deletion process. Taken together, these results imply inconsistent tree inference under mild conditions. For example, if some positions in a sequence are constrained to be free of substitutions and insertion/deletion events while the remaining sites evolve with independent substitutions and insertion/deletion events, then the distances obtained by treating gaps as missing data can support an incorrect tree topology even given an unlimited amount of data.Comment: 29 pages, 3 figure

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.

Marine tethysuchian crocodyliform from the ?Aptian-Albian (Lower Cretaceous) of the Isle of Wight, UK

A marine tethysuchian crocodyliform from the Isle of Wight, most likely from the Upper Greensand Formation (upper Albian, Lower Cretaceous), is described. However, we cannot preclude it being from the Ferruginous Sands Formation (upper Aptian), or more remotely, the Sandrock Formation (upper Aptian-upper Albian). The specimen consists of the anterior region of the right dentary, from the tip of the dentary to the incomplete fourth alveolus. This specimen increases the known geological range of marine tethysuchians back into the late Lower Cretaceous. Although we refer it to Tethysuchia incertae sedis, there are seven anterior dentary characteristics that suggest a possible relationship with the Maastrichtian-Eocene clade Dyrosauridae. We also review ‘middle’ Cretaceous marine tethysuchians, including putative Cenomanian dyrosaurids. We conclude that there is insufficient evidence to be certain that any known Cenomanian specimen can be safely referred to Dyrosauridae, as there are some cranial similarities between basal dyrosaurids and Cenomanian–Turonian marine ‘pholidosaurids’. Future study of middle Cretaceous tethysuchians could help unlock the origins of Dyrosauridae and improve our understanding of tethysuchian macroevolutionary trends

Paragangliomas of the sellar region: report of two cases

Journal ArticleTWO CASES OF paraganglioma arising from the parasellar region are presented. Both occurred in middle-aged women who sought treatment of headaches but who had no endocrinological dysfunction; one case was associated with ophthalmoplegia from cavernous sinus involvement. Diagnosis in both cases was confirmed by typical histological appearance and cytochemical demonstration of immunoreactive chromogranin in tumor cells. The pathological features and possible pathogenesis of parasellar paragangliomas are discussed

Phylogenetic Flexibility via Hall-Type Inequalities and Submodularity

Given a collection τ of subsets of a finite set X, we say that τ is phylogenetically flexible if, for any collection R of rooted phylogenetic trees whose leaf sets comprise the collection τ , R is compatible (i.e. there is a rooted phylogenetic X-tree that displays each tree in R). We show that τ is phylogenetically flexible if and only if it satisfies a Hall-type inequality condition of being ‘slim’. Using submodularity arguments, we show that there is a polynomial-time algorithm for determining whether or not τ is slim. This ‘slim’ condition reduces to a simpler inequality in the case where all of the sets in τ have size 3, a property we call ‘thin’. Thin sets were recently shown to be equivalent to the existence of an (unrooted) tree for which the median function provides an injective mapping to its vertex set; we show here that the unrooted tree in this representation can always be chosen to be a caterpillar tree. We also characterise when a collection τ of subsets of size 2 is thin (in terms of the flexibility of total orders rather than phylogenies) and show that this holds if and only if an associated bipartite graph is a forest. The significance of our results for phylogenetics is in providing precise and efficiently verifiable conditions under which supertree methods that require consistent inputs of trees can be applied to any input trees on given subsets of species