Polyhedral computational geometry for averaging metric phylogenetic trees

This paper investigates the computational geometry relevant to calculations of the Frechet mean and variance for probability distributions on the phylogenetic tree space of Billera, Holmes and Vogtmann, using the theory of probability measures on spaces of nonpositive curvature developed by Sturm. We show that the combinatorics of geodesics with a specified fixed endpoint in tree space are determined by the location of the varying endpoint in a certain polyhedral subdivision of tree space. The variance function associated to a finite subset of tree space has a fixed $C^\infty$ algebraic formula within each cell of the corresponding subdivision, and is continuously differentiable in the interior of each orthant of tree space. We use this subdivision to establish two iterative methods for producing sequences that converge to the Frechet mean: one based on Sturm's Law of Large Numbers, and another based on descent algorithms for finding optima of smooth functions on convex polyhedra. We present properties and biological applications of Frechet means and extend our main results to more general globally nonpositively curved spaces composed of Euclidean orthants.Comment: 43 pages, 6 figures; v2: fixed typos, shortened Sections 1 and 5, added counter example for polyhedrality of vistal subdivision in general CAT(0) cubical complexes; v1: 43 pages, 5 figure

Investigating Extra Hepatic Steroid And Eicosanoid Metabolizing Enzymes In Cattle

Steroid and eicosanoid metabolism occurs in two phases and primarily within hepatic tissues, but localized metabolism has been examined in several extra-hepatic tissues in humans and rodents. Phase I of metabolism is performed by Cytochrome P450s (CYP) that add hydroxyl groups to the carbon ring structure which is further metabolized by phase II UDP-glucuronosyltransferase (UGT). The overall objectives of the following experiments were to: 1) determine the amount of extra-hepatic steroid metabolism within reproductive tissues of cattle across the estrous cycle; 2) determine the amount of extra-hepatic steroid metabolism and an oxylipin profile within reproductive tissues of cattle based on pregnancy status; and 3) determine the amount of endometrial blood perfusion in cattle using a novel laser Doppler technique. Activity of CYP1A was found within corpora lutea (CL) tissues of both pregnant and non-pregnant cattle, but not within endometrial tissues. Endometrial perfusion, measured using a novel laser Doppler technique, was also validated by measuring angiogenic factors in close proximity to the location of perfusion. A positive correlation (r = 0.28; P = 0.04) was observed between endometrial perfusion and nitrite concentration, an angiogenic factor. Endometrial blood perfusion was affected by the proximity to the CL, but not by the proximity of the dominant follicle. In addition, UGT was categorized across the estrous cycle and the activity was dependent upon the proximity of the CL. Oxylipins, including eicosanoids, were also profiled in CL of cattle that were non-pregnant and pregnant with 5 out of 39 oxylipins differentially expressed. The activity and oxylipin products of steroid and eicosanoid enzymes were not correlated with serum or luteal progesterone. Through these experiments, we have verified that there is localized metabolism of steroids and eicosanoids within reproductive tissues of cattle as well as fetal tissues. Also, we have achieved a full oxylipin profile of non-pregnant and pregnant cattle CL with five oxylipins contained in various amounts between pregnancy status

A Note on the Unsolvability of the Weighted Region Shortest Path Problem

Let S be a subdivision of the plane into polygonal regions, where each region has an associated positive weight. The weighted region shortest path problem is to determine a shortest path in S between two points s, t in R^2, where the distances are measured according to the weighted Euclidean metric-the length of a path is defined to be the weighted sum of (Euclidean) lengths of the sub-paths within each region. We show that this problem cannot be solved in the Algebraic Computation Model over the Rational Numbers (ACMQ). In the ACMQ, one can compute exactly any number that can be obtained from the rationals Q by applying a finite number of operations from +, -, \times, \div, \sqrt[k]{}, for any integer k >= 2. Our proof uses Galois theory and is based on Bajaj's technique.Comment: 6 pages, 1 figur