Hyperplane arrangements, interval orders, and trees.

A hyperplane arrangement is a finite set of hyperplanes in a real affine space. An especially important arrangement is the braid arrangement, which is the set of all hyperplanes xi - xj = 1, 1 trees. For instance, the number of labeled interval orders that can be obtained from n intervals I1,..., In of generic lengths is counted. There is also discussed an arrangement due to N. Linial whose number of regions is the number of alternating (or intransitive) trees, as defined by Gelfand, Graev, and Postnikov [Gelfand, I. M., Graev, M. I., and Postnikov, A. (1995), preprint]. Finally, a refinement is given, related to counting labeled trees by number of inversions, of a result of Shi [Shi, J.-Y. (1986), Lecture Notes in Mathematics, no. 1179, Springer-Verlag] that a certain deformation of the braid arrangement has (n + 1)n-1 regions.

Root of the universal tree of life based on ancient aminoacyl-tRNA synthetase gene duplications.

Universal trees based on sequences of single gene homologs cannot be rooted. Iwabe et al. [Iwabe, N., Kuma, K.-I., Hasegawa, M., Osawa, S. & Miyata, T. (1989) Proc. Natl. Acad. Sci. USA 86, 9355-9359] circumvented this problem by using ancient gene duplications that predated the last common ancestor of all living things. Their separate, reciprocally rooted gene trees for elongation factors and ATPase subunits showed Bacteria (eubacteria) as branching first from the universal tree with Archaea (archaebacteria) and Eucarya (eukaryotes) as sister groups. Given its topical importance to evolutionary biology and concerns about the appropriateness of the ATPase data set, an evaluation of the universal tree root using other ancient gene duplications is essential. In this study, we derive a rooting for the universal tree using aminoacyl-tRNA synthetase genes, an extensive multigene family whose divergence likely preceded that of prokaryotes and eukaryotes. An approximately 1600-bp conserved region was sequenced from the isoleucyl-tRNA synthetases of several species representing deep evolutionary branches of eukaryotes (Nosema locustae), Bacteria (Aquifex pyrophilus and Thermotoga maritima) and Archaea (Pyrococcus furiosus and Sulfolobus acidocaldarius). In addition, a new valyl-tRNA synthetase was characterized from the protist Trichomonas vaginalis. Different phylogenetic methods were used to generate trees of isoleucyl-tRNA synthetases rooted by valyl- and leucyl-tRNA synthetases. All isoleucyl-tRNA synthetase trees showed Archaea and Eucarya as sister groups, providing strong confirmation for the universal tree rooting reported by Iwabe et al. As well, there was strong support for the monophyly (sensu Hennig) of Archaea. The valyl-tRNA synthetase gene from Tr. vaginalis clustered with other eukaryotic ValRS genes, which may have been transferred from the mitochondrial genome to the nuclear genome, suggesting that this amitochondrial trichomonad once harbored an endosymbiotic bacterium.

Matchings and phylogenetic trees

This paper presents a natural coordinate system for phylogenetic trees using a correspondence with the set of perfect matchings in the complete graph. This correspondence produces a distance between phylogenetic trees, and a way of enumerating all trees in a minimal step order. It is useful in randomized algorithms because it enables moves on the space of trees that make random optimization strategies “mix” quickly. It also promises a generalization to intermediary trees when data are not decisive as to their choice of tree, and a new way of constructing Bayesian priors on tree space.

Shape anisotropy of a single random-walk polymer

Random walks have been used to describe a wide variety of systems ranging from cell colonies to polymers. Sixty-five years ago, Kuhn [Kuhn, W. (1934) Kolloid-Z. 68, 2–11] made the prediction, backed later by computer simulations, that the overall shape of a random-walk polymer is aspherical, yet no experimental work has directly tested Kuhn's general idea and subsequent computer simulations. By using fluorescence microscopy, we monitored the conformation of individual, long, random-walk polymers (fluorescently labeled DNA molecules) at equilibrium. We found that a polymer most frequently adopts highly extended, nonfractal structures with a strongly anisotropic shape. The ensemble-average ratio of the lengths of the long and short axes of the best-fit ellipse of the polymer was much larger than unity.