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.

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 of certain deformations of the braid arrangement are surveyed. In particular, there are unexpected connections with the theory of interval orders and with the enumeration of 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.

From patterns to processes: Phase and density dependencies in the Canadian lynx cycle

Across the boreal forest of North America, lynx populations undergo 10-year cycles. Analysis of 21 time series from 1821 to the present demonstrates that these fluctuations are generated by nonlinear processes with regulatory delays. Trophic interactions between lynx and hares cause delayed density-dependent regulation of lynx population growth. The nonlinearity, in contrast, appears to arise from phase dependencies in hunting success by lynx through the cycle. Using a combined approach of empirical, statistical, and mathematical modeling, we highlight how shifts in trophic interactions between the lynx and the hare generate the nonlinear process primarily by shifting functional response curves during the increase and the decrease phases.

A phylogenomic approach to microbial evolution

To study the origin and evolution of biochemical pathways in microorganisms, we have developed methods and software for automatic, large-scale reconstructions of phylogenetic relationships. We define the complete set of phylogenetic trees derived from the proteome of an organism as the phylome and introduce the term phylogenetic connection as a concept that describes the relative relationships between taxa in a tree. A query system has been incorporated into the system so as to allow searches for defined categories of trees within the phylome. As a complement, we have developed the pyphy system for visualising the results of complex queries on phylogenetic connections, genomic locations and functional assignments in a graphical format. Our phylogenomics approach, which links phylogenetic information to the flow of biochemical pathways within and among microbial species, has been used to examine more than 8000 phylogenetic trees from seven microbial genomes. The results have revealed a rich web of phylogenetic connections. However, the separation of Bacteria and Archaea into two separate domains remains robust.

Genes, peoples, and languages

The genetic history of a group of populations is usually analyzed by reconstructing a tree of their origins. Reliability of the reconstruction depends on the validity of the hypothesis that genetic differentiation of the populations is mostly due to population fissions followed by independent evolution. If necessary, adjustment for major population admixtures can be made. Dating the fissions requires comparisons with paleoanthropological and paleontological dates, which are few and uncertain. A method of absolute genetic dating recently introduced uses mutation rates as molecular clocks; it was applied to human evolution using microsatellites, which have a sufficiently high mutation rate. Results are comparable with those of other methods and agree with a recent expansion of modern humans from Africa. An alternative method of analysis, useful when there is adequate geographic coverage of regions, is the geographic study of frequencies of alleles or haplotypes. As in the case of trees, it is necessary to summarize data from many loci for conclusions to be acceptable. Results must be independent from the loci used. Multivariate analyses like principal components or multidimensional scaling reveal a number of hidden patterns and evaluate their relative importance. Most patterns found in the analysis of human living populations are likely to be consequences of demographic expansions, determined by technological developments affecting food availability, transportation, or military power. During such expansions, both genes and languages are spread to potentially vast areas. In principle, this tends to create a correlation between the respective evolutionary trees. The correlation is usually positive and often remarkably high. It can be decreased or hidden by phenomena of language replacement and also of gene replacement, usually partial, due to gene flow.