Multiple genome rearrangement methodology facilitates the inference of animal phylogeny from gene orders on the mitochondrial genome. Breakpoint distance is preferable to other, highly correlated but computationally more difficult, genomic distances when applied to these data. A number of theories of metazoan evolution are compared to phylogenies reconstructed by ancestral genome optimization, using a minimal total breakpoints criterion. The notion of unambiguously reconstructed segments is introduced as a way of extracting the invariant aspects of multiple solutions for a given ancestral genome; this enables a detailed reconstruction of the evolution of non-tRNA mitochondrial gene order. 1 Introduction. In comparative genomics, the quantitative comparison of gene order differences can be used for phylogenetic inference about a set of organisms. This generally involves methods based on distance matrices (e.g. Sankoff et al., 1992), though it would be of more interest to empl...

We review the combinatorial optimization problems in calculating edit distances between genomes and phylogenetic inference based on minimizing gene order changes. With a view to avoiding the computational cost and the "long branches attract" artifact of some tree-building methods, we explore the probabilization of genome rearrangment models prior to developing a methodology based on branch-length invariants. We characterize probabilistically the evolution of the structure of the gene adjacency set for inversions on unsigned circular genomes and, using a non-trivial recurrence relation, inversions on signed genomes. Concepts from the theory of invariants developed for the phylogenetics of homologous gene sequences can be used to derive a complete set of linear invariants for unsigned inversions, as well as for a mixed rearrangement model for signed genomes, though not for pure transposition nor pure signed inversion models. The invariants are based on an extended Jukes-Cantor semigroup....

The method of phylogenetic invariants was developed to apply to aligned sequence data generated, according to a stochastic substitution model, for N species related through an unknown phylogenetic tree. The invariants are functions of the probabilities of the observable N-tuples, which are identically zero, over all choices of branch length, for some trees. Evaluating the invariants associated with all possible trees, using observed N-tuple frequencies over all sequence positions, enables us to rapidly infer the generating tree. An aspect of evolution at the genomic level much studied recently is the rearrangements of gene order along the chromosome from one species to another. Instead of the substitutions responsible for sequence evolution, we examine the non-local processes responsible for genome rearrangements such as inversion of arbitrarily long segments of chromosomes. By treating the potential adjacency of each possible pair of genes as a "position", an appropriate "substitution" model can be recognized as governing the rearrangement process, and a probabilistically principled phylogenetic inference can be set up. We calculate the invariants for this process for N = 5, and apply them to mitochondrial genome data from coelomate metazoans, showing how they resolvekey aspects of branching order.

Abstract. Phylogenetic invariants are certain polynomials in the joint probability distribution of a Markov model on a phylogenetic tree. Such polynomials are of theoretical interest in the field of algebraic statistics and they are also of practical interest—they can be used to construct phylogenetic trees. This paper is a self-contained introduction to the algebraic, statistical, and computational challenges involved in the practical use of phylogenetic invariants. We survey the relevant literature and provide some partial answers and many open problems.

Abstract. Erikkson showed that singular value decomposition(SVD) of flatten-ings determined a partition of a phylogenetic tree to be a split ([7]). In this paper, based on his work, we develop new statistically consistent algorithms fit for grid computing to construct a phylogenetic tree by computing SVD of flattenings with the small fixed number of rows. 1.