Conference Agenda

Since late eighties, algebraic tools have been present in phylogenetic theory and have been crucial in understanding the limitations of models and methods and in proposing improvements to the existing tools. In this session we intend to present some of the most recent work in this area.

Under the coalescent model, the dominant quartet should match the topology on the species tree. However, in practice we only have a finite sample of gene trees from which to estimate the dominant quartet. We introduce a quartet weighting system which enables accurate species tree reconstruction when combined with a quartet amalgamation algorithm such as MaxCut. The weighting system also provides a mechanism for determining which data should be included in their analysis.

Modeling evolution is the first and one of the most fundamental steps in phylogenetics and it is where the crucial hypotheses that should lead to reconstruct the evolutionary history are assumed. There are several approaches one might take and it is natural to ask which conditions must be required for evolutionary models to fit the real evolutionary processes. For instance, under a Markov process, which are the plausible structures for transition matrices in a given nucleotide or amino acid substitution model? Should any Markov matrix with such structure be considered as a biologically realistic transition matrix? These and similar questions lead to the study of the "Embedding problem for Markov matrices" (Elfving, 1937), which attempts to characterize those Markov matrices that are consistent with a homogeneous continuous-time approach of evolution. In this talk, we focus on the differences between continuous time and discrete time approaches in the design of models for nucleotide substitution. We will review some facts about the connection between stochastic and rate matrices and present some new results related to some popular evolutionary models. In the most general case, we will characterize the embeddability and the identifiability of the rates in a full-dimension relatively open subspace of the set of 4x4 Markov matrices. This is an ongoing work in collaboration with Marta Casanellas and Jordi Roca-Lacostena.

This work concerns a compelling example of the mathematics of phylogenetics leading to a novel algebraic/combinatorial structure. The motivation for this work comes from a simple model of aminoacyl-tRNA synthetase (aaRS) evolution devised by Julia Shore (UTAS) and Peter Wills (U Auckland). Starting with a proposed rooted tree describing the specialization of aaRS through evolution of the genetic code, their model produces a space of symmetric Markov rate matrices that form a commutative algebra under matrix multiplication. We refer to each of these as a `tree-algebra'.

From their construction, one most naturally expects that the tree-algebras occur as special instances of association schemes (which are well-studied in algebraic combinatorics). However, this is incorrect as one finds that a tree-algebra corresponds to an association scheme only in a highly degenerate case. In fact, further study has revealed that both the tree-algebras and association schemes can be conceived of as occurring as special cases of a novel class of combinatorial structures, which we (possibly imperfectly) refer to as `Markov association schemes'.

In this talk, I will describe our attempts thus far to characterize Markov association schemes. In particular, I will present two natural binary operations of `sum' and `product' on the class of schemes and show that the tree algebras arise precisely from repeatedly applying the sum operation to the trivial scheme.

A ranked gene tree topology is a labeled gene tree topology together with a temporal ordering (a ranking) of its coalescence events. A species tree is a labeled species tree topology considered with a set of lengths for its branches that naturally induces a ranking of the coalescence events present in the tree. Disregarding the ordering of the internal nodes of a ranked tree yields a leaf labeled tree topology, which is the unranked topology of the tree. When exactly one gene copy is sampled for each species, we consider ranked gene tree topologies realized in a ranked species tree under the multispecies coalescent model, and study the unranked topology of the ranked gene tree topologies with the largest conditional probability. We show that among the ranked gene tree topologies that are maximally probable, there is always at least one whose unranked topology matches that of the species tree. We also show that not all of the maximally probable ranked gene tree topologies have a concordant unranked topology.