Conference Agenda

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Session Overview
Session
MS177, part 2: Algebraic and combinatorial phylogenetics
Time:
Wednesday, 10/Jul/2019:
10:00am - 12:00pm

Location: Unitobler, F011
30 seats, 59m^2

Presentations
10:00am - 12:00pm

Algebraic and combinatorial phylogenetics

Chair(s): Marta Casanellas (Universitat Politècnica de Catalunya), Jane Coons (North Carolina State University), Seth Sullivant (North Carolina State University)

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.

 

(25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise)

 

Weighting the Coalescent

Joseph Rusinko
Hobart and William Smith Colleges

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.

 

Identifiability of 2-tree mixtures for the Kimura 3ST model

Jesús Fernández-Sánchez1, Marta Casanellas1, Alessandro Oneto2
1Universitat Politècnica de Catalunya, 2BGSMath and Universitat Politècnica de Catalunya

The inference of evolutionary trees from molecular sequence data relies on modeling site substitutions by a Markov process on a phylogenetic tree, or by a mixture of such processes in a number of trees (not necessarily distinct). The identifiability of the parameters of the models is a crucial feature for this inference process to be consistent. From an algebraic geometry perspective, the unmixed substitution models can be described in terms of some algebraic varieties associated to the tree topologies while the mixtures of these models correspond to the join of these varieties (secant varieties, when the trees considered are the same).

The identifiability of the 2-tree mixtures (mixed models obtained from two tree topologies) under the so-called group-based models with 4 states has been deeply studied and, in particular, Allman et al. 2011 proved that under the JC and K2P models, it is possible to distinguish unmixed models from mixtures obtained from two trees. A key point in the proof is the existence of linear constraints that allow us to distinguish between different tree topologies. Unfortunately, such linear equations do not exist for the more general K3P model.

In this talk, we will recall some general facts on mixed models and state some results of our joint work with Marta Casanellas and Alessandro Oneto. In particular, we will present some advances related to the generic identifiability of tree parameters under the K3P model.

 

Markov association schemes

Jeremy Sumner
University of Tasmania

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.

 

Existence of maximally probable ranked gene tree topologies with a matching unranked topology

Filippo Disanto1, Pasquale Miglionico2, Guido Narduzzi2
1University of Pisa, 2Scuola Normale Superiore, Pisa

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.