Identifiability of Two-Tree Mixtures for Group-Based Models

Abstract

Phylogenetic data arising on two possibly different tree topologies might be mixed through several biological mechanisms, including incomplete lineage sorting or horizontal gene transfer in the case of different topologies, or simply different substitution processes on characters in the case of the same topology. Recent work on a 2-state symmetric model of character change showed that for 4 taxa, such a mixture model has nonidentifiable parameters, and thus, it is theoretically impossible to determine the two tree topologies from any amount of data under such circumstances. Here, the question of identifiability is investigated for two-tree mixtures of the 4-state group-based models, which are more relevant to DNA sequence data. Using algebraic techniques, we show that the tree parameters are identifiable for the JC and K2P models. We also prove that generic substitution parameters for the JC mixture models are identifiable, and for the K2P and K3P models obtain generic identifiability results for mixtures on the same tree. This indicates that the full phylogenetic signal remains in such mixtures, and the 2-state symmetric result is thus a misleading guide to the behavior of other models.

Tying Up Loose Strands: Defining Equations of the Strand Symmetric Model

“…In this paper, we show that these 50 equations in fact generate the entire ideal of the SSM for K 1,3 . First, we use the parameterization of the model after the matrix-valued Fourier transform and the tropical secant dimension technique of Draisma [5] to determine the dimension of the variety of probability distributions arising from the model.…”

“…In the Fourier coordinates, it becomes evident that the parameterization of the model for K 1,3 is a coordinate projection of the secant variety of the Segre embedding of P 3 × P 3 × P 3 . From this observation, the same authors were able to find 32 degree three and 18 degree four invariants of the homogenous ideal for K 1,3 and to show that these invariants generate the ideal up to degree four. Whether or not these equations generate the entire ideal was heretofore unknown.…”

“…[2]). Results in [6] imply that to determine generators of the ideal of phylogenetic invariants for any trivalent tree, it suffices to determine them for the claw tree, K 1,3 .…”

Invariant based quartet puzzling

“…The computational power required to complete these tasks makes the direct use of invariants impractical for large data sets. As such, the resurgence of interest in invariant based phylogenies has been focused both around the development of new and exciting mathematics, and as a theoretical framework for proving that certain types of phylogenetic reconstructions are possible [9,14,15]. …”

“…Unlike distance based methods, they can account for any conceivable evolutionary rate, and any number of differences per site. They have been used, for example, to demonstrate when it is possible to determine if a mixture model can be inferred from a data set [14]. However, practicing biologists have widely dismissed invariant based algorithms because there has not been a clear algorithm for applying them to realistic phylogenetic reconstruction problems.…”