Markov invariants for phylogenetic rate matrices derived from embedded submodels

TL;DR

This paper introduces a new class of phylogenetic models based on embedded rate matrices, identifies Markov invariants for these models, and demonstrates their effectiveness in inferring evolutionary distances with potential advantages over existing invariants.

Contribution

It provides a representation-theoretic method to identify and count Markov invariants for symmetric embedded models, including explicit enumeration for low-dimensional cases.

Findings

Two quadratic invariants for 2 taxa in the 2->3 embedded model

Invariants can directly infer pairwise distances from sequences

Simulations show invariants outperform standard cubic invariants

Abstract

We consider novel phylogenetic models with rate matrices that arise via the embedding of a progenitor model on a small number of character states, into a target model on a larger number of character states. Adapting representation-theoretic results from recent investigations of Markov invariants for the general rate matrix model, we give a prescription for identifying and counting Markov invariants for such `symmetric embedded' models, and we provide enumerations of these for low-dimensional cases. The simplest example is a target model on 3 states, constructed from a general 2 state model; the `2->3' embedding. We show that for 2 taxa, there exist two invariants of quadratic degree, that can be used to directly infer pairwise distances from observed sequences under this model. A simple simulation study verifies their theoretical expected values, and suggests that, given the appropriateness of the model class, they have greater statistical power than the standard (log) Det invariant (which is of cubic degree for this case).