Markov Invariants for Phylogenetic Rate Matrices Derived from Embedded Submodels

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 the first few cases with a small number of character states. The simplest example is a target model on three 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 superior statistical properties than the standard (log) Det invariant (which is of cubic degree for this case).

Matrix group structure and Markov invariants in the strand symmetric phylogenetic substitution model

“…The following discussion adopts the notation and adapts the results of [29,32,18], and especially [19]. The required background on symmetric function manipulations can be found in the classic text [25].…”

Lie-Markov Models Derived from Finite Semigroups

“…The geometric consequences of the identification of the Lie algebra underlying this model are explored in [15]. We will not review the construction, but an interesting 3-state model arises from the 2-state general Markov model using the method given in [7]. This model has rate matrices given by…”

“…Since these two models define two-dimensional Lie algebras satisfying the same commutator relations, they are isomorphic as Lie algebras. In fact, for any number of character states k, the method given in [7] produces a two-dimensional Lie-Markov model which is isomorphic, as a Lie algebra, to the 2-state general Markov model. This illustrates that Lie algebra isomorphism is not (in itself) a useful tool for identifying distinct Lie-Markov models.…”

Adventures in invariant theory

“…There are many more gems to be examined in hunting down Markov invariants for different models and subgroups [14,13], with potential practical and theoretical interest. As one instance of as-yet unexplored terrain, for K = 3 we have evidence [25,26] at degree 8 for stochastic tangle ('stangle') invariants with mixed weight, since it turns out that ) .…”