Computing phylogenetic invariants for time-reversible models: from TN93 to its submodels

TL;DR

This paper develops a method to derive phylogenetic invariants for time-reversible models, including submodels of the Tamura-Nei model, using algebraic geometry and basis transformations, aiding phylogenetic analysis.

Contribution

It extends the derivation of phylogenetic invariants to non-equivariant time-reversible models, providing explicit equations for submodels of Tamura-Nei.

Findings

Varieties of submodels coincide with intersections with the general model

Binomial equations suffice for defining the model's variety

Explicit equations for local complete intersections are provided

Abstract

Phylogenetic invariants are equations that vanish on algebraic varieties associated with Markov processes that model molecular substitutions on phylogenetic trees. For practical applications, it is essential to understand these equations across a wide range of substitution models. Recent work has shown that, for equivariant models, phylogenetic invariants can be derived from those of the general Markov model by restricting to the linear space defined by the model (namely, the space of mixtures of distributions on the model). Following this philosophy, we describe the space of mixtures and phylogenetic invariants for time-reversible models that are not equivariant. Specifically, we study two submodels of the Tamura-Nei nucleotide substitution model (Felsenstein 81 and 84) using an orthogonal change of basis recently introduced for algebraic time-reversible models. For tripods, we prove that the algebraic variety of each submodel coincides with the variety of Tamura-Nei intersected with the linear space of the submodel. In the case of quartets, we show that it is an irreducible component of this intersection. Moreover, we demonstrate that it suffices to consider only the binomial equations defining the linear space, which correspond to the natural symmetries of the model in the new coordinates. For each submodel, we explicitly provide equations defining a local complete intersection that characterizes the phylogenetic variety on a dense open subset containing the biologically relevant points.