A novel algebraic approach to time-reversible evolutionary models

TL;DR

This paper introduces a new algebraic framework for studying time-reversible evolutionary models, enabling the derivation of phylogenetic invariants for practical models like Tamura-Nei.

Contribution

It develops a novel algebraic approach that generalizes existing methods, allowing for better analysis of widely used time-reversible models in phylogenetics.

Findings

Provides algebraic invariants for Tamura-Nei model

Generalizes Fourier transform for time-reversible models

Enhances algebraic tools for phylogenetic analysis

Abstract

In the last years, algebraic tools have been proven useful in phylogenetic reconstruction and model selection through the study of phylogenetic invariants. However, up to now, the models studied from an algebraic viewpoint are either too general or too restrictive (as group-based models with a uniform stationary distribution) to be used in practice. In this paper we provide a new framework to study time-reversible models, which are the most widely used by biologists. In our approach we consider algebraic time-reversible models on phylogenetic trees (as defined by Allman and Rhodes) and introduce a new inner product to make all transition matrices of the process diagonalizable through the same orthogonal eigenbasis. This framework generalizes the Fourier transform widely used to work with group-based models and recovers some of the well known results. As illustration, we combine our technique with algebraic geometry tools to provide relevant phylogenetic invariants for trees evolving under the Tamura-Nei model of nucleotide substitution.