Lie Markov Models

TL;DR

This paper introduces Lie Markov models in phylogenetics, which are characterized by their rate matrices forming a Lie algebra, unifying various existing models and providing a hierarchy of models based on symmetry and parameter complexity.

Contribution

It presents a systematic way to generate Lie Markov models using nucleotide permutation symmetries, unifying group-based and equivariant models within a hierarchical framework.

Findings

Full list of Lie Markov models with maximal symmetry for 2, 3, and 4 states.

Identification of models that are neither group-based nor equivariant.

Lie Markov models include well-known models and new interesting examples.

Abstract

Recent work has discussed the importance of multiplicative closure for the Markov models used in phylogenetics. For continuous-time Markov chains, a sufficient condition for multiplicative closure of a model class is ensured by demanding that the set of rate-matrices belonging to the model class form a Lie algebra. It is the case that some well-known Markov models do form Lie algebras and we refer to such models as "Lie Markov models". However it is also the case that some other well-known Markov models unequivocally do not form Lie algebras. In this paper, we will discuss how to generate Lie Markov models by demanding that the models have certain symmetries under nucleotide permutations. We show that the Lie Markov models include, and hence provide a unifying concept for, "group-based" and "equivariant" models. For each of two, three and four character states, the full list of Lie Markov models with maximal symmetry is presented and shown to include interesting examples that are neither group-based nor equivariant. We also argue that our scheme is pleasing in the context of applied phylogenetics, as, for a given symmetry of nucleotide substitution, it provides a natural hierarchy of models with increasing number of parameters.