The algebra of the general Markov model on phylogenetic trees and networks

TL;DR

This paper develops an algebraic framework to extend the general Markov model from phylogenetic trees to arbitrary splits and networks, enabling more accurate modeling of complex evolutionary processes including convergent evolution.

Contribution

It introduces an algebraic method to extend the general Markov model to networks, surpassing previous models limited to trees, and highlights properties like convergent evolution.

Findings

Extension of the Markov model to arbitrary splits and networks.

Distinguishes the new approach from previous models on incompatible splits.

Supports modeling of convergent evolution in phylogenetics.

Abstract

It is known that the Kimura 3ST model of sequence evolution on phylogenetic trees can be extended quite naturally to arbitrary split systems. However, this extension relies heavily on mathematical peculiarities of the K3ST model, and providing an analogous augmentation of the general Markov model has thus far been elusive. In this paper we rectify this shortcoming by showing how to extend the general Markov model on trees to to include arbitrary splits; and even further to more general network models. This is achieved by exploring the algebra of the generators of the continuous-time Markov chain together with the "splitting" operator that generates the branching process on phylogenetic trees. For simplicity we proceed by discussing the two state case and note that our results are easily extended to more states with little complication. Intriguingly, upon restriction of the two state general Markov model to the parameter space of the binary symmetric model, our extension is indistinguishable from the previous approach only on trees; as soon as any incompatible splits are introduced the two approaches give rise to differing probability distributions with disparate structure. Through exploration of a simple example, we give a tentative argument that our approach to extending to more general networks has desirable properties that the previous approaches do not share. In particular, our construction allows for the possibility of convergent evolution of previously divergent lineages; a property that is of significant interest for biological applications.