Invariants for level-1 phylogenetic networks under the random walk 4-state Markov model

TL;DR

This paper identifies quadratic invariants for level-1 phylogenetic networks under a 4-state Markov model, advancing methods for network inference and parameter identifiability in evolutionary studies.

Contribution

It determines all quadratic invariants for sunlet networks under the random walk 4-state model and introduces a new class of invariants for level-1 networks using toric fiber products.

Findings

All quadratic invariants for sunlet networks are characterized.

A new class of invariants for level-1 networks is constructed.

An efficient method for finding invariants surpassing previous approaches.

Abstract

Phylogenetic networks can represent evolutionary events that cannot be described by phylogenetic trees, such as hybridization, introgression, and lateral gene transfer. Studying phylogenetic networks under a statistical model of DNA sequence evolution can aid the inference of phylogenetic networks. Most notably Markov models like the Jukes-Cantor or Kimura-3 model can been employed to infer a phylogenetic network using phylogenetic invariants. In this article we determine all quadratic invariants for sunlet networks under the random walk 4-state Markov model, which includes the aforementioned models. Taking toric fiber products of trees and sunlet networks, we obtain a new class of invariants for level-1 phylogenetic networks under the same model. Furthermore, we apply our results to the identifiability problem of a network parameter. In particular, we prove that our new class of invariants of the studied model is not sufficient to derive identifiability of quarnets (4-leaf networks). Moreover, we provide an efficient method that is faster and more reliable than the state-of-the-art in finding a significant number of invariants for many level-1 phylogenetic networks.