Algebraic invariants for inferring 4-leaf semi-directed phylogenetic networks

A core goal of phylogenomics is to determine the evolutionary history of a set of species from biological sequence data. Phylogenetic networks are able to describe more complex evolutionary phenomena than phylogenetic trees but are more difficult to accurately reconstruct. Recently, there has been growing interest in developing methods to infer semi-directed phylogenetic networks. As computing such networks can be computationally intensive, one approach to building such networks is to puzzle together smaller networks. Thus, it is essential to have robust methods for inferring semi-directed phylogenetic networks on small numbers of taxa. In this paper, we investigate an algebraic method for performing phylogenetic network inference from nucleotide sequence data on 4-leaved semi-directed phylogenetic networks by analysing the distribution of leaf-pattern probabilities. On simulated data, we found that we can correctly identify with high accuracy semi-directed networks as sequences approach 10Mbp in length, and that we are able to use our approach to identify tree-like evolution and determine the underlying tree. We also applied our approach to published transcriptome data from swordtail fish to compare its performance with a pseudolikelihood method for inferring semi-directed networks.