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-leaf 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 the undirected phylogenetic network for sequences of
length at least 10kbp. We found that identifying the semi-directed network
is more challenging and requires sequences of length approaching 10Mbp. We
are also able to use our approach to identify tree-like evolution and
determine the underlying tree. Finally, we employ our method on a real
dataset from the Xiphophorus species and use the results to build a
phylogenetic network.