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

With the increasing availability of genomic data, biologists aim to find more accurate descriptions of evolutionary histories influenced by secondary contact, where diverging lineages reconnect before diverging again. Such reticulate evolutionary events can be more accurately represented in phylogenetic networks than in phylogenetic trees. Since the root location of phylogenetic networks cannot be inferred from biological data under several evolutionary models, we consider semi-directed (phylogenetic) networks: partially directed graphs without a root in which the directed edges represent reticulate evolutionary events. By specifying a known outgroup, the rooted topology can be recovered from such networks. We introduce the algorithm Squirrel (Semi-directed Quarnet-based Inference to Reconstruct Level-1 Networks) which constructs a semi-directed level-1 network from a full set of quarnets (four-leaf semi-directed networks). Our method also includes a heuristic to construct such a quarnet set directly from sequence alignments. We demonstrate Squirrel’s performance through simulations and on real sequence data sets, the largest of which contains 29 aligned sequences close to 1.7 Mb long. The resulting networks are obtained on a standard laptop within a few minutes. Lastly, we prove that Squirrel is combinatorially consistent: given a full set of quarnets coming from a triangle-free semi-directed level-1 network, it is guaranteed to reconstruct the original network. Squirrel is implemented in Python, has an easy-to-use graphical user interface that takes sequence alignments or quarnets as input, and is freely available [...]