Beyond representing orthology relations by trees

Reconstructing the evolutionary past of a family of genes is an important aspect of many genomic studies. To help with this, simple operations on a set of sequences called orthology relations may be employed. In addition to being interesting from a practical point of view they are also attractive from a theoretical perspective in that e. g. a characterization is known for when such a relation is representable by a certain type of phylogenetic tree. For an orthology relation inferred from real biological data it is however generally too much to hope for that it satisfies that characterization. Rather than trying to correct the data in some way or another which has its own drawbacks, as an alternative, we propose to represent an orthology relation $\delta$ in terms of a structure more general than a phylogenetic tree called a phylogenetic network. To compute such a network in the form of a level-1 representation for $\delta$, we introduce the novel {\sc Network-Popping} algorithm which has several attractive properties. In addition, we characterize orthology relations $\delta$ on some set $X$ that have a level-1 representation in terms of eight natural properties for $\delta$ as well as in terms for level-1 representations of orthology relations on certain subsets of $X$.