Convergence-divergence models: Generalizations of phylogenetic trees modeling gene flow over time

Phylogenetic trees are simple models of evolutionary processes. They describe conditionally independent divergent evolution of taxa from common ancestors. Phylogenetic trees commonly do not have enough flexibility to adequately model all evolutionary processes. For example, introgressive hybridization, where genes can flow from one taxon to another. Phylogenetic networks model evolution not fully described by a phylogenetic tree. However, many phylogenetic network models assume ancestral taxa merge instantaneously to form ``hybrid'' descendant taxa. In contrast, our convergence-divergence models retain a single underlying ``principal'' tree, but permit gene flow over arbitrary time frames. Alternatively, convergence-divergence models can describe other biological processes leading to taxa becoming more similar over a time frame, such as replicated evolution. Here we present novel maximum likelihood-based algorithms to infer most aspects of $N$-taxon convergence-divergence models, many consistently, using a quartet-based approach. The algorithms can be applied to multiple sequence alignments restricted to genes or genomic windows or to gene presence/absence datasets.