Supplementary material for: Impact of ghost introgression on coalescent-based species tree inference and estimation of divergence time

The species studied in any evolutionary investigation generally constitute
a small proportion of all the species currently existing or that have gone
extinct. It is therefore likely that introgression, which is widespread
across the tree of life, involves “ghosts,” i.e., unsampled, unknown, or
extinct lineages. However, the impact of ghost introgression on
estimations of species trees has rarely been studied and is poorly
understood. Here, we use mathematical analysis and simulations to examine
the robustness of species tree methods based on the multispecies
coalescent model to introgression from a ghost or extant lineage. We found
that many results originally obtained for introgression between extant
species can easily be extended to ghost introgression, such as the
strongly interactive effects of incomplete lineage sorting (ILS) and
introgression on the occurrence of anomalous gene trees (AGTs). The
relative performance of the summary species tree method (ASTRAL) and the
full-likelihood method (*BEAST) varies under different introgression
scenarios, with the former being more robust to gene flow between
non-sister species whereas the latter performing better under certain
conditions of ghost introgression. When an outgroup ghost (defined as a
lineage that diverged before the most basal species under investigation)
acts as the donor of the introgressed genes, the time of root divergence
among the investigated species generally was overestimated, whereas
ingroup introgression, as commonly perceived, can only lead to
underestimation. In many cases of ingroup introgression that may or may
not involve ghost lineages, the stronger the ILS, the higher the accuracy
achieved in estimating the time of root divergence, although the topology
of the species tree is more prone to be biased by the effect of
introgression.