Exploring the space of gene/species reconciliations with transfers

Reconciliations between gene and species trees have important applications in the study of genome evolution (e.g. sequence orthology prediction or quantification of transfer events). While numerous methods have been proposed to infer them, little has been done to study the underlying reconciliation space. In this paper, we characterise the reconciliation space for two evolutionary models: the [Formula: see text] (duplication, loss and transfer) model and a variant of it-the no-[Formula: see text] model-which does not allow [Formula: see text] events (a transfer immediately followed by a loss). We provide formulae to compute the size of the corresponding spaces and define a set of transformation operators sufficient to explore the entire reconciliation space. We also define a distance between two reconciliations as the minimal number of operations needed to transform one into the other and prove that this distance is easily computable in the no-[Formula: see text] model. Computing this distance in the [Formula: see text] model is more difficult and it is an open question whether it is NP-hard or not. This work constitutes an important step toward reconciliation space characterisation and reconciliation comparison, needed to better assess the performance of reconciliation inference methods through simulations.