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 presented.) (duplication, loss and transfer) model and a variant of it—the no-(Formula presented.) model—which does not allow (Formula presented.) 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 presented.) model. Computing this distance in the (Formula presented.) 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