Forbidden Time Travel: Characterization of Time-Consistent Tree Reconciliation Maps

Motivation: In the absence of horizontal gene transfer it is possible to reconstruct the historyof gene families from empirically determined orthology relations, which are equivalent to event-labeled gene trees. Knowledge of the event labels considerably simplifies the problem of recon-ciling a gene tree T with a species trees S, relative to the reconciliation problem without priorknowledge of the event types. It is well-known that optimal reconciliations in the unlabeled casemay violate time-consistency and thus are not biologically feasible. Here we investigate the math-ematical structure of the event labeled reconciliation problem with horizontal transfer.Results: We investigate the issue of time-consistency for the event-labeled version of the reconcil-iation problem, provide a convenient axiomatic framework, and derive a complete characterizationof time-consistent reconciliations. This characterization depends on certain weak conditions onthe event-labeled gene trees that reflect conditions under which evolutionary events are observ-able at least in principle. We give an O(|V (T )| log(|V (S)|))-time algorithm to decide whether atime-consistent reconciliation map exists. It does not require the construction of explicit timingmaps, but relies entirely on the comparably easy task of checking whether a small auxiliary graphis acyclic.Significance: The combinatorial characterization of time consistency and thus biologically fea-sible reconciliation is an important step towards the inference of gene family histories with hor-izontal transfer from orthology data, i.e., without presupposed gene and species trees. The fastalgorithm to decide time consistency is useful in a broader context because it constitutes an at-tractive component for all tools that address tree reconciliation problems.