Time-Consistent Reconciliation Maps and Forbidden Time Travel

Abstract

Background: In the absence of horizontal gene transfer it is possible to reconstruct the history of 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 reconciling a gene tree T with a species trees S, relative to the reconciliation problem without prior knowledge of the event types. It is well-known that optimal reconciliations in the unlabeled case may violate time-consistency and thus are not biologically feasible. Here we investigate the mathematical 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 reconciliation problem, provide a convenient axiomatic framework, and derive a complete characterization of time-consistent reconciliations. This characterization depends on certain weak conditions on the event-labeled gene trees that reflect conditions under which evolutionary events are observable at least in principle. We give an O(|V (T )| log(|V (S)|))-time algorithm to decide whether a time-consistent reconciliation map exists. It does not require the construction of explicit timing maps, but relies entirely on the comparably easy task of checking whether a small auxiliary graph is acyclic. The algorithms are implemented in C++ using the boost graph library and are freely available at https://github.com/Nojgaard/tc-recon. Significance: The combinatorial characterization of time consistency and thus biologically feasible reconciliation is an important step towards the inference of gene family histories with horizontal transfer from orthology data, i.e., without presupposed gene and species trees. The fast algorithm to decide time consistency is useful in a broader context because it constitutes an attractive component for all tools that address tree reconciliation problems.

Reconstruction of time-consistent species trees

“…0. These conditions are also called "observability-axioms" and are exhaustively discussed in [3] and [4]. We repeat here shortly the arguments to justify Condition (O1)-(O3).…”

“…otherwise, i.e., at least one of t(x) and t(y) is a speciation s, µ(x) ≺ S µ(y). We call µ the reconciliation map from (T ;t, σ ) to S. The provided definition of a reconciliation map coincides with the one as given in [3,4,47] and is a natural generalization of the maps as in [35,37,38,48] for the case where no HGT took place.…”

“…In Section An Algorithm for the GTC Problem, we provide a solution to the GTC problem. For the design of this algorithm, we will utilize an auxiliary directed graph A(T, S) based on a given event-labeled gene tree T and a given species tree S. This type of graph was established in [4]. The authors showed that there is time-consistent map between T and S if and only if A(T, S) is acyclic.…”

“…In the latter approximation, we have used the fact that distinct trees T i and T j have disjoint sets of leaf sets (cf. [4,Lemma 1]). Thus, ∑ T i ∈T n i ≤ n and ∑ T j :T i →T j n i ≤ n. In summary, R(T ;t, σ ) can be computed in O(n 3 ) time.…”

Reconstruction of time-consistent species trees

“…Contrary, if one of the copies of a gene "jumps" into a different branch of the species tree, then a HGT event happened. The latter can be annotated in the gene tree by associating a label to the edge that points from the horizontal transfer event to the transferred copy [3][4][5][6][7]. Since both HGT and duplication events occur in between different speciation events, such vertices of the gene trees are usually mapped to the edges of the species tree.…”