Orthology relations, symbolic ultrametrics, and cographs

Abstract

Orthology detection is an important problem in comparative and evolutionary genomics and, consequently, a variety of orthology detection methods have been devised in recent years. Although many of these methods are dependent on generating gene and/or species trees, it has been shown that orthology can be estimated at acceptable levels of accuracy without having to infer gene trees and/or reconciling gene trees with species trees. Thus, it is of interest to understand how much information about the gene tree, the species tree, and their reconciliation is already contained in the orthology relation on the underlying set of genes. Here we shall show that a result by Böcker and Dress concerning symbolic ultrametrics, and subsequent algorithmic results by Semple and Steel for processing these structures can throw a considerable amount of light on this problem. More specifically, building upon these authors' results, we present some new characterizations for symbolic ultrametrics and new algorithms for recovering the associated trees, with an emphasis on how these algorithms could be potentially extended to deal with arbitrary orthology relations. In so doing we shall also show that, somewhat surprisingly, symbolic ultrametrics are very closely related to cographs, graphs that do not contain an induced path on any subset of four vertices. We conclude with a discussion on how our results might be applied in practice to orthology detection.

Orthology and paralogy constraints: satisfiability and consistency

“…The problem of constraint graph satisfiability has been addressed in [17] in the restricted case of a full set of constraints. The following theorem is a reformulation of one of the main results of this paper.…”

“…Theorem 2 ( [17] ) A full constraint graph R is satisfiable if and only if R is P 4 - free (or equivalently, iff $\stackrel{MathClass-op\; ¯}{R}$ is P 4 -free since P 4 is self-complementary), meaning that no four vertices of R induce a path of length 4. …”

“…Recent studies have considered the connection of trees and orthology from the angle of reconstructing phylogenies from orthology relations [17-19]: How much information about the gene tree, the species tree and their reconciliation is already contained in the orthology relation between genes? In other words, having a set $\mathcal{C}$ of full pairwise orthology/paralogy relations (each pair of genes is constrained), can one reconstruct the gene and species trees?…”

“…Similarly to gene tree editing, the first question to be asked is whether the orthology/paralogy constraints can simultaneously co-exist in a history of the gene family. Interestingly, by making the link with symbolic ultrametrics and co-graphs, a simple characterization of satisfiability (symbolic ultrametric) for full paralogy/orthology relations is given in [17], where satisfiability relates to the existence of an event-labeled gene tree G (symbolic representation) leading to $\mathcal{C}$. Notice that satisfiability does not mean the possibility for orthology/paralogy relations to co-exist in a true history, as the triplet phylogenies contained in G are not necessarily consistent (included in a species tree).…”

“…Here, we consider the general case for $\mathcal{C}$: in contrast with [17,18], we do not require $\mathcal{C}$ to be full, i.e., to involve a constraint for each pair of genes. We introduce the notations and problems in the following section.…”