Orthology and paralogy constraints: satisfiability and consistency

Abstract

BackgroundA variety of methods based on sequence similarity, reconciliation, synteny or functional characteristics, can be used to infer orthology and paralogy relations between genes of a given gene family G. But is a given set C of orthology/paralogy constraints possible, i.e., can they simultaneously co-exist in an evolutionary history for G? While previous studies have focused on full sets of constraints, here we consider the general case where C does not necessarily involve a constraint for each pair of genes. The problem is subdivided in two parts: (1) Is C satisfiable, i.e. can we find an event-labeled gene tree G inducing C? (2) Is there such a G which is consistent, i.e., such that all displayed triplet phylogenies are included in a species tree?ResultsPrevious results on the Graph sandwich problem can be used to answer to (1), and we provide polynomial-time algorithms for satisfiability and consistency with a given species tree. We also describe a new polynomial-time algorithm for the case of consistency with an unknown species tree and full knowledge of pairwise orthology/paralogy relationships, as well as a branch-and-bound algorithm in the case when unknown relations are present. We show that our algorithms can be used in combination with ProteinOrtho, a sequence similarity-based orthology detection tool, to extract a set of robust orthology/paralogy relationships.

On the consistency of orthology relationships

“…T ) along with the species assignment s , is consistent if there exists a species tree S such that ( G,s ) (resp. ( T,s )) is consistent with S [9]. …”

“…In this paper, we extend the results of [9] by giving a O ( n 3 ) time algorithm to decide whether is consistent, even when the species tree is not known and is not full, and show an application on real data. Thus the problems of deciding satisfiability, deciding consistency given a species tree, and deciding consistency with an unknown species tree, are all polynomial-time solvable.…”

“…In the last years, the decision problems associated with these questions have been extensively studied, both when is full , i.e., involves a constraint for each pair of genes in V [7, 8], and when it is not [9]. …”

“…In [9], the authors give O ( n 3 ) time algorithms to decide whether is satisfiable and consistent under the assumption that the species tree is known – where n =| V |. These results hold whether is a full set of constraints or not.…”

“…In light of this result, Hellmuth et al [10] gave a framework for finding the DS-tree and species tree for which the maximum number of triplets are displayed, using Integer Linear Programming. Lafond and El-Mabrouk [9] improved the result of [7] by showing that it is enough to consider only the triplets in t r ( T , s ) that have a speciation node as the root node and a duplication node as the other internal node. This can expressed in terms of the consistency of an orthology graph in the following way.…”