Best match graphs arise naturally as the first processing intermediate in algorithms for orthology detection. Let T be a phylogenetic (gene) tree T and an assignment of leaves of T to species. The best match graph is a digraph that contains an arc from x to y if the genes x and y reside in different species and y is one of possibly many (evolutionary) closest relatives of x compared to all other genes contained in the species . Here, we characterize best match graphs and show that it can be decided in cubic time and quadratic space whether derived from a tree in this manner. If the answer is affirmative, there is a unique least resolved tree that explains , which can also be constructed in cubic time.

A wide variety of problems in computational biology, most notably the assessment of orthology, are solved with the help of reciprocal best matches. Using an evolutionary definition of best matches that captures the intuition behind the concept we clarify rigorously the relationships between reciprocal best matches, orthology, and evolutionary events under the assumption of duplication/loss scenarios. We show that the orthology graph is a subgraph of the reciprocal best match graph (RBMG). We furthermore give conditions under which an RBMG that is a cograph identifies the correct orthlogy relation. Using computer simulations we find that most false positive orthology assignments can be identified as so-called good quartets -and thus corrected -in the absence of horizontal transfer. Horizontal transfer, however, may introduce also false-negative orthology assignments.

Background: Many of the commonly used methods for orthology detection start from mutually most similar pairs of genes (reciprocal best hits) as an approximation for evolutionary most closely related pairs of genes (reciprocal best matches). This approximation of best matches by best hits becomes exact for ultrametric dissimilarities, i.e., under the Molecular Clock Hypothesis. It fails, however, whenever there are large lineage specific rate variations among paralogous genes. In practice, this introduces a high level of noise into the input data for best-hit-based orthology detection methods.Results: If additive distances between genes are known, then evolutionary most closely related pairs can be identified by considering certain quartets of genes provided that in each quartet the outgroup relative to the remaining three genes is known. A priori knowledge of underlying species phylogeny greatly facilitates the identification of the required outgroup. Although the workflow remains a heuristic since the correct outgroup cannot be determined reliably in all cases, simulations with lineage specific biases and rate asymmetries show that nearly perfect results can be achieved. In a realistic setting, where distances data have to be estimated from sequence data and hence are noisy, it is still possible to obtain highly accurate sets of best matches.Conclusion: Improvements of tree-free orthology assessment methods can be expected from a combination of the accurate inference of best matches reported here and recent mathematical advances in the understanding of (reciprocal) best match graphs and orthology relations.Availability: Accompanying software is available at https ://githu b.com/david -schal ler/Asymm eTree .

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Two errors in the article Best Match Graphs (Geiß et al. in JMB 78: 2015–2057, 2019) are corrected. One concerns the tacit assumption that digraphs are sink-free, which has to be added as an additional precondition in Lemma 9, Lemma 11, Theorem 4. Correspondingly, Algorithm 2 requires that its input is sink-free. The second correction concerns an additional necessary condition in Theorem 9 required to characterize best match graphs. The amended results simplify the construction of least resolved trees for n-cBMGs, i.e., Algorithm 1. All other results remain unchanged and are correct as stated.

Clustering genes in similarity graphs is a popular approach for orthology prediction. Most algorithms group genes without considering their species, which results in clusters that contain several paralogous genes. Moreover, clustering is known to be problematic when in-paralogs arise from ancient duplications. Recently, we proposed a two-step process that avoids these problems. First, we infer clusters of only orthologs (i.e. with only genes from distinct species), and second, we infer the missing inter-cluster orthologs. In this paper, we focus on the first step, which leads to a problem we call Colorful Clustering . In general, this is as hard as classical clustering. However, in similarity graphs, the number of species is usually small, as well as the neighborhood size of genes in other species. We therefore study the problem of clustering in which the number of colors is bounded by [Formula: see text], and each gene has at most [Formula: see text] neighbors in another species. We show that the well-known cluster editing formulation remains NP-hard even when [Formula: see text] and [Formula: see text]. We then propose a fixed-parameter algorithm in [Formula: see text] to find the single best cluster in the graph. We implemented this algorithm and included it in the aforementioned two-step approach. Experiments on simulated data show that this approach performs favorably to applying only an unconstrained clustering step.