Background

Phylogenetic networks are generalizations of phylogenetic trees, that are used to model evolutionary events in various contexts. Several different methods and criteria have been introduced for reconstructing phylogenetic trees. Maximum Parsimony is a character-based approach that infers a phylogenetic tree by minimizing the total number of evolutionary steps required to explain a given set of data assigned on the leaves. Exact solutions for optimizing parsimony scores on phylogenetic trees have been introduced in the past.

Results

In this paper, we define the parsimony score on networks as the sum of the substitution costs along all the edges of the network; and show that certain well-known algorithms that calculate the optimum parsimony score on trees, such as Sankoff and Fitch algorithms extend naturally for networks, barring conflicting assignments at the reticulate vertices. We provide heuristics for finding the optimum parsimony scores on networks. Our algorithms can be applied for any cost matrix that may contain unequal substitution costs of transforming between different characters along different edges of the network. We analyzed this for experimental data on 10 leaves or fewer with at most 2 reticulations and found that for almost all networks, the bounds returned by the heuristics matched with the exhaustively determined optimum parsimony scores.

Conclusion

The parsimony score we define here does not directly reflect the cost of the best tree in the network that displays the evolution of the character. However, when searching for the most parsimonious network that describes a collection of characters, it becomes necessary to add additional cost considerations to prefer simpler structures, such as trees over networks. The parsimony score on a network that we describe here takes into account the substitution costs along the additional edges incident on each reticulate vertex, in addition to the substitution costs along the other edges which are common to all the branching patterns introduced by the reticulate vertices. Thus the score contains an in-built cost for the number of reticulate vertices in the network, and would provide a criterion that is comparable among all networks. Although the problem of finding the parsimony score on the network is believed to be computationally hard to solve, heuristics such as the ones described here would be beneficial in our efforts to find a most parsimonious network.

Motivation: Identifying orthologous genes in multiple genomes is a fundamental task in comparative genomics. Construction of intergenomic symmetrical best matches (SymBets) and joining them into clusters is a popular method of ortholog definition, embodied in several software programs. Despite their wide use, the computational complexity of these programs has not been thoroughly examined.

Results: In this work, we show that in the standard approach of iteration through all triangles of SymBets, the memory scales with at least the number of these triangles, O(g3) (where g = number of genomes), and construction time scales with the iteration through each pair, i.e. O(g6). We propose the EdgeSearch algorithm that iterates over edges in the SymBet graph rather than triangles of SymBets, and as a result has a worst-case complexity of only O(g3log g). Several optimizations reduce the run-time even further in realistically sparse graphs. In two real-world datasets of genomes from bacteriophages (POGs) and Mollicutes (MOGs), an implementation of the EdgeSearch algorithm runs about an order of magnitude faster than the original algorithm and scales much better with increasing number of genomes, with only minor differences in the final results, and up to 60 times faster than the popular OrthoMCL program with a 90% overlap between the identified groups of orthologs.

Availability and implementation: C++ source code freely available for download at ftp.ncbi.nih.gov/pub/wolf/COGs/COGsoft/

Contact: dmk@stowers.org

Supplementary information: Supplementary materials are available at Bioinformatics online.