A tight kernel for computing the tree bisection and reconnection distance between two phylogenetic trees

In 2001 Allen and Steel showed that, if subtree and chain reduction rules have been applied to two unrooted phylogenetic trees, the reduced trees will have at most 28k taxa where k is the TBR (Tree Bisection and Reconnection) distance between the two trees. Here we reanalyse Allen and Steel's kernelization algorithm and prove that the reduced instances will in fact have at most 15k-9 taxa. Moreover we show, by describing a family of instances which have exactly 15k-9 taxa after reduction, that this new bound is tight. These instances also have no common clusters, showing that a third commonly-encountered reduction rule, the cluster reduction, cannot further reduce the size of the kernel in the worst case. To achieve these results we introduce and use "unrooted generators" which are analogues of rooted structures that have appeared earlier in the phylogenetic networks literature. Using similar argumentation we show that, for the minimum hybridization problem on two rooted trees, 9k-2 is a tight bound (when subtree and chain reduction rules have been applied) and 9k-4 is a tight bound (when, additionally, the cluster reduction has been applied) on the number of taxa, where k is the hybridization number of the two trees.Comment: One figure added, two small typos fixed. This version to appear in SIDMA (SIAM Journal on Discrete Mathematics

Maximum parsimony distance on phylogenetic trees: A linear kernel and constant factor approximation algorithm

Maximum parsimony distance is a measure used to quantify the dissimilarity of two unrooted phylogenetic trees. It is NP-hard to compute, and very few positive algorithmic results are known due to its complex combinatorial structure. Here we address this shortcoming by showing that the problem is fixed parameter tractable. We do this by establishing a linear kernel i.e., that after applying certain reduction rules the resulting instance has size that is bounded by a linear function of the distance. As powerful corollaries to this result we prove that the problem permits a polynomial-time constant-factor approximation algorithm; that the treewidth of a natural auxiliary graph structure encountered in phylogenetics is bounded by a function of the distance; and that the distance is within a constant factor of the size of a maximum agreement forest of the two trees, a well studied object in phylogenetics

Analysis of the impact on phylogenetic inference of non-reversible nucleotide substitution models

Most phylogenetic trees are inferred using time-reversible evolutionary models that assume that the relative rates of substitution for any given pair of nucleotides are the same regardless of the direction of the substitutions. However, there is no reason to assume that the underlying biochemical mutational processes that cause substitutions are similarly symmetrical. Here, we evaluate the effect on phylogenetic inference in empirical viral and simulated data of incorporating non-reversibility into models of nucleotide substitution processes. I consider two non-reversible nucleotide substitution models: (1) a 6-rate nonreversible model (NREV6) that is applicable to analyzing mutational processes in double-stranded genomes in that complementary substitutions occur at identical rates; and (2) a 12-rate non-reversible model (NREV12) that is applicable to analyzing mutational processes in single-stranded (ss) genomes in that all substitution types are free to occur at different rates. Using likelihood ratio and Akaike Information Criterion-based model tests, we show that, surprisingly, NREV12 provided a significantly better fit than the General Time Reversible (GTR) and NREV6 models to 21/31 dsRNA and 20/30 dsDNA datasets. As expected, however, NREV12 provided a significantly better fit to 24/33 ssDNA and 40/47 ssRNA datasets. I tested how non-reversibility impacts the accuracy with which phylogenetic trees are inferred. As simulated degrees of non-reversibility (DNR) increased, the tree topology inferences using both NREV12 and GTR became more accurate, whereas inferred tree branch lengths became less accurate. I conclude that while non-reversible models should be helpful in the analysis of mutational processes in most virus species, there is no pressing need to use these models for routine phylogenetic inference. Finally, I introduce a web application, RpNRM, that roots phylogenetic trees using a non-reversible nucleotide substitution model. The phylogenetic tree is rooted on every branch and the likelihoods of each rooting are determined and compared with the highest likelihood tree being identified as that with the most plausible rooting. The rooting accuracy of RpNRM was compared to that of the outgroup rooting method, the midpoint rooting method and another non-reversible model-based rooting method implemented in the program IQTREE. I find that although the RpNRM and IQTREE reversible model-based methods are not as accurate on their own as outgroup or midpoint rooting methods, they nevertheless provide an independent means of verifying the root locations that are inferred by these other methods