On the elusiveness of clusters

A Resolution of the Static Formulation Question for the Problem of Computing the History Bound

Evolutionary data has been traditionally modeled via phylogenetic trees; however, branching alone cannot model conflicting phylogenetic signals, so networks are used instead. Ancestral recombination graphs (ARGs) are used to model the evolution of incompatible sets of SNP data, allowing each site to mutate only once. The model often aims to minimize the number of recombinations. Similarly, incompatible cluster data can be represented by a reticulation network that minimizes reticulation events. The ARG literature has traditionally been disjoint from the reticulation network literature. By building on results from the reticulation network literature, we resolve an open question of interest to the ARG community. We explicitly prove that the History Bound, a lower bound on the number of recombinations in an ARG for a binary matrix, which was previously only defined procedurally, is equal to the minimum number of reticulation nodes in a network for the corresponding cluster data. To facilitate the proof, we give an algorithm that constructs this network using intermediate values from the procedural History Bound definition. We then develop a top-down algorithm for computing the History Bound, which has the same worst-case runtime as the known dynamic program, and show that it is likely to run faster in typical cases.

On the fixed parameter tractability of agreement-based phylogenetic distances

Three important and related measures for summarizing the dissimilarity in phylogenetic trees are the minimum number of hybridization events required to fit two phylogenetic trees onto a single phylogenetic network (the hybridization number), the (rooted) subtree prune and regraft distance (the rSPR distance) and the tree bisection and reconnection distance (the TBR distance) between two phylogenetic trees. The respective problems of computing these measures are known to be NP-hard, but also fixed-parameter tractable in their respective natural parameters. This means that, while they are hard to compute in general, for cases in which a parameter (here the hybridization number and rSPR/TBR distance, respectively) is small, the problem can be solved efficiently even for large input trees. Here, we present new analyses showing that the use of the "cluster reduction" rule-already defined for the hybridization number and the rSPR distance and introduced here for the TBR distance-can transform any [Formula: see text]-time algorithm for any of these problems into an [Formula: see text]-time one, where n is the number of leaves of the phylogenetic trees, p is the natural parameter and k is a much stronger (that is, smaller) parameter: the minimum level of a phylogenetic network displaying both trees.

Efficient algorithms for reconciling gene trees and species networks via duplication and loss events

Reconciliation methods explain topology differences between a species tree and a gene tree by evolutionary events other than speciations. However, not all phylogenies are trees: hybridization can occur and create new species and this results into reticulate phylogenies. Here, we consider the problem of reconciling a gene tree with a species network via duplication and loss events. Two variants are proposed and solved with effcient algorithms: the first one finds the best tree in the network with which to reconcile the gene tree, and the second one finds the best reconciliation between the gene tree and the whole network.

A simple fixed parameter tractable algorithm for computing the hybridization number of two (not necessarily binary) trees

Here, we present a new fixed parameter tractable algorithm to compute the hybridization number r of two rooted, not necessarily binary phylogenetic trees on taxon set Χ in time (6(r)r!) · poly(n), where n = |Χ|. The novelty of this approach is its use of terminals, which are maximal elements of a natural partial order on Χ, and several insights from the softwired clusters literature. This yields a surprisingly simple and practical bounded-search algorithm and offers an alternative perspective on the underlying combinatorial structure of the hybridization number problem.

Quartets and unrooted phylogenetic networks

Phylogenetic networks were introduced to describe evolution in the presence of exchanges of genetic material between coexisting species or individuals. Split networks in particular were introduced as a special kind of abstract network to visualize conflicts between phylogenetic trees which may correspond to such exchanges. More recently, methods were designed to reconstruct explicit phylogenetic networks (whose vertices can be interpreted as biological events) from triplet data. In this article, we link abstract and explicit networks through their combinatorial properties, by introducing the unrooted analog of level-k networks. In particular, we give an equivalence theorem between circular split systems and unrooted level-1 networks. We also show how to adapt to quartets some existing results on triplets, in order to reconstruct unrooted level-k phylogenetic networks. These results give an interesting perspective on the combinatorics of phylogenetic networks and also raise algorithmic and combinatorial questions.