Polynomial-Time Algorithms for Phylogenetic Inference Problems

Abstract

A common problem in phylogenetics is to try to infer a species phylogeny from gene trees. We consider different variants of this problem. The first variant, called Unrestricted Minimal Episodes Inference, aims at inferring a species tree based on a model of speciation and duplication where duplications are clustered in duplication episodes. The goal is to minimize the number of such episodes. The second variant, Parental Hybridization, aims at inferring a species network based on a model of speciation and reticulation. The goal is to minimize the number of reticulation events. It is a variant of the wellstudied Hybridization Number problem with a more generous view on which gene trees are consistent with a given species network. We show that these seemingly different problems are in fact closely related and can, surprisingly, both be solved in polynomial time, using a structure we call "beaded trees". However, we also show that methods based on these problems have to be used with care because the optimal species phylogenies always have some restricted form. We discuss several possibilities to overcome this problem.

Weakly displaying trees in temporal tree-child network

“…In Section 4, we then consider the weak hybrid number of two trees. In particular, we determine the weak hybrid number for a specific pair of phylogenetic trees and show that for this pair of trees we get a different number to the analogous hybrid number defined in [14]. This example shows that the beaded trees introduced in [14] can lead to a quite different solution when aiming to find a network in which to embed the given trees.…”

“…In this case we shall say that the pair of trees is rigidly displayed by the network. Note that related problems were recently considered in [20] (the Parental Tree Network Problem) and in [14] (The Beaded Tree Problem). In the Parental Tree Network Problem the aim is to find a network with a minimum number of reticulation vertices that weakly displays all trees in a given set of phylogenetic trees; in the Beaded Tree Problem, however, networks with parallel edges are permitted and a different concept of displaying is used which can lead to different solutions (see Section 4 for more details).…”

Heading in the right direction? Using head moves to traverse phylogenetic network space

“…The extra structure of phylogenetic networks makes them harder to reconstruct than their tree counterparts. For certain models, it is still possible to quickly construct networks from data (e.g., Van Iersel et al, 2018). Reconstruction is harder in most other cases, depending on the kind of data and the model of reconstruction.…”

The rigid hybrid number for two phylogenetic trees

“…3 for the definition). In biological terms, as nicely explained in van Iersel et al (2018), "this means that different lineages of the gene tree may "travel down" the same branch of the network, as long as any branching node in the tree coincides with a branching node in the network". In this paper, we focus on the special situation where two phylogenetic trees T and T are weakly displayed by a temporal tree-child network under the assumption that there exist simultaneous embeddings of both trees that do not permit more than three branches of T and T to come together at a reticulation vertex.…”

“…In Sect. 8, we consider the relationship between the rigid hybrid number and three closely related hybrid numbers: the weak, beaded, and temporal hybrid numbers (the beaded hybrid number was implicitly defined in van Iersel et al (2018)). In particular, in Theorem 4, we first show that there is a pair of phylogenetic trees on a set X with |X | arbitrarily large, so that the difference between the temporal and rigid hybrid numbers for these two trees is at least |X | 4 − 3.…”