Rearrangement operations on unrooted phylogenetic networks

Abstract

Rearrangement operations transform a phylogenetic tree into another one and hence induce a metric on the space of phylogenetic trees. Popular operations for unrooted phylogenetic trees are NNI (nearest neighbour interchange), SPR (subtree prune and regraft), and TBR (tree bisection and reconnection). Recently, these operations have been extended to unrooted phylogenetic networks, which are generalisations of phylogenetic trees that can model reticulated evolutionary relationships.Here, we study global and local properties of spaces of phylogenetic networks under these three operations. In particular, we prove connectedness and asymptotic bounds on the diameters of spaces of different classes of phylogenetic networks, including tree-based and level-k networks. We also examine the behaviour of shortest TBR-sequence between two phylogenetic networks in a class, and whether the TBR-distance changes if intermediate networks from other classes are allowed: for example, the space of phylogenetic trees is an isometric subgraph of the space of phylogenetic networks under TBR. Lastly, we show that computing the TBR-distance and the PR-distance of two phylogenetic networks is NP-hard.

Exploring spaces of semi-directed phylogenetic networks

“…Moreover, we have introduced two additional operations, CET + and CET − , that allow to move between semi-directed phylogenetic networks and between semi-directed level-1 networks with a fixed leaf set and an arbitrary number of reticulations. While CET moves have a similar flavor as SPR and rSPR moves on unrooted, respectively rooted phylogenetic trees and networks [1,6,7,11], we have also shown that any CET can be translated into a sequence of more local CET 1 moves, which are similar to NNI moves studied on phylogenetic trees and networks [11,16,21,28]. Such CET 1 moves essentially coincide with moves that are used in the popular network inference software PhyloNetworks [29,30] up to a slight relaxation of one of their moves.…”

“…The set X is called the leaf set of N r . As with other publications on spaces of phylogenetic networks [7,21], we allow edges to be in parallel or, equivalently, underlying cycles of length two. A vertex with in-degree two and out-degree one is called a reticulation, and a vertex with in-degree one and out-degree two is called a tree vertex.…”

“…In other words, can every phylogenetic network of a space of networks (e.g., all semi-directed phylogenetic networks on a fixed leaf set) be reached from any other phylogenetic network in the space by applying a sequence of these rearrangement operations such that the resulting network after each operation is also in the space? This question has been analyzed for various spaces of unrooted and rooted phylogenetic trees (e.g., [1,6,14]), unrooted phylogenetic networks (e.g., [16,17,10,21]) and rooted phylogenetic networks (e.g., [7,9,11,20,22,23]), and several rearrangement moves to traverse these spaces have been introduced. However, these results do not immediately carry over to spaces of semi-directed phylogenetic networks.…”

Exploring spaces of semi-directed level-1 networks

“…As with other publications on spaces of phylogenetic networks (Bordewich et al. 2017 ; Janssen and Klawitter 2019 ), we allow edges to be in parallel or, equivalently, underlying cycles of length two. Although we do allow edges to be in parallel in a rooted phylogenetic network, we note that we do not allow them in rooted level-1 networks as defined later in this section.…”

“…In other words, can every phylogenetic network of a space of networks (e.g., all semi-directed phylogenetic networks on a fixed leaf set) be reached from any other phylogenetic network in the space by applying a sequence of these rearrangement operations such that the resulting network after each operation is also in the space? This question has been analyzed for various spaces of unrooted and rooted phylogenetic trees (e.g., Allen and Steel 2001 ; Bordewich and Semple 2005 ; Hein et al 1996 ), unrooted phylogenetic networks (e.g., Huber et al 2015 , 2016 ; Francis et al 2017 ; Janssen and Klawitter 2019 ) and rooted phylogenetic networks (e.g., Bordewich et al 2017 ; Erdős et al 2021 ; Gambette et al 2017 ; Janssen 2021a ; Janssen et al 2018 ; Klawitter 2018 ), and several rearrangement moves to traverse these spaces have been introduced. We also refer the reader to two excellent PhD theses on the topic by Janssen ( 2021b ) and Klawitter ( 2020 ).…”