Rearrangement moves on rooted phylogenetic networks

Abstract

Phylogenetic tree reconstruction is usually done by local search heuristics that explore the space of the possible tree topologies via simple rearrangements of their structure. Tree rearrangement heuristics have been used in combination with practically all optimization criteria in use, from maximum likelihood and parsimony to distance-based principles, and in a Bayesian context. Their basic components are rearrangement moves that specify all possible ways of generating alternative phylogenies from a given one, and whose fundamental property is to be able to transform, by repeated application, any phylogeny into any other phylogeny. Despite their long tradition in tree-based phylogenetics, very little research has gone into studying similar rearrangement operations for phylogenetic network—that is, phylogenies explicitly representing scenarios that include reticulate events such as hybridization, horizontal gene transfer, population admixture, and recombination. To fill this gap, we propose “horizontal” moves that ensure that every network of a certain complexity can be reached from any other network of the same complexity, and “vertical” moves that ensure reachability between networks of different complexities. When applied to phylogenetic trees, our horizontal moves—named rNNI and rSPR—reduce to the best-known moves on rooted phylogenetic trees, nearest-neighbor interchange and rooted subtree pruning and regrafting. Besides a number of reachability results—separating the contributions of horizontal and vertical moves—we prove that rNNI moves are local versions of rSPR moves, and provide bounds on the sizes of the rNNI neighborhoods. The paper focuses on the most biologically meaningful versions of phylogenetic networks, where edges are oriented and reticulation events clearly identified. Moreover, our rearrangement moves are robust to the fact that networks with higher complexity usually allow a better fit with the data. Our goal is to provide a solid basis for practical phylogenetic network reconstruction.

Generating normal networks via leaf insertion and nearest neighbor interchange

“…The resulting network is denoted by C-Rotate(N, u, r). Note that a child rotation is a special case of the rNNI rearrangement introduced by Gambette et al in [12]. Let T CN k (n) denote the set of TCNs with k reticulations on [1, n].…”

“…As such, a phylogenetic reconstruction method often uses nearest neighbor interchanges (NNIs) or other rearrangement operations to search for an optimal tree or network [8,9]. Recently, different variants of NNI have been proposed for RPNs [10,11,12,13,14,15,16].…”

Exploring the Tiers of Rooted Phylogenetic Network Space Using Tail Moves

“…Gambette et al. ( 2017 ) focused on rooted phylogenetic networks, i.e. phylogenetic networks where the underlying graph is a rooted directed acyclic graph, and introduced generalizations of rNNI and rSPR moves for rooted phylogenetic networks.…”

“… 2014 ) correspond, respectively, to head and tail moves (see Definitions 2.5 and 2.6 ) and their union correspond to the rSPR moves defined in (Gambette et al. 2017 ), c.f. next section.…”

The SNPR neighbourhood of tree-child networks

“…They showed that the size depends on structures of the network like the number of cycles of size three and four. Gambette et al [12] extended this with an upper bound for the neighbourhood size of an unrooted phylogenetic network and NNI, but restricted to networks with a fixed number of reticulations. Similarly, Francis et al [9] gave an upper bound for the same networks but for SPR instead of for NNI.…”