Abstract
The evolutionary relationships among organisms have traditionally been represented using rooted phylogenetic trees. However, due to reticulate processes such as hybridization or lateral gene transfer, evolution cannot always be adequately represented by a phylogenetic tree, and rooted phylogenetic networks that describe such complex processes have been introduced as a generalization of rooted phylogenetic trees. In fact, estimating rooted phylogenetic networks from genomic sequence data and analyzing their structural properties is one of the most important tasks in contemporary phylogenetics. Over the last two decades, several subclasses of rooted phylogenetic networks (characterized by certain structural constraints) have been introduced in the literature, either to model specific biological phenomena or to enable tractable mathematical and computational analyses. In the present manuscript, we provide a thorough review of these network classes, as well as provide a biological interpretation of the structural constraints underlying these networks where possible. In addition, we discuss how imposing structural constraints on the network topology can be used to address the scalability and identifiability challenges faced in the estimation of phylogenetic networks from empirical data.
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Notes
Note that there might be several complete sequences of cherry reductions that reduce an orchard network to a single leaf and the order in which cherry reductions are performed does not matter (Erdős et al 2019, Proposition 4.1).
Note that Huber et al (2016) originally simply called such networks stable. However, as the word “stable” is used in various contexts in the phylogenetic networks literature, we refer to FU-stable networks to avoid any ambiguity.
A rooted edge-weighted phylogenetic network \({\mathcal {N}}\) is called ultrametric if every directed path from the root of \({\mathcal {N}}\) to any leaf has the same length.
We remark that sometimes the notion of display is defined as follows: A phylogenetic network \({\mathcal {N}}\) displays a phylogenetic tree \({\mathcal {T}}\) if \({\mathcal {T}}\) can be obtained from \({\mathcal {N}}\) be deleting arcs and vertices, and suppressing any resulting in-degree one and out-degree one vertices. In this case, the roots of \({\mathcal {N}}\) and \({\mathcal {T}}\) are not required to coincide. Note, however, that the two definitions can be used interchangeably. If a tree \({\mathcal {T}}\) is displayed by \({\mathcal {N}}\) in the first sense, it is also displayed by \({\mathcal {N}}\) in the second sense, and vice versa.
In a semi-binary phylogenetic network all reticulation vertices have in-degree precisely two, but tree vertices may have an out-degree strictly greater than two. Note that every binary phylogenetic network is in particular a semi-binary phylogenetic network.
Let \({\mathcal {N}}\) be a phylogenetic network. Then, a crown is an (undirected) cycle in \({\mathcal {N}}\) consisting only of reticulation arcs.
Note that Willson (2013) considered the problem of reconstructing a phylogenetic network \({\mathcal {N}}\) given the tree-average distance (Willson 2012) between any pair of taxa, which is the expected value of their distance in the trees displayed by \({\mathcal {N}}\), where each displayed tree has a certain probability obtained from assigning inheritance probabilities to the reticulation arcs of \({\mathcal {N}}\).
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Acknowledgements
JCP was supported by the Ministerio de Ciencia e Innovación (MCI), the Agencia Estatal de Investigación (AEI) and the European Regional Development Funds (ERDF); through project PGC2018-096956-B-C43 (FEDER/MICINN/AEI). KW was supported by The Ohio State University’s President’s Postdoctoral Scholars Program. All authors thank two anonymous reviewers for detailed comments on an earlier version of this manuscript.
Funding
JCP was supported by the Ministerio de Ciencia e Innovación (MCI), the Agencia Estatal de Investigación (AEI) and the European Regional Development Funds (ERDF); through project PGC2018-096956-B-C43 (FEDER/MICINN/AEI). KW was supported by The Ohio State University’s President’s Postdoctoral Scholars Program.
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Kong, S., Pons, J.C., Kubatko, L. et al. Classes of explicit phylogenetic networks and their biological and mathematical significance. J. Math. Biol. 84, 47 (2022). https://doi.org/10.1007/s00285-022-01746-y
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DOI: https://doi.org/10.1007/s00285-022-01746-y