The Space of Tree-Based Phylogenetic Networks

Abstract

Phylogenetic networks are generalizations of phylogenetic trees that allow the representation of reticulation events such as horizontal gene transfer or hybridization, and can also represent uncertainty in inference. A subclass of these, tree-based phylogenetic networks, have been introduced to capture the extent to which reticulate evolution nevertheless broadly follows tree-like patterns. Several important operations that change a general phylogenetic network have been developed in recent years and are important for allowing algorithms to move around spaces of networks; a vital ingredient in finding an optimal network given some biological data. A key such operation is the nearest neighbour interchange, or NNI. While it is already known that the space of unrooted phylogenetic networks is connected under NNI, it has been unclear whether this also holds for the subspace of tree-based networks. In this paper, we show that the space of unrooted tree-based phylogenetic networks is indeed connected under the NNI operation. We do so by explicitly showing how to get from one such network to another one without losing tree-basedness along the way. Moreover, we introduce some new concepts, for instance “shoat networks”, and derive some interesting aspects concerning tree-basedness. Last, we use our results to derive an upper bound on the size of the space of tree-based networks.

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Notes

  1. 1.

    Because of their close resemblance to the juvenile boars that frequent the streets of northern Germany.

  2. 2.

    Note that a binary phylogenetic tree with n leaves has precisely \(2n-2\) vertices (Semple and Steel 2003). Any of the extra k edges added to such a tree will induce two new vertices. In total, this is \(2n-2+2k = 2(n+k-1)\) vertices.

  3. 3.

    We used the computer algebra system Mathematica (Wolfram 2017) to verify that the shortest path from network (i) in Fig. 7 to a network isomorphic to (ii) requires 5 NNI moves. While this is tricky to see, it is combinatorially rather easy to see that network (i) has 24 1-step NNI neighbours, which can be divided into two classes (i.e. the NNI neighbourhood of network (i) contains only two non-isomorphic networks): those isomorphic to (i), and those isomorphic to a specific different network, which is in fact tree-based. So network (ii) cannot be in the 1-step neighbourhood of (i).

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Acknowledgements

MF wishes to thank the DAAD for conference travel funding to the Annual New Zealand Phylogenomics Meeting, where partial results leading to this manuscript were achieved.

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Correspondence to Andrew Francis.

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Fischer, M., Francis, A. The Space of Tree-Based Phylogenetic Networks. Bull Math Biol 82, 70 (2020). https://doi.org/10.1007/s11538-020-00744-9

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