Bulletin of Mathematical Biology

, Volume 72, Issue 2, pp 340–358 | Cite as

Properties of Normal Phylogenetic Networks

  • Stephen J. WillsonEmail author
Original Article


A phylogenetic network is a rooted acyclic digraph with vertices corresponding to taxa. Let X denote a set of vertices containing the root, the leaves, and all vertices of outdegree 1. Regard X as the set of vertices on which measurements such as DNA can be made. A vertex is called normal if it has one parent, and hybrid if it has more than one parent. The network is called normal if it has no redundant arcs and also from every vertex there is a directed path to a member of X such that all vertices after the first are normal. This paper studies properties of normal networks.

Under a simple model of inheritance that allows homoplasies only at hybrid vertices, there is essentially unique determination of the genomes at all vertices by the genomes at members of X if and only if the network is normal. This model is a limiting case of more standard models of inheritance when the substitution rate is sufficiently low.

Various mathematical properties of normal networks are described. These properties include that the number of vertices grows at most quadratically with the number of leaves and that the number of hybrid vertices grows at most linearly with the number of leaves.


Normal network Hybrid Recombination Speciation Genome Ancestral reconstruction 


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  1. Bandelt, H.-J., Dress, A., 1992. Split decomposition: a new and useful approach to phylogenetic analysis of distance data. Mol. Phylogenet. Evol. 1, 242–252. CrossRefGoogle Scholar
  2. Baroni, M., Steel, M., 2006. Accumulation phylogenies. Ann. Comb. 10, 19–30. zbMATHCrossRefMathSciNetGoogle Scholar
  3. Baroni, M., Semple, C., Steel, M., 2004. A framework for representing reticulate evolution. Ann. Comb. 8, 391–408. zbMATHCrossRefMathSciNetGoogle Scholar
  4. Baroni, M., Semple, C., Steel, M., 2006. Hybrids in real time. Syst. Biol. 55, 46–56. CrossRefGoogle Scholar
  5. Bordewich, M., Semple, C., 2007. Computing the minimum number of hybridization events for a consistent evolutionary history. Discrete Appl. Math. 155, 914–928. zbMATHCrossRefMathSciNetGoogle Scholar
  6. Cardona, G., Rosselló, F., Valiente, G., 2007. Comparison of tree-child phylogenetic networks. To appear in IEEE/ACM Trans. Comput. Biol. Bioinform. doi: 10.1109/TCBB.2007.70270
  7. Cardona, G., Llabrés, M., Rosselló, F., Valiente, G., 2008a. A distance metric for a class of tree-sibling phylogenetic networks. Bioinformatics 24, 1481–1488. CrossRefGoogle Scholar
  8. Cardona, G., Rosselló, F., Valiente, G., 2008b. Tripartitions do not always discriminate phylogenetic networks. Math. Biosci. 211, 356–370. zbMATHCrossRefMathSciNetGoogle Scholar
  9. Cardona, G., Llabrés, M., Rosselló, F., Valiente, G., 2009a. Metrics for phylogenetic networks I: Generalizations of the Robinson–Foulds metric. IEEE/ACM Trans. Comput. Biol. Bioinform. 6, 46–61. CrossRefGoogle Scholar
  10. Cardona, G., Llabrés, M., Rosselló, F., Valiente, G., 2009b. Metrics for phylogenetic networks II: Nodal and triplets metrics. IEEE/ACM Trans. Comput. Biol. Bioinform. 6, 454–469. CrossRefGoogle Scholar
  11. Gusfield, D., Eddhu, S., Langley, C., 2004. Optimal, efficient reconstruction of phylogenetic networks with constrained recombination. J. Bioinform. Comput. Biol. 2, 173–213. CrossRefGoogle Scholar
  12. Hein, J., 1990. Reconstructing evolution of sequences subject to recombination using parsimony. Math. Biosci. 98, 185–200. zbMATHCrossRefMathSciNetGoogle Scholar
  13. Jin, G., Nakhleh, L., Snir, S., Tuller, T., 2006. Maximum likelihood of phylogenetic networks. Bioinformatics 22, 2604–2611. CrossRefGoogle Scholar
  14. Jin, G., Nakhleh, L., Snir, S., Tuller, T., 2007. Efficient parsimony-based methods for phylogenetic network reconstruction. Bioinformatics 23, e123–e128. CrossRefGoogle Scholar
  15. Kimura, M., 1980. A simple model for estimating evolutionary rates of base substitutions through comparative studies of nucleotide sequences. J. Mol. Evol. 16, 111–120. CrossRefGoogle Scholar
  16. Moret, B.M.E., Nakhleh, L., Warnow, T., Linder, C.R., Tholse, A., Padolina, A., Sun, J., Timme, R., 2004. Phylogenetic networks: modeling, reconstructibility, and accuracy. IEEE/ACM Trans. Comput. Biol. Bioinform. 1, 13–23. CrossRefGoogle Scholar
  17. Nakhleh, L., Warnow, T., Linder, C.R., 2004. Reconstructing reticulate evolution in species-theory and practice. In: Bourne, P.E., Gusfield, D. (Eds.), Proceedings of the Eighth Annual International Conference on Computational Molecular Biology, RECOMB ’04, March 27–31, 2004, San Diego, California, pp. 337–346. ACM, New York. Google Scholar
  18. Semple, C., Steel, M., 2003. Phylogenetics. Oxford University Press, Oxford. zbMATHGoogle Scholar
  19. van Iersel, L., Keijsper, J., Kelk, S., Stougie, L., 2007. Constructing level-2 phylogenetic networks from triplets. arXiv:0707.2890v1 [q-bio.PE].
  20. Wang, L., Zhang, K., Zhang, L., 2001. Perfect phylogenetic networks with recombination. J. Comput. Biol. 8, 69–78. CrossRefGoogle Scholar
  21. Willson, S.J., 2007a. Unique determination of some homoplasies at hybridization events. Bull. Math. Biol. 69, 1709–1725. zbMATHCrossRefMathSciNetGoogle Scholar
  22. Willson, S.J., 2007b. Reconstruction of some hybrid pylogenetic networks with homoplasies from distances. Bull. Math. Biol. 62, 2561–2590. CrossRefMathSciNetGoogle Scholar
  23. Willson, S.J., 2008. Reconstruction of certain phylogenetic networks from the genomes at their leaves. J. Theor. Biol. 252, 338–349. CrossRefGoogle Scholar

Copyright information

© Society for Mathematical Biology 2009

Authors and Affiliations

  1. 1.Department of MathematicsIowa State UniversityAmesUSA

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