Algorithmica

, Volume 66, Issue 3, pp 714–738 | Cite as

Encoding and Constructing 1-Nested Phylogenetic Networks with Trinets

Article

Abstract

Phylogenetic networks are a generalization of phylogenetic trees that are used in biology to represent reticulate or non-treelike evolution. Recently, several algorithms have been developed which aim to construct phylogenetic networks from biological data using triplets, i.e. binary phylogenetic trees on 3-element subsets of a given set of species. However, a fundamental problem with this approach is that the triplets displayed by a phylogenetic network do not necessarily uniquely determine or encode the network. Here we propose an alternative approach to encoding and constructing phylogenetic networks, which uses phylogenetic networks on 3-element subsets of a set, or trinets, rather than triplets. More specifically, we show that for a special, well-studied type of phylogenetic network called a 1-nested network, the trinets displayed by a 1-nested network always encode the network. We also present an efficient algorithm for deciding whether a dense set of trinets (i.e. one that contains a trinet on every 3-element subset of a set) can be displayed by a 1-nested network or not and, if so, constructs that network. In addition, we discuss some potential new directions that this new approach opens up for constructing and comparing phylogenetic networks.

Keywords

Phylogenetic network Triplets Trinets Reticulate evolution 

References

  1. 1.
    Aho, A.V., Sagiv, Y., Szymanski, T.G., Ullman, J.D.: Inferring a tree from lowest common ancestors with an application to the optimization of relational expressions. SIAM J. Comput. 10, 405–421 (1981) MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Bryant, D.: Building trees, hunting for trees, and comparing trees: theory and methods in phylogenetic analysis. PhD Thesis, University of Canterbury, Christchurch, New Zealand (1997) Google Scholar
  3. 3.
    Byrka, J., Guillemot, S., Jansson, J.: New results on optimizing rooted triplets consistency. Discrete Appl. Math. 158, 1136–1147 (2010) MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Cardona, G., Llabrés, M., Rosselló, F., Valiente, G.: A distance metric for a class of tree-sibling phylogenetic networks. Bioinformatics 24, 1481–1488 (2008) CrossRefGoogle Scholar
  5. 5.
    Cardona, G., Llabrés, M., Rosselló, F., Valiente, G.: Metrics for phylogenetic networks I: generalization of the Robinson-Foulds metric. IEEE/ACM Trans. Comput. Biol. Bioinform. 6, 46–61 (2009) CrossRefGoogle Scholar
  6. 6.
    Cardona, G., Llabrés, M., Rosselló, F., Valiente, G.: Metrics for phylogenetic networks II: nodal and triplets metrics. IEEE/ACM Trans. Comput. Biol. Bioinform. 6, 454–469 (2009) CrossRefGoogle Scholar
  7. 7.
    Cardona, G., Llabrés, M., Rosselló, F., Valiente, G.: Comparison of galled trees. IEEE/ACM Trans. Comput. Biol. Bioinform. 8, 410–427 (2011) CrossRefGoogle Scholar
  8. 8.
    Cardona, G., Rosselló, F., Valiente, G.: Tripartitions do not always discriminate phylogenetic networks. Math. Biosci. 211, 356–370 (2008) MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Choy, C., Jansson, J., Sadakane, K., Sung, W.K.: Computing the maximum agreement of phylogenetic networks. Theor. Comput. Sci. 335, 93–107 (2005) MathSciNetCrossRefGoogle Scholar
  10. 10.
    Gambette, P., Huber, K.T.: On encodings of phylogenetic networks of bounded level. J. Math. Biol. 65(1), 157–180 (2012) MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Gasieniec, L., Jansson, J., Lingas, A., Ostlin, A.: On the complexity of computing evolutionary trees. In: COCOON, pp. 134–145 (1997) Google Scholar
  12. 12.
    Gasieniec, L., Jansson, J., Lingas, A., Ostlin, A.: On the complexity of constructing evolutionary trees. J. Comb. Optim. 3(2–3), 183–197 (1999) MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Huber, K.T., van Iersel, L., Kelk, S., Suchecki, R.: A practical algorithm for reconstructing level-1 phylogenetic networks. IEEE/ACM Trans. Comput. Biol. Bioinform. 8, 635–649 (2011) CrossRefGoogle Scholar
  14. 14.
    Huson, D., Rupp, R., Scornavacca, C.: Phylogenetic Networks. Cambridge University Press, Cambridge (2011) Google Scholar
  15. 15.
    van Iersel, L., Keijsper, J., Kelk, S., Stougie, L., Hagen, F., Boekhout, T.: Constructing level-2 phylogenetic networks form triplets. IEEE/ACM Trans. Comput. Biol. Bioinform. 6, 667–681 (2009) CrossRefGoogle Scholar
  16. 16.
    van Iersel, L.J.J., Kelk, S.M.: Constructing the simplest possible phylogenetic network from triplets. Algorithmica 60, 207–235 (2011) MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Jansson, J.: On the complexity of inferring rooted evolutionary trees. Electron. Notes Discrete Math. 7, 50–53 (2001) MathSciNetCrossRefGoogle Scholar
  18. 18.
    Jansson, J., Nguyen, N.B., Sung, W.K.: Algorithms for combining rooted triplets into a galled phylogenetic network. SIAM J. Comput. 35, 1098–1121 (2006) MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Jansson, J., Sung, W.K.: Inferring a level-1 phylogenetic network from a dense set of rooted triplets. Theor. Comput. Sci. 363, 60–68 (2006) MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Jin, G., Nakhleh, L., Snir, S., Tuller, T.: Maximum likelihood of phylogenetic networks. Bioinformatics 22, 2604–2611 (2006) CrossRefGoogle Scholar
  21. 21.
    Jin, G., Nakhleh, L., Snir, S., Tuller, T.: Parsimony score of phylogenetic networks: hardness results and a linear-time heuristic. IEEE/ACM Trans. Comput. Biol. Bioinform. 6, 495–505 (2009) CrossRefGoogle Scholar
  22. 22.
    Koblmüller, S., Duftner, N., Sefc, K.M., Aibara, M., Stipacek, M., Blanc, M., Egger, B., Sturmbauer, C.: Reticulate phylogeny of gastropod-shell-breeding cichlids from lake tanganyika—the result of repeated introgressive hybridization. BMC Evol. Biol. 7, 7 (2007) CrossRefGoogle Scholar
  23. 23.
    Lengauer, T., Tarjan, R.E.: A fast algorithm for finding dominators in a flowgraph. ACM Trans. Program. Lang. Syst. 1, 121–141 (1979) MATHCrossRefGoogle Scholar
  24. 24.
    Lott, M., Spillner, A., Huber, K.T., Petri, A., Oxelman, B., Moulton, V.: Inferring polyploid phylogenies from multi-labeled gene trees. BMC Evol. Biol. 9, 216 (2009) CrossRefGoogle Scholar
  25. 25.
    Nakhleh, L.: Evolutionary phylogenetic networks: models and issues. In: Heath, L.S., Ramakrishnan, N. (eds.) Problem Solving Handbook in Computational Biology and Bioinformatics. Springer, Berlin (2011) Google Scholar
  26. 26.
    Page, R.D.M.: Modified mincut supertrees. In: Proceedings of Workshop on Algorithms in Bioinformatics (WABI 2002). Lecture Notes in Computer Science, vol. 2452, pp. 537–552 (2002) CrossRefGoogle Scholar
  27. 27.
    Planet, P.J., Kachlany, S.C., Fine, D.H., DeSalle, R., Figurski, D.H.: The widespread colonization island of actinobacillus actinomycetemcomitans. Nat. Genet. 34, 193–198 (2003) CrossRefGoogle Scholar
  28. 28.
    Semple, C., Steel, M.: A supertree method for rooted trees. Discrete Appl. Math. 105, 147–158 (2000) MathSciNetMATHCrossRefGoogle Scholar
  29. 29.
    Semple, C., Steel, M.: Phylogenetics. Oxford Lecture Series in Mathematics and its Applications, vol. 24. Oxford University Press, Oxford (2003) MATHGoogle Scholar
  30. 30.
    Song, Y.S., Hein, J.: Constructing minimal ancestral recombination graphs. J. Comput. Biol. 12, 147–169 (2005) CrossRefGoogle Scholar
  31. 31.
    Strimmer, K., von Haeseler, A.: Quartet puzzling: a quartet maximum likelihood method for reconstructing tree topologies. Mol. Biol. Evol. 13, 964–969 (1996) CrossRefGoogle Scholar
  32. 32.
    To, T.H., Habib, M.: Level-k phylogenetic networks are constructable from a dense triplet set in polynomial time. In: Combinatorial Pattern Matching (CPM). Lecture Notes in Computer Science, vol. 5577, pp. 275–288 (2009) CrossRefGoogle Scholar
  33. 33.
    Tusserkani, R., Eslahchi, C., Poormohammadi, H., Azadi, A.: TripNet: a method for constructing phylogenetic networks from triplets. arXiv:1104.4720v1 [cs.CE]
  34. 34.
    Willson, S.: Regular networks can be uniquely constructed from their trees. IEEE/ACM Trans. Comput. Biol. Bioinform. 8, 785–796 (2011) CrossRefGoogle Scholar
  35. 35.
    Wu, B.Y.: Constructing the maximum consensus tree from rooted triples. J. Comb. Optim. 8, 29–39 (2004) MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.School of Computing SciencesUniversity of East AngliaNorwichUK

Personalised recommendations