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Mixed-up Trees: the Structure of Phylogenetic Mixtures

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In this paper, we apply new geometric and combinatorial methods to the study of phylogenetic mixtures. The focus of the geometric approach is to describe the geometry of phylogenetic mixture distributions for the two state random cluster model, which is a generalization of the two state symmetric (CFN) model. In particular, we show that the set of mixture distributions forms a convex polytope and we calculate its dimension; corollaries include a simple criterion for when a mixture of branch lengths on the star tree can mimic the site pattern frequency vector of a resolved quartet tree. Furthermore, by computing volumes of polytopes we can clarify how “common” non-identifiable mixtures are under the CFN model. We also present a new combinatorial result which extends any identifiability result for a specific pair of trees of size six to arbitrary pairs of trees. Next we present a positive result showing identifiability of rates-across-sites models. Finally, we answer a question raised in a previous paper concerning “mixed branch repulsion” on trees larger than quartet trees under the CFN model.

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  • Allman, E.S., Rhodes, J.A., 2006. The identifiability of tree topology for phylogenetic models, including covarion and mixture models. J. Comput. Biol. 13(5), 1101–1113.

    Article  MathSciNet  Google Scholar 

  • Bandelt, H.J., Dress, A.W.M., 1992. A canonical decomposition theory for metrics on a finite set. Adv. Math. 92, 47–105.

    Article  MathSciNet  MATH  Google Scholar 

  • Felsenstein, J., 2004. Inferring Phylogenies. Sinauer Press, Sunderland.

    Google Scholar 

  • Gawrilow, E., Joswig, M., 2005. Geometric reasoning with polymake. arXiv:math.CO/0507273.

  • Grünbaum, B., 2003. Convex Polytopes. Springer, Berlin.

    Google Scholar 

  • Kaibel, V., Pfetsch, M.E., 2003. Some algorithmic problems in polytope theory. In: Algebra, Geometry, and Software Systems, pp. 23–47. Springer, Berlin.

    Google Scholar 

  • Matsen, F.A., Steel, M., 2007. Phylogenetic mixtures on a single tree can mimic a tree of another topology. arXiv:0704.2260v1 [q-bio.PE].

  • Meacham, C.A., 1983. Theoretical and computational considerations of the compatibility of qualitative taxonomic characters. In: J. Felsenstein (Ed.), Numerical taxonomy, NATO ASI Series, vol. G1, pp. 304–314. Springer, Berlin.

    Google Scholar 

  • Mossel, E., Steel, M., 2004. A phase transition for a random cluster model on phylogenetic trees. Math. Biosci. 187(4), 189–203.

    Article  MathSciNet  MATH  Google Scholar 

  • Ochman, H., Lawrence, J.G., Groisman, E.A., 2000. Lateral gene transfer and the nature of bacterial innovation. Nature 405(6784), 299–304.

    Article  Google Scholar 

  • Rokas, A., Williams, B.L., King, N., Carroll, S.B., 2003. Genome-scale approaches to resolving incongruence in molecular phylogenies. Nature 425(6960), 798–804.

    Article  Google Scholar 

  • Semple, C., Steel, M., 2003. Phylogenetics. Oxford Lecture Series in Mathematics and its Applications, vol. 24. Oxford University Press, Oxford.

    MATH  Google Scholar 

  • Simon, C., Nigro, L., Sullivan, J., Holsinger, K., Martin, A., Grapputo, A., Franke, A., McIntosh, C., 1996. Large differences in substitutional pattern and evolutionary rate of 12S ribosomal RNA genes. Mol. Biol. Evol. 13(7), 923–932.

    Google Scholar 

  • Steel, M.A., Szekely, L.A., Hendy, M.D., 1994. Reconstructing trees when sequence sites evolve at variable rates. J. Comput. Biol. 1(2), 153–163.

    Article  Google Scholar 

  • Štefankovič, D., Vigoda, E., 2007. Phylogeny of mixture models: robustness of maximum likelihood and non-identifiable distributions. J. Comput. Biol. 14(2), 156–189.

    Article  MathSciNet  Google Scholar 

  • Ziegler, G.M., 1994. Lectures on Convex Polytopes. Springer, Berlin.

    Google Scholar 

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Correspondence to Frederick A. Matsen.

Additional information

F.A. Matsen’s and M. Steel’s research was supported by the Allan Wilson Centre for Molecular Ecology and Evolution.

E. Mossel’s research was supported by a Sloan fellowship in Mathematics, NSF awards DMS 0528488 and DMS 0548249 (CAREER) and by ONR grant N0014-07-1-05-06.

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Matsen, F.A., Mossel, E. & Steel, M. Mixed-up Trees: the Structure of Phylogenetic Mixtures. Bull. Math. Biol. 70, 1115–1139 (2008).

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