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Dimensions of Group-Based Phylogenetic Mixtures

  • Special Issue: Algebraic Methods in Phylogenetics
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Abstract

Mixtures of group-based Markov models of evolution correspond to joins of toric varieties. In this paper, we establish a large number of cases for which these phylogenetic join varieties realize their expected dimension, meaning that they are nondefective. Nondefectiveness is not only interesting from a geometric point-of-view, but has been used to establish combinatorial identifiability for several classes of phylogenetic mixture models. Our focus is on group-based models where the equivalence classes of identified parameters are orbits of a subgroup of the automorphism group of the abelian group defining the model. In particular, we show that for these group-based models, the variety corresponding to the mixture of r trees with n leaves is nondefective when \(n \ge 2r+5\). We also give improved bounds for claw trees and give computational evidence that 2-tree and 3-tree mixtures are nondefective for small n.

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Acknowledgements

This work began at the 2016 AMS Mathematics Research Community on “Algebraic Statistics,” which was supported by the National Science Foundation under Grant number DMS-1321794. RD was supported by NSF DMS-1401591. EG was supported by NSF DMS-1620109. RW was supported by a NSF GRF under Grant number PGF-031543, NSF RTG Grant 0943832, and a Ford Foundation Dissertation Fellowship. HB was supported in part by a research assistantship, funded by the National Institutes of Health Grant R01 GM117590. PEH was partially supported by NSF Grant DMS-1620202.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Alaska Fairbanks, Fairbanks, AK, 99775, US

    Hector Baños

  2. Department of Mathematics and Statistics, University of Minnesota Duluth, Duluth, MN, 55812, USA

    Nathaniel Bushek

  3. Department of Mathematics, University of Illinois Urbana-Champaign, Urbana, IL, 61801, USA

    Ruth Davidson

  4. Department of Mathematics, University of Hawai’i at Mānoa, Honolulu, HI, 96822, USA

    Elizabeth Gross

  5. Department of Mathematics and Statistics, Williams College, Williamstown, MA, 01267, USA

    Pamela E. Harris

  6. Department of Mathematics, University of California Davis, Davis, CA, 95616, USA

    Robert Krone

  7. Mathematical Biosciences Institute, The Ohio State University, Columbus, OH, 43210, USA

    Colby Long

  8. Department of Mathematics, Seattle University, Seattle, WA, 98122, USA

    Allen Stewart

  9. Department of Mathematics, University of Michigan, Ann Arbor, MI, 48109, USA

    Robert Walker

Authors
  1. Hector Baños
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  2. Nathaniel Bushek
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  3. Ruth Davidson
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  4. Elizabeth Gross
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  5. Pamela E. Harris
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  6. Robert Krone
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  7. Colby Long
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  8. Allen Stewart
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  9. Robert Walker
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Corresponding author

Correspondence to Elizabeth Gross.

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Baños, H., Bushek, N., Davidson, R. et al. Dimensions of Group-Based Phylogenetic Mixtures. Bull Math Biol 81, 316–336 (2019). https://doi.org/10.1007/s11538-018-0489-0

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  • DOI: https://doi.org/10.1007/s11538-018-0489-0

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