Skip to main content
SpringerLink
Log in

Lassoing and Corralling Rooted Phylogenetic Trees

  • Original Article
  • Published:
Bulletin of Mathematical Biology Aims and scope Submit manuscript

Abstract

The construction of a dendogram on a set of individuals is a key component of a genomewide association study. However, even with modern sequencing technologies the distances on the individuals required for the construction of such a structure may not always be reliable making it tempting to exclude them from an analysis. This, in turn, results in an input set for dendogram construction that consists of only partial distance information, which raises the following fundamental question. For what (proper) subsets of a dendogram’s leaf set can we uniquely reconstruct the dendogram from the distances that it induces on the elements of such a subset? By formalizing a dendogram in terms of an edge-weighted, rooted, phylogenetic tree on a pre-given finite set X with |X|≥3 whose edge-weighting is equidistant and subsets Y of X for which the distances between every pair of elements in Y is known in terms of sets of 2-subsets of X, we investigate this problem from the perspective of when such a tree is lassoed, that is, uniquely determined by the elements in . For this, we consider four different formalizations of the idea of “uniquely determining” giving rise to four distinct types of lassos. We present characterizations for all of them in terms of the child-edge graphs of the interior vertices of such a tree. Our characterizations imply in particular that in case the tree in question is binary, then all four types of lasso must coincide.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

Notes

  1. The definition of a topological lasso for an unrooted phylogenetic tree on X is the same as that of a topological lasso for an X-tree, but with the requirement dropped that the two proper edge-weightings mentioned in that definition are equidistant.

References

  • De Soete, G. (1984). Ultrametric tree representations of incomplete dissimilarity data. J. Classif., 1, 235–242.

    Article  Google Scholar 

  • Deza, M. M., & Rosenberg, I. G. (2000). n-semimetrics. Eur. J. Comb., 21(6), 797–806.

    Article  MathSciNet  MATH  Google Scholar 

  • Diestel, R. (2005). Graph theory. Heidelberg: Springer. Electronic Edition.

    MATH  Google Scholar 

  • Dress, A., Huber, K. T., Koolen, J., Moulton, V., & Spillner, A. (2012a). Basic phylogenetic combinatorics. Cambridge: Cambridge University Press.

    MATH  Google Scholar 

  • Dress, A. W. M., Huber, K. T., & Steel, M. (2012b). ‘Lassoing’ a phylogenetic tree I: basic properties, shellings and covers. J. Math. Biol., 65, 77–105.

    Article  MathSciNet  MATH  Google Scholar 

  • Felsenstein, F. (2003). Inferring phylogenies. Sunderland: Sinauer Associates.

    Google Scholar 

  • Harper, A. L., Trick, M., Higgins, J., Fraser, F., Clissold, L., Wells, R., Hattori, C., Werner, P., & Bancroft, I. (2012). Associative transcriptomics of traits in the polyploid crop species Brassica napus. Nat. Biotechnol., 30(8), 798–802.

    Article  Google Scholar 

  • Herrmann, S., Huber, K. T., Moulton, V., & Spillner, A. (2012). Recognizing treelike k-dissimilarities. J. Classif., 29(3), 321–340.

    Article  MathSciNet  Google Scholar 

  • Huber, K. T., & Steel, M. (2013). Reconstructing fully-resolved trees from triplet cover distances. Submitted.

  • Muir, W. M., Wong, G. K. S., Zhang, Y., Wang, J., Groenen, M. A. M., Crooijmans, R. P. M. A., Megens, H. J., Zhang, H., Okimoto, R., Vereijken, A., Jungerius, A., Albers, G. A. A., Lawley, C. T., Delany, M. E., MacEachern, S., & Cheng, H. H. (2008). Genome-wide assessment of worldwide chicken SNP genetic diversity indicates significant absence of rare alleles in commercial breeds. Proc. Natl. Acad. Sci., 105(45), 17312–17317.

    Article  Google Scholar 

  • Pachter, L., & Speyer, D. (2004). Reconstructing trees from subtree weights. Appl. Math. Lett., 17, 615–621.

    Article  MathSciNet  MATH  Google Scholar 

  • Philippe, H., Snell, E. A., Bapteste, E., Lopez, P., Holland, P. H., & Casane, D. (2004). Phylogenomics of eukaryotes: impact of missing data on large alignments. Mol. Biol. Evol., 21, 1740–1752.

    Article  Google Scholar 

  • Sanderson, M. J., McMahon, M. M., & Steel, M. (2010). Phylogenomics with incomplete taxon coverage: the limits to inference. BMC Evol. Biol., 10, 155.

    Article  Google Scholar 

  • Semple, C., & Steel, M. (2003). Phylogenetics. Oxford: Oxford University Press.

    MATH  Google Scholar 

  • Steel, M., & Sanderson, M. J. (2010). Characterizing phylogenetically decisive taxon coverage. Appl. Math. Lett., 23, 82–86.

    Article  MathSciNet  MATH  Google Scholar 

  • Warrens, M. J. (2010). n-way metrics. J. Classif., 27(2), 173–190.

    Article  MathSciNet  Google Scholar 

  • Zhang, Z., Ersoz, E., Lai, C. Q., Todhunter, R. J., Tiwari, H. K., Gore, M. A., Bradbury, P. J., Yu, J., Arnett, D. K., Ordovas, J. M., & Buckler, E. (2010). Mixed linear model approach adapted for genome-wide association studies. Nat. Genet., 42(4), 355–360.

    Article  Google Scholar 

Download references

Acknowledgement

A.-A. Popescu thanks the Norwich Research Park (NRP) for support. The authors thank the referees for their helpful comments.

Author information

Authors and Affiliations

  1. School of Computing Sciences, University of East Anglia, Norwich, NR4 7TJ, UK

    Katharina T. Huber & Andrei-Alin Popescu

Authors
  1. Katharina T. Huber
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Andrei-Alin Popescu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Katharina T. Huber.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Huber, K.T., Popescu, AA. Lassoing and Corralling Rooted Phylogenetic Trees. Bull Math Biol 75, 444–465 (2013). https://doi.org/10.1007/s11538-013-9815-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11538-013-9815-8

Keywords

Search

Navigation