CSDL Home IEEE/ACM Transactions on Computational Biology and Bioinformatics 2009 vol.6 Issue No.01 - January-March

Subscribe

Issue No.01 - January-March (2009 vol.6)

pp: 89-95

Frederick A. Matsen , UC Berkeley, Berkeley

DOI Bookmark: http://doi.ieeecomputersociety.org/10.1109/TCBB.2008.68

ABSTRACT

Phylogenetic invariants are not the only constraints on site-pattern frequency vectors for phylogenetic trees. A mutation matrix, by its definition, is the exponential of a matrix with non-negative off-diagonal entries; this positivity requirement implies non-trivial constraints on the site-pattern frequency vectors. We call these additional constraints "edge-parameter inequalities." In this paper, we first motivate the edge-parameter inequalities by considering a pathological site-pattern frequency vector corresponding to a quartet tree with a negative internal edge. This site-pattern frequency vector nevertheless satisfies all of the constraints described up to now in the literature. We next describe two complete sets of edge-parameter inequalities for the group-based models; these constraints are square-free monomial inequalities in the Fourier transformed coordinates. These inequalities, along with the phylogenetic invariants, form a complete description of the set of site-pattern frequency vectors corresponding to \emph{bona fide} trees. Said in mathematical language, this paper explicitly presents two finite lists of inequalities in Fourier coordinates of the form "monomial $\leq 1$," each list characterizing the phylogenetically relevant semialgebraic subsets of the phylogenetic varieties.

INDEX TERMS

Impact of VLSI on system design, Backup/recovery

CITATION

Frederick A. Matsen, "Fourier Transform Inequalities for Phylogenetic Trees",

*IEEE/ACM Transactions on Computational Biology and Bioinformatics*, vol.6, no. 1, pp. 89-95, January-March 2009, doi:10.1109/TCBB.2008.68REFERENCES

- [7] J. Felsenstein,
Inferring Phylogenies. Sinauer Press, 2004.- [8] E. Allman and J. Rhodes, “Phylogenetic Invariants,”
Reconstructing Evolution: New Mathematical and Computational Advances, O.Gascuel and M. Steel, eds., Oxford Univ. Press, 2007.- [10] N. Eriksson, “Using Invariants for Phylogenetic Tree Construction,” to be published in
Emerging Applications of Algebraic Geometry, M. Putinar and S.Sullivant, eds., Springer, arXiv:0709.2890, 2008.- [12] D.W. Stroock,
An Introduction to Markov Processes. Springer-Verlag, 2005.- [13] C. Semple and M. Steel,
Phylogenetics. Oxford Univ. Press, 2003.- [15] D. Bryant, “Extending Tree Models to Split Networks,”
Algebraic Statistics for Computational Biology, L. Pachter and B. Sturmfels, eds., Cambridge Univ. Press, 2005. |