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Fourier Transform Inequalities for Phylogenetic Trees
January-March 2009 (vol. 6 no. 1)
pp. 89-95
Frederick A. Matsen, UC Berkeley, Berkeley
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.

[1] E.S. Allman and J.A. Rhodes, “Phylogenetic Invariants for the General Markov Model of Sequence Mutation,” Math. Biosciences, vol. 186, no. 2, pp.113-144, 2003.
[2] M. Casanellas and J. Fernandez-Sanchez, “Geometry of the Kimura 3-Parameter Model,” Advances in Applied Math., vol. 41, no. 3, pp. 265-292, 2008.
[3] J.A. Cavender and J. Felsenstein, “Invariants of Phylogenies in a Simple Case with Discrete States,” J. Classification, vol. 4, pp. 57-71, 1987.
[4] J. Kim, “Slicing Hyperdimensional Oranges: The Geometry of Phylogenetic Estimation,” Molecular Phylogenetics and Evolution, vol. 17, no. 1, pp. 58-75, Oct. 2000.
[5] B. Sturmfels and S. Sullivant, “Toric Ideals of Phylogenetic Invariants,” J.Computational Biology, vol. 12, no. 2, pp. 204-228, Mar. 2005.
[6] E.S. Allman and J.A. Rhodes, “Phylogenetic Ideals and Varieties for the General Markov Model,” Advances in Applied Math., vol. 40, no. 2, pp. 127-148, 2008.
[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.
[9] S. Hoşten, A. Khetan, and B. Sturmfels, “Solving the Likelihood Equations,” Foundations of Computational Math., vol. 5, no. 4, pp.389-407, 2005.
[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.
[11] M. Kimura, “A Simple Method for Estimating Evolutionary Rates of Base Substitutions through Comparative Studies of Nucleotide Sequences,” J. Molecular Evolution, vol. 16, no. 2, pp. 111-120, Dec. 1980.
[12] D.W. Stroock, An Introduction to Markov Processes. Springer-Verlag, 2005.
[13] C. Semple and M. Steel, Phylogenetics. Oxford Univ. Press, 2003.
[14] L.A. Székely, M.A. Steel, and P.L. Erds, “Fourier Calculus on Evolutionary Trees,” Advances in Applied Math., vol. 14, no. 2, pp.200-210, 1993.
[15] D. Bryant, “Extending Tree Models to Split Networks,” Algebraic Statistics for Computational Biology, L. Pachter and B. Sturmfels, eds., Cambridge Univ. Press, 2005.
[16] V. Moulton and M. Steel, “Peeling Phylogenetic “Oranges”,” Advances in Applied Math., vol. 33, no. 4, pp. 710-727, 2004.
[17] M.D. Hendy and D. Penny, “A Framework for the Quantitative Study of Evolutionary Trees,” Systematic Zoology, vol. 38, no. 4, pp.297-309, 1989.
[18] M.D. Hendy, “The Relationship between Simple Evolutionary Tree Models and Observable Sequence Data,” Systematic Zoology, vol. 38, no. 4, pp. 301-321, 1989.
[19] S.N. Evans and T.P. Speed, “Invariants of Some Probability Models Used in Phylogenetic Inference,” Annals of Statistics, vol. 21, no. 1, pp. 355-377, 1993.
[20] F.A. Matsen and M. Steel, “Phylogenetic Mixtures on a Single Tree Can Mimic a Tree of Another Topology,” Systematic Biology, vol. 56, no. 5, pp.767-775, Oct. 2007.

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Frederick A. Matsen, "Fourier Transform Inequalities for Phylogenetic Trees," IEEE/ACM Transactions on Computational Biology and Bioinformatics, vol. 6, no. 1, pp. 89-95, Jan.-March 2009, doi:10.1109/TCBB.2008.68
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