Toric Ideals of Phylogenetic Invariants
Top Cited Papers
- 1 March 2005
- journal article
- Published by Mary Ann Liebert Inc in Journal of Computational Biology
- Vol. 12 (2), 204-228
- https://doi.org/10.1089/cmb.2005.12.204
Abstract
Statistical models of evolution are algebraic varieties in the space of joint probability distributions on the leaf colorations of a phylogenetic tree. The phylogenetic invariants of a model are the polynomials which vanish on the variety. Several widely used models for biological sequences have transition matrices that can be diagonalized by means of the Fourier transform of an abelian group. Their phylogenetic invariants form a toric ideal in the Fourier coordinates. We determine generators and Gröbner bases for these toric ideals. For the Jukes-Cantor and Kimura models on a binary tree, our Gröbner bases consist of certain explicitly constructed polynomials of degree at most four.Keywords
This publication has 15 references indexed in Scilit:
- Phylogenetic invariants for the general Markov model of sequence mutationMathematical Biosciences, 2003
- Gröbner Bases and Polyhedral Geometry of Reducible and Cyclic ModelsJournal of Combinatorial Theory, Series A, 2002
- Multiple maxima of likelihood in phylogenetic trees: an analytic approach.Molecular Biology and Evolution, 2000
- Determining the Number and Structure of Phylogenetic InvariantsAdvances in Applied Mathematics, 2000
- Constructing and Counting Phylogenetic InvariantsJournal of Computational Biology, 1998
- A remarkable nonlinear invariant for evolution with heterogeneous ratesMathematical Biosciences, 1996
- Complete Families of Linear Invariants for Some Stochastic Models of Sequence Evolution, with and without the Molecular Clock AssumptionJournal of Computational Biology, 1996
- Phylogenetic invariants for more general evolutionary modelsJournal of Theoretical Biology, 1995
- Invariants of Some Probability Models Used in Phylogenetic InferenceThe Annals of Statistics, 1993
- Invariants of phylogenies in a simple case with discrete statesJournal of Classification, 1987