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Phylogenetic complexity of the Kimura 3-parameter model. (English) Zbl 1412.14031
The part of computational biology that models evolution is called phylogenetics. An important object with crucial role in phylogenetics is a tree model. It consists of a tree, a finite set of states \(S\) and a family \(\mathcal{M}\) of transition matrices. The most important case is when \(S=\{A,C,G,T\}\) where the basis elements correspond to the four nucleobases of DNA. The models for which \(\mathcal{M}\) is a proper subspace of matrices reflect symmetries among elements of \(S\), where the symmetries are encoded by the action of a finite group \(G\) on \(S\).
The Kimura 3-parameter model is one of the most interesting phylogenetic models, which is given by \(S=G=\mathbb{Z}_2\times \mathbb{Z}_2\). In this model we have that \(S=\{A,C,G,T\}\) and the action of \(G\) reflects the pairing between \((A,G)\) and \((C,T)\). B. Sturmfels and S. Sullivant [J. Comput. Biol. 12, No. 2, 204–228 (2005; Zbl 1391.13058)] conjectured that the ideals of the algebraic varieties associated to the Kimura 3-parameter model are generated by polynomials of degree at most four. In the present article, the authors prove the above conjecture.
MSC:
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
13P25 Applications of commutative algebra (e.g., to statistics, control theory, optimization, etc.)
52B20 Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry)
05C90 Applications of graph theory
92E10 Molecular structure (graph-theoretic methods, methods of differential topology, etc.)
92D15 Problems related to evolution
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