Geometry of the Kimura 3-parameter model

Advances in Applied Mathematics

Volumn 41, Issue 3, 2008, Pages 265-292

  Casanellas, M ^a   Fernandez Sanchez J ^a  

^a UNIVERSITAT POLITÈCNICA DE CATALUNYA   (Spain)

Author keywords

Algebraic statistics; Kimura evolutionary model; Local complete intersection; Phylogenetic invariants

Indexed keywords

MATRIX ALGEBRA;

AFFINE SPACE; ALGEBRAIC GEOMETRY; ALGEBRAIC STATISTICS; ALGEBRAIC VARIETIES; ARBITRARY NUMBERS; FINITE GROUPS; KIMURA EVOLUTIONARY MODEL; LOCAL COMPLETE INTERSECTION; MINIMAL GENERATORS; PARAMETER MODELLING; PHYLOGENETIC INVARIANTS; PHYLOGENETIC RECONSTRUCTION; PHYLOGENETICS;

BOOLEAN ALGEBRA;

EID: 50549095109     PISSN: 01968858     EISSN: 10902074     Source Type: Journal    
DOI: 10.1016/j.aam.2007.09.003     Document Type: Article

References (23)

1. Allman E.S., and Rhodes J.A. Phylogenetic invariants for the general Markov model of sequence mutation. Math. Biosci. 186 (2003) 113-144
2. Allman E.S., and Rhodes J.A. Mathematical Models in Biology, an Introduction (2004), Cambridge Univ. Press, Cambridge
3. Allman E.S., and Rhodes J.A. Phylogenetic ideals and varieties for the general Markov model. preprint, available at:. http://arxiv.org/abs/math.AG/0410604v2 Adv. in Appl. Math., in press
4. Buczynska W., and Wisniewski J.A. On phylogenetic trees-a geometer's view. preprint, available at:. http://arxiv.org/abs/math.AG/0601357v1 J. Eur. Math. Soc., in press
5. Casanellas M., and Fernández-Sánchez J. Performance of a new invariants method on homogeneous and nonhomogeneous quartet trees. Mol. Biol. Evol. 24 (2007) 288-293
6. Casanellas M., Garcia L.D., and Sullivant S. Catalog of small trees. In: Pachter L., and Sturmfels B. (Eds). Algebraic Statistics for Computational Biology (2005), Cambridge Univ. Press, Cambridge 291-304
7. Casanellas M., and Sullivant S. The strand symmetric model. In: Pachter L., and Sturmfels B. (Eds). Algebraic Statistics for Computational Biology (2005), Cambridge Univ. Press, Cambridge 305-321
8. Cattani E., Curran R., and Dickenstein A. Complete intersections in toric ideals. Proc. Amer. Math. Soc. 135 (2007) 329-335
9. Chang J.T. Full reconstruction of Markov models on evolutionary trees: Identifiability and consistency. Math. Biosci. 137 (1996) 51-73
10. Dolgachev I. Lectures on Invariant Theory. London Math. Soc. Lecture Note Ser. (2003), Cambridge Univ. Press, Cambridge
11. Eriksson N. Tree construction using singular value decomposition. In: Pachter L., and Sturmfels B. (Eds). Algebraic Statistics for Computational Biology (2005), Cambridge Univ. Press, Cambridge 347-358
12. Eriksson N., Ranestad K., Sturmfels B., and Sullivant S. Phylogenetic algebraic geometry. In: Ciliberto C., Geramita A., Harbourne B., Roig R.M., and Ranestad K. (Eds). Projective Varieties with Unexpected Properties (2005), de Gruyter, Berlin 237-255
13. Evans S., and Speed T. Invariants of some probability models used in phylogenetic inference. Ann. Statist. 21 (1993) 355-377
14. Hagedorn T.R. Determining the number and structure of phylogenetic invariants. Adv. in Appl. Math. 24 (2000) 1-21
15. Hoşten S., and Shapiro J. Primary decomposition of lattice basis ideals. J. Symbolic Comput. 29 (2000) 625-639
16. Jukes T.H., and Cantor C.R. Evolution of protein molecules. Mammalian Protein Metabolism (1969), Academic Press, New York 21-132
17. Kimura M. A simple method for estimating evolutionary rates of base substitution through comparative studies of nucleotide sequences. J. Mol. Evol. 16 (1980) 111-120
18. Kimura M. Estimation of evolutionary sequences between homologous nucleotide sequences. Proc. Natl. Acad. Sci. USA 78 (1981) 454-458
19. Miller E., and Sturmfels B. Combinatorial Commutative Algebra. Grad. Texts in Math. vol. 227 (2005), Springer-Verlag, New York
20. Pachter L., and Sturmfels B. Tropical geometry of statistical models. Proc. Natl. Acad. Sci. USA 101 (2004) 16132-16137
21. In: Pachter L., and Sturmfels B. (Eds). Algebraic Statistics for Computational Biology (2005), Cambridge Univ. Press, Cambridge
22. Steel M.A., Székely L.A., Erdo{combining double acute accent}s P.L., and Waddell P. A complete family of phylogenetic invariants for any number of taxa under Kimura's 3st model. New Zealand J. Bot. 31 (1993) 289-296
23. Sturmfels B., and Sullivant S. Toric ideals of phylogenetic invariants. J. Comput. Biol. 12 (2005) 204-228