Abstract
We aim to fit 4-state sequences of DNA characters from three species to a tripod tree, whose evolutionary model is Jukes-Cantor. For this purpose, we adapt the closest tree method used in the fit of 2-state sequences coming from four species to a quartet, where the states are purines and pyrimidines and the evolutionary model is CFN. The adaptation requires a multi stage methodology called ‘reduction process’. We take the frequencies of 2-state character patterns on the quartet as parameters and search for solutions to the fit.
References
R. B. Bapat. Linear algebra and linear models. Third. Universitext. Springer, London; Hindustan Book Agency, New Delhi, 2012, viii+167. DOI: 10.1007/978-1-4471-2739-0.
B. Chor, M. Hendy, and S. Snir. Maximum likelihood Jukes-Cantor triplets: analytic solutions. Molecular biology and evolution 23(2006), no. 3, 626–632. DOI: 10.1093/molbev/msj069.
D. A. Cox, J. Little, and D. O’Shea. Ideals, varieties, and algorithms. Fourth. Undergraduate Texts in Mathematics. An introduction to computational algebraic geometry and commutative algebra. Springer, Cham, 2015, xvi+646. DOI: 10.1007/978-3-319-16721-3.
S. N. Evans. Fourier analysis and phylogenetic trees. Modern signal processing. Vol. 46. Math. Sci. Res. Inst. Publ. Cambridge Univ. Press, Cambridge, 2004, 117–136. eprint: http://library.msri.org/books/Book46/index.html.
G.-M. Greuel and G. Pfister. A Singular introduction to commutative algebra. Springer-Verlag, Berlin, 2002, xviii+588. DOI: 10.1007/978-3- 662-04963-1.
M. Hendy. The Relationship Between Simple Evolutionary Tree Models and Observable Sequence Data. Systematic Biology 38(Dec. 1989), no. 4, 310–321. DOI: 10.2307/2992397.
M. Hendy and S. Snir. Hadamard conjugation for the Kimura 3st model: combinatorial proof using path sets. IEEE/ACM Transactions on Computational Biology and Bioinformatics 5(2008), no. 3, 461–471. DOI: 10.1109/TCBB.2007.70227.
M. Steel, M. Hendy, L. Szekely, and P. Erdos. Spectral analysis and a closest tree method for genetic sequences. Applied mathematics letters 5(1992), no. 6, 63–67. DOI: 10.1016/0893-9659(92)90016-3.
M. Steel, M. Hendy, and D. Penny. Reconstructing phylogenies from nucleotide pattern probabilities: a survey and some new results. Discrete Applied Mathematics 88(1998), no. 1-3, 367–396. DOI: 10.1016/S0166-218X(98)00080-8.
Comments
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.
Copyright (c) 2023 Revista de Matemática: Teoría y Aplicaciones