Subscribe
Issue No.01 - January-March (2009 vol.6)
pp: 76-88
Elizabeth S. Allman , University of Alaska Fairbanks, Fairbanks
John A. Rhodes , University of Alaska Fairbanks, Fairbanks
ABSTRACT
Covarion models of character evolution describe inhomogeneities in substitution processes through time. In phylogenetics, such models are used to describe changing functional constraints or selection regimes during the evolution of biological sequences. In this work the identifiability of such models for generic parameters on a known phylogenetic tree is established, provided the number of covarion classes does not exceed the size of the observable state space. `Generic parameters' as used here means all parameters except possibly those in a set of measure zero within the parameter space. Combined with earlier results, this implies both the tree and generic numerical parameters are identifiable if the number of classes is strictly smaller than the number of observable states.
INDEX TERMS
phylogenetics, Markov processes on trees, statistical consistency
CITATION
Elizabeth S. Allman, John A. Rhodes, "The Identifiability of Covarion Models in Phylogenetics", IEEE/ACM Transactions on Computational Biology and Bioinformatics, vol.6, no. 1, pp. 76-88, January-March 2009, doi:10.1109/TCBB.2008.52
REFERENCES
 [1] E.S. Allman, C. Ané, and J.A. Rhodes, “Identifiability of a Markovian Model of Molecular Evolution with Gamma-Distributed Rates,” Advances in Applied Probability, vol. 40, no. 1, pp. 229-249, arXiv:0709.0531, 2008. [2] E.S. Allman and J.A. Rhodes, “The Identifiability of Tree Topology for Phylogenetic Models, Including Covarion and Mixture Models,” J. Computational Biology, vol. 13, no. 5, pp. 1101-1113, arXiv:q-bio.PE/0511009, 2006. [3] E.S. Allman and J.A. Rhodes, “Identifying Evolutionary Trees and Substitution Parameters for the General Markov Model with Invariable Sites,” Math. Biosciences, vol. 211, no. 1, pp. 18-33, arXiv:q-bio.PE/0702050, 2008. [4] E.S. Allman and J.A. Rhodes, “Phylogenetic Ideals and Varieties for the General Markov Model,” Advances in Applied Math., vol. 40, no. 2, arXiv:math.AG/0410604, 2008. [5] J.T. Chang, “Full Reconstruction of Markov Models on Evolutionary Trees: Identifiability and Consistency,” Math. Biosciences, vol. 137, no. 1, pp. 51-73, 1996. [6] W.M. Fitch and E. Markowitz, “An Improved Method for Determining Codon Variability in a Gene and Its Application to the Rate of Fixation of Mutations in Evolution,” Biochemical Genetics, vol. 4, pp. 579-593, 1970. [7] N. Galtier, “Maximum-Likelihood Phylogenetic Analysis under a Covarion-Like Model,” Molecular Biology and Evolution, vol. 18, no. 5, pp. 866-873, 2001. [8] N. Galtier and A. Jean-Marie, “Markov-Modulated Markov Chains and the Covarion Process of Molecular Evolution,” J. Computational Biology, vol. 11, no. 4, pp. 727-733, 2004. [9] O. Gascuel and S. Guindon, “Modelling the Variability of Evolutionary Processes,” Reconstructing Evolution: New Mathematics and Computational Advances, O. Gascuel and M. Steel, eds., pp. 65-107, Oxford Univ. Press, 2007. [10] S. Guindon, A.G. Rodrigo, K.A. Dyer, and J.P. Huelsenbeck, “Modeling the Site-Specific Variation of Selection Patterns across Lineages,” Proc. Nat'l Academy of Sciences USA, vol. 101, pp. 12957-12962, 2004. [11] R.A. Horn and C.R. Johnson, Matrix Analysis. Cambridge Univ. Press, 1985. [12] J. Huelsenbeck, “Testing a Covariotide Model of DNA Substitution,” Molecular Biology and Evolution, vol. 19, pp. 698-707, 2002. [13] J.B. Kruskal, “Rank, Decomposition, and Uniqueness for 3-Way and $N$ -Way Arrays,” Multiway Data Analysis, pp.7-18, North-Holland, 1989. [14] J.B. Kruskal, “More Factors than Subjects, Tests and Treatments: An Indeterminacy Theorem for Canonical Decomposition and Individual Differences Scaling,” Psychometrika, vol. 41, no. 3, pp. 281-293, 1976. [15] J.B. Kruskal, “Three-Way Arrays: Rank and Uniqueness of Trilinear Decompositions, with Application to Arithmetic Complexity and Statistics,” Linear Algebra and Its Applications, vol. 18, no. 2, pp. 95-138, 1977. [16] T. Petrie, “Probabilistic Functions of Finite State Markov Chains,” Annals of Math. Statistics, vol. 40, pp. 97-115, 1969. [17] J.S. Rogers, “Maximum Likelihood Estimation of Phylogenetic Trees is Consistent When Substitution Rates Vary According to the Invariable Sites Plus Gamma Distribution,” Systematic Biology, vol. 50, no. 5, pp. 713-722, 2001. [18] C. Semple and M. Steel, Phylogenetics, vol. 24, Oxford Lecture Series in Mathematics and Its Applications, Oxford Univ. Press, 2003. [19] C. Tuffley and M. Steel, “Modeling the Covarion Hypothesis of Nucleotide Substitution,” Math. Biosciences, vol. 147, no. 1, pp. 63-91, 1998. [20] H.-C. Wang, M. Spencer, E. Susko, and A. Roger, “Testing for Covarion-Like Evolution in Protein Sequences,” Molecular Biology and Evolution, vol. 24, no. 1, pp. 294-305, 2007. [21] S. Whelan, Spatial and Temporal Heterogeneity in Nucleotide Evolution, preprint, 2008.