In a homogeneous continuous-time Markov chain on a finite state
space, two states that jump to every other state with the same
rate are called similar. By partitioning states into similarity
classes, the algebraic derivation of the transition matrix can be
simplified, using hidden holding times and lumped Markov chains.
When the rate matrix is reversible, the transition matrix is
explicitly related in an intuitive way to that of the lumped
chain. The theory provides a unified derivation for a whole range
of useful DNA base substitution models, and a number of amino acid
substitution models.
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References
Ball, F. and Yeo, G. F. (1993). Lumpability and marginalisability for continuous-time Markov chains. J. Appl. Prob. 30, 518--528.
Ewens, W. J. and Grant, G. R. (2005). Statistical Methods in Bioinformatics: An Introduction, 2nd edn. Springer, New York.
Feller, W. (1971). An Introduction to Probability Theory and Its Applications, Vol. 2, 2nd edn. John Wiley, New York.
Mathematical Reviews (MathSciNet):
MR270403
Felsenstein, J. (1981). Evolutionary trees from DNA sequences: a maximum likelihood approach. J. Molec. Evol. 17, 368--376.
Felsenstein, J. (2004). Inferring Phylogenies. Sinauer, New York.
Graur, D. and Li, W.-H. (2000). Fundamentals of Molecular Evolution. Sinauer, Sunderland, MA.
Hasegawa, M. and Fujiwara, M. (1993). Relative efficiencies of the maximum likelihood, maximum parsimony, and neighbor-joining methods for estimating protein phylogeny. Molec. Phylogenet. Evol. 2, 1--5.
Hasegawa, M., Kishino, H. and Yano, T. (1985). Phylogenetic relationships among eukaryotic kingdoms inferred from ribosomal RNA sequences. J. Molec. Evol. 22, 32--38.
Jukes, T. H. and Cantor, C. (1969). Evolution of protein molecules. In Mammalian Protein Metabolism. Academic Press, New York, pp. 21--132.
Keilson, J. (1979). Markov Chain Models---Rarity and Exponentiality (Appl. Math. Sci. 28). Springer, New York.
Mathematical Reviews (MathSciNet):
MR528293
Kelly, F. P. (1979). Reversibility and Stochastic Networks. John Wiley, Chichester.
Mathematical Reviews (MathSciNet):
MR554920
Kimura, M. (1980). A simple method for estimating evolutionary rates of base substitutions through comparative studies of nucleotide sequences. J. Molec. Evol. 16, 111--120.
Kishino, H. and Hasegawa, M. (1989). Evaluation of the maximum likelihood estimate of the evolutionary tree topologies from DNA sequence data, and the branching order in Hominoidea. J. Molec. Evol. 29, 170--179.
Norris, J. R. (1997). Markov Chains. Cambridge University Press.
Schadt, E. E., Sinsheimer, J. S. and Lange, K. (1989). Computational advances in maximum likelihood methods for molecular phylogeny. Genome Res. 8, 222--233.
Tamura, K. (1992). Estimation of the number of nucleotide substitutions when there are strong transition-transversion and G+C content biases. Molec. Biol. Evol. 9, 678--687.
Tamura, K. and Nei, M. (1993). Estimation of the number of nucleotide substitutions in the control region of mitochondrial DNA in humans and chimpanzees. Molec. Biol. Evol. 10, 512--526.
