Abstract
Continuous-time Markov chains are a standard tool in phylogenetic inference. If homogeneity is assumed, the chain is formulated by specifying time-independent rates of substitutions between states in the chain. In applications, there are usually extra constraints on the rates, depending on the situation. If a model is formulated in this way, it is possible to generalise it and allow for an inhomogeneous process, with time-dependent rates satisfying the same constraints. It is then useful to require that, under some time restrictions, there exists a homogeneous average of this inhomogeneous process within the same model. This leads to the definition of “Lie Markov models” which, as we will show, are precisely the class of models where such an average exists. These models form Lie algebras and hence concepts from Lie group theory are central to their derivation. In this paper, we concentrate on applications to phylogenetics and nucleotide evolution, and derive the complete hierarchy of Lie Markov models that respect the grouping of nucleotides into purines and pyrimidines—that is, models with purine/pyrimidine symmetry. We also discuss how to handle the subtleties of applying Lie group methods, most naturally defined over the complex field, to the stochastic case of a Markov process, where parameter values are restricted to be real and positive. In particular, we explore the geometric embedding of the cone of stochastic rate matrices within the ambient space of the associated complex Lie algebra.
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Notes
The reader may notice that we have changed the terminology of Sumner et al. (2012a) and we refer to the desired property as “locally multiplicative closure” instead of “multiplicative closure”. The problem of global multiplicative closure for a continuous-time Markov model is a deep problem related to the convergence of the Baker–Campbell–Hausdorff formula (see Blanes and Casas 2004). Notice that this is not a serious drawback as the nature of the problem is local.
Note this group is isomorphic to the dihedral group \({\mathbf {D}}_4\), which describes the symmetries of a square. However, it also admits a more natural description in our setting as \({\mathfrak {S}}_2 \wr {\mathfrak {S}}_2 \), the wreath product of \({\mathfrak {S}}_2\) with itself (see Rotman 1995, Chapter VII).
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Acknowledgments
JFS was partially supported by Ministerio de Educación y Ciencia MTM2009-14163-C02-02, MTM2012-38122-C03-01 and Generalitat de Catalunya, 2009 SGR 1284. JGS and PDJ were partially supported by Australian Research Council grant DP0877447. MDW was partially supported by Australian Research Council Grant FT100100031.
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Fernández-Sánchez, J., Sumner, J.G., Jarvis, P.D. et al. Lie Markov models with purine/pyrimidine symmetry. J. Math. Biol. 70, 855–891 (2015). https://doi.org/10.1007/s00285-014-0773-z
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DOI: https://doi.org/10.1007/s00285-014-0773-z