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Lie-Markov Models Derived from Finite Semigroups


We present and explore a general method for deriving a Lie-Markov model from a finite semigroup. If the degree of the semigroup is k, the resulting model is a continuous-time Markov chain on k-states and, as a consequence of the product rule in the semigroup, satisfies the property of multiplicative closure. This means that the product of any two probability substitution matrices taken from the model produces another substitution matrix also in the model. We show that our construction is a natural generalization of the concept of group-based models.

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  • Casanellas M, Sullivant S (2005) The strand symmetric model. In: Algebraic statistics for computational biology. Cambridge University Press, New York, pp 305–321

    Book  MATH  Google Scholar 

  • Draisma J, Kuttler J (2009) On the ideals of equivariant tree models. Math Ann 344(3):619–644

    MathSciNet  Article  MATH  Google Scholar 

  • Felsenstein J (1981) Evolutionary trees from DNA sequences: a maximum likelihood approach. J Mol Evol 17(6):368–376

    Article  Google Scholar 

  • Fernández-Sánchez J, Sumner JG, Jarvis PD, Woodhams MD (2015) Lie markov models with purine/pyrimidine symmetry. J Math Biol 70(4):855–891

    MathSciNet  Article  MATH  Google Scholar 

  • Forsythe GE (1955) SWAC computes 126 distinct semigroups of order 4. Proc Am Math Soc 6(3):443–447

    MathSciNet  MATH  Google Scholar 

  • Hall BC (2015) Lie groups, Lie algebras, and representations: an elementary introduction. Springer, Berlin

    Book  MATH  Google Scholar 

  • Hasegawa M, Kishino H, Yano T (1985) Dating of human-ape splitting by a molecular clock of mitochondrial DNA. J Mol Evol 22:160–174

    Article  Google Scholar 

  • Hendy MD, Penny D, Steel MA (1994) A discrete Fourier analysis for evolutionary trees. Proc Natl Acad Sci 91(8):3339–3343

    Article  MATH  Google Scholar 

  • Jarvis P, Sumner J (2012) Markov invariants for phylogenetic rate matrices derived from embedded submodels. IEEE/ACM Trans Comput Biol Bioinform 9(3):828–836

    Article  Google Scholar 

  • Jarvis PD, Sumner JG (2016) Matrix group structure and markov invariants in the strand symmetric phylogenetic substitution model. J Math Biol 73(2):259–282.

    MathSciNet  Article  MATH  Google Scholar 

  • Kimura M (1980) A simple method for estimating evolutionary rates of base substitutions through comparative studies of nucleotide sequences. J Mol Evol 16(2):111–120

    Article  Google Scholar 

  • Kimura M (1981) Estimation of evolutionary distances between homologous nucleotide sequences. Proc Natl Acad Sci 78(1):454–458

    Article  MATH  Google Scholar 

  • Kingman JFC (1962) The imbedding problem for finite Markov chains. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete 1(1):14–24

    MathSciNet  Article  MATH  Google Scholar 

  • Kolmogorov A (1936) Zur Theorie der Markoffschen Ketten. Math Ann 112:155–160

    MathSciNet  Article  Google Scholar 

  • Semple C, Steel MA (2003) Phylogenetics, vol 24. Oxford University Press, Oxford

    MATH  Google Scholar 

  • Steel M (2016) Phylogeny: discrete and random processes in evolution. CBMS-NSF regional conference series on mathematics, vol 89. SIAM, 293 pp

  • Stillwell J (2008) Naive lie theory. Springer, Berlin

    Book  MATH  Google Scholar 

  • Sturmfels B, Sullivant S (2005) Toric ideals of phylogenetic invariants. J Comput Biol 12(2):204–228

    Article  MATH  Google Scholar 

  • Sumner JG (2013) Lie geometry of \(2\times 2\) markov matrices. J Theor Biol 327(21):88–90

    Article  MATH  Google Scholar 

  • Sumner JG (2017) Multiplicatively closed markov models must form Lie algebras. ANZIAM J 59(2):240–246

    MathSciNet  Article  MATH  Google Scholar 

  • Sumner JG, Fernández-Sánchez J, Jarvis PD (2012a) Lie Markov models. J Theor Biol 298:16–31

    MathSciNet  Article  MATH  Google Scholar 

  • Sumner JG, Jarvis PD, Fernández-Sánchez J, Kaine BT, Woodhams MD, Holland BR (2012b) Is the general time-reversible model bad for molecular phylogenetics? Syst Biol 61(6):1069–1074

    Article  Google Scholar 

  • Székely LA, Steel MA, Erdős PL (1993) Fourier calculus on evolutionary trees. Adv Appl Math 14(2):200–216

    MathSciNet  Article  MATH  Google Scholar 

  • Tavaré S (1986) Some probabilistic and statistical problems in the analysis of DNA sequences. Lect Math Life Sci (Am Soc) 17:57–86

    MathSciNet  MATH  Google Scholar 

  • Woodhams MD, Fernández-Sánchez J, Sumner JG (2015) A new hierarchy of phylogenetic models consistent with heterogeneous substitution rates. Syst Biol 64(4):638–650

    Article  Google Scholar 

  • Yap VB, Pachter L (2004) Identification of evolutionary hotspots in the rodent genomes. Genome Res 14(4):574–579

    Article  Google Scholar 

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This research was supported by Australian Research Council (ARC) Discovery Grant DP150100088. We would like to thank Des FitzGerald for helpful comments on an early draft, and the anonymous reviewers for their careful reading and suggestions that lead to a substantially improved manuscript.

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Correspondence to Jeremy G. Sumner.

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Sumner, J.G., Woodhams, M.D. Lie-Markov Models Derived from Finite Semigroups. Bull Math Biol 81, 361–383 (2019).

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  • Lie algebras
  • Continuous-time Markov chains
  • Group-based models
  • Phylogenetics