Bulletin of Mathematical Biology

, Volume 81, Issue 2, pp 361–383 | Cite as

Lie-Markov Models Derived from Finite Semigroups

  • Jeremy G. SumnerEmail author
  • Michael D. Woodhams
Special Issue: Algebraic Methods in Phylogenetics


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.


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



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.


  1. Casanellas M, Sullivant S (2005) The strand symmetric model. In: Algebraic statistics for computational biology. Cambridge University Press, New York, pp 305–321CrossRefzbMATHGoogle Scholar
  2. Draisma J, Kuttler J (2009) On the ideals of equivariant tree models. Math Ann 344(3):619–644MathSciNetCrossRefzbMATHGoogle Scholar
  3. Felsenstein J (1981) Evolutionary trees from DNA sequences: a maximum likelihood approach. J Mol Evol 17(6):368–376CrossRefGoogle Scholar
  4. Fernández-Sánchez J, Sumner JG, Jarvis PD, Woodhams MD (2015) Lie markov models with purine/pyrimidine symmetry. J Math Biol 70(4):855–891MathSciNetCrossRefzbMATHGoogle Scholar
  5. Forsythe GE (1955) SWAC computes 126 distinct semigroups of order 4. Proc Am Math Soc 6(3):443–447MathSciNetzbMATHGoogle Scholar
  6. Hall BC (2015) Lie groups, Lie algebras, and representations: an elementary introduction. Springer, BerlinCrossRefzbMATHGoogle Scholar
  7. Hasegawa M, Kishino H, Yano T (1985) Dating of human-ape splitting by a molecular clock of mitochondrial DNA. J Mol Evol 22:160–174CrossRefGoogle Scholar
  8. Hendy MD, Penny D, Steel MA (1994) A discrete Fourier analysis for evolutionary trees. Proc Natl Acad Sci 91(8):3339–3343CrossRefzbMATHGoogle Scholar
  9. Jarvis P, Sumner J (2012) Markov invariants for phylogenetic rate matrices derived from embedded submodels. IEEE/ACM Trans Comput Biol Bioinform 9(3):828–836CrossRefGoogle Scholar
  10. 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. MathSciNetCrossRefzbMATHGoogle Scholar
  11. Kimura M (1980) A simple method for estimating evolutionary rates of base substitutions through comparative studies of nucleotide sequences. J Mol Evol 16(2):111–120CrossRefGoogle Scholar
  12. Kimura M (1981) Estimation of evolutionary distances between homologous nucleotide sequences. Proc Natl Acad Sci 78(1):454–458CrossRefzbMATHGoogle Scholar
  13. Kingman JFC (1962) The imbedding problem for finite Markov chains. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete 1(1):14–24MathSciNetCrossRefzbMATHGoogle Scholar
  14. Kolmogorov A (1936) Zur Theorie der Markoffschen Ketten. Math Ann 112:155–160MathSciNetCrossRefGoogle Scholar
  15. Semple C, Steel MA (2003) Phylogenetics, vol 24. Oxford University Press, OxfordzbMATHGoogle Scholar
  16. Steel M (2016) Phylogeny: discrete and random processes in evolution. CBMS-NSF regional conference series on mathematics, vol 89. SIAM, 293 ppGoogle Scholar
  17. Stillwell J (2008) Naive lie theory. Springer, BerlinCrossRefzbMATHGoogle Scholar
  18. Sturmfels B, Sullivant S (2005) Toric ideals of phylogenetic invariants. J Comput Biol 12(2):204–228CrossRefzbMATHGoogle Scholar
  19. Sumner JG (2013) Lie geometry of \(2\times 2\) markov matrices. J Theor Biol 327(21):88–90CrossRefzbMATHGoogle Scholar
  20. Sumner JG (2017) Multiplicatively closed markov models must form Lie algebras. ANZIAM J 59(2):240–246MathSciNetCrossRefzbMATHGoogle Scholar
  21. Sumner JG, Fernández-Sánchez J, Jarvis PD (2012a) Lie Markov models. J Theor Biol 298:16–31MathSciNetCrossRefzbMATHGoogle Scholar
  22. 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):1069–1074CrossRefGoogle Scholar
  23. Székely LA, Steel MA, Erdős PL (1993) Fourier calculus on evolutionary trees. Adv Appl Math 14(2):200–216MathSciNetCrossRefzbMATHGoogle Scholar
  24. Tavaré S (1986) Some probabilistic and statistical problems in the analysis of DNA sequences. Lect Math Life Sci (Am Soc) 17:57–86MathSciNetzbMATHGoogle Scholar
  25. Woodhams MD, Fernández-Sánchez J, Sumner JG (2015) A new hierarchy of phylogenetic models consistent with heterogeneous substitution rates. Syst Biol 64(4):638–650CrossRefGoogle Scholar
  26. Yap VB, Pachter L (2004) Identification of evolutionary hotspots in the rodent genomes. Genome Res 14(4):574–579CrossRefGoogle Scholar

Copyright information

© Society for Mathematical Biology 2018

Authors and Affiliations

  1. 1.School of Physical SciencesUniversity of TasmaniaHobartAustralia

Personalised recommendations