Journal of Mathematical Biology

, Volume 56, Issue 3, pp 391–412 | Cite as

Counting labeled transitions in continuous-time Markov models of evolution



Counting processes that keep track of labeled changes to discrete evolutionary traits play critical roles in evolutionary hypothesis testing. If we assume that trait evolution can be described by a continuous-time Markov chain, then it suffices to study the process that counts labeled transitions of the chain. For a binary trait, we demonstrate that it is possible to obtain closed-form analytic solutions for the probability mass and probability generating functions of this evolutionary counting process. In the general, multi-state case we show how to compute moments of the counting process using an eigen decomposition of the infinitesimal generator, provided the latter is a diagonalizable matrix. We conclude with two examples that demonstrate the utility of our results.


Counting processes Continuous-time Markov chains Evolution Phylogenetics 

Mathematics Subject Classification (2000)

60J27 92D15 92D20 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Adachi J. and Hasegawa M. (1996). Model of amino acid substitution in proteins encoded by mitochondrial DNA. J. Mol. Evol. 42: 459–468 Google Scholar
  2. 2.
    Ball F. (1997). Empirical clustering of bursts of openings in Markov and semi-Markov models of single channel gating incorporating time interval omission. Adv. Appl. Probab. 29: 909–946 MATHCrossRefGoogle Scholar
  3. 3.
    Ball F. and Milne R.K. (2005). Simple derivations of properties of counting processes associated with Markov renewal processes. J. Appl. Probab. 42: 1031–1043 MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Chatfield C. (2004). The Analysis of Time Series: An Introduction. Chapman & Hall, London MATHGoogle Scholar
  5. 5.
    Darroch J.N and Morris K.W. (1967). Some passage-time generating functions for discrete-time and continuous-time finite Markov chains. J. Appl. Probab. 4: 496–507 MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Felsenstein J. (1981). Evolutionary trees from DNA sequences: a maximum likelihood approach. J. Mol. Evol. 13: 93–104 Google Scholar
  7. 7.
    Felsenstein J. (2004). Inferring Phylogenies. Sinauer Associates Inc., Sunderland Google Scholar
  8. 8.
    Fitch W.M., Bush R.M., Bender C.A. and Cox N.J. (1997). Long term trends in the evolution of H(3) HA1 human influenza type A. Proc. Natl. Acad. Sci. USA 94: 7712–7718 CrossRefGoogle Scholar
  9. 9.
    Guttorp P. (1995). Stochastic Modeling of Scientific Data. Chapman & Hall, Suffolk MATHGoogle Scholar
  10. 10.
    Hasegawa M., Kishino H. and Yano T. (1985). Dating the human-ape splitting by a molecular clock of mitochondrial DNA. J. Mol. Evol. 22: 160–174 CrossRefGoogle Scholar
  11. 11.
    Henikoff S. and Henikoff J.G. (1992). Amino acid substitution matrices from protein blocks. Proc. Natl. Acad. Sci. USA 89: 10915–10919 CrossRefGoogle Scholar
  12. 12.
    Hobolth, A., Jensen, J.L.: Statistical inference in evolutionary models of DNA sequences via the EM algorithm. Stat. Appl. Gen. Mol. Biol. 4, Article 18 (2005)Google Scholar
  13. 13.
    Kass R.E. and Raftery A.E. (1995). Bayes factors. J. Am. Stat. Assoc. 90: 773–795 MATHCrossRefGoogle Scholar
  14. 14.
    Meng X.L. (1994). Posterior predictive P-values. Ann. Stat. 22: 1142–1160 MATHGoogle Scholar
  15. 15.
    Narayana S. and Neuts M.F. (1992). The first two moment matrices of the counts for the Markovian arrival process. Stoch. Models 8: 459–477 MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Neuts M.F. (1979). A versatile Markovian point process. J. Appl. Probab. 16: 764–779 MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Neuts M.F. (1992). Models based on the Markovian arrival process. IEICE Trans. Commun. E75-B: 1255–1265 Google Scholar
  18. 18.
    Neuts M.F. (1995). Algorithmic Probability: a Collection of Problems. Chapman and Hall, London MATHGoogle Scholar
  19. 19.
    Nielsen R. (2002). Mapping mutations on phylogenies. Syst. Biol. 51: 729–739 CrossRefGoogle Scholar
  20. 20.
    Nielsen R. (2005). Molecular signatures of natural selection. Ann. Rev. Gen. 39: 197–218 CrossRefGoogle Scholar
  21. 21.
    Oakley T.H. and Cunningham C.W. (2002). Molecular phylogenetic evidence for the independent evolutionary origin of an arthropod compound eye. Proc. Natl. Acad. Sci. USA 99: 1426–1430 CrossRefGoogle Scholar
  22. 22.
    Pagel M. (1994). Detecting correlated evolution on phylogenies: a general method for the comparative analysis of discrete characters. Proc. R. Soc. B 255: 37–45 CrossRefGoogle Scholar
  23. 23.
    Pagel M. (1999). The maximum likelihood approach to reconstructing ancestral character states of discrete characters on phylogenies. Syst. Biol. 48: 612–622 CrossRefGoogle Scholar
  24. 24.
    Pollock D.D., Taylor W.R. and Goldman N. (1999). Coevolving protein residues: maximum likelihood identification and relationship to structure. J. Mol. Biol. 287: 187–198 CrossRefGoogle Scholar
  25. 25.
    Rambaut A. and Grassly N.C. (1997). Seq-Gen: An application for the Monte Carlo simulation of DNA sequence evolution along phylogenetic trees. Comput. Appl. Biosci. 13: 235–238 Google Scholar
  26. 26.
    Schadt E. and Lange K. (2002). Codon and rate variation models in molecular phylogeny. Mol. Biol. Evol. 19: 1534–1549 Google Scholar
  27. 27.
    Schadt E.E., Sinsheimer J.S. and Lange K. (1998). Computational advances in maximum likelihood methods for molecular phylogeny. Genome Res. 8: 222–233 Google Scholar
  28. 28.
    Siepel, A., Pollard, K.S., Haussler, D.: New methods for detecting lineage-specific selection. In: Proceedings of the 10th international conference on research in computational molecular biology, pp. 190–205 (2006)Google Scholar
  29. 29.
    Suchard M.A., Weiss R.E., Dorman K.S. and Sinsheimer J.S. (2002). Inferring spatial phylogenetic variation along nucleotide sequences: a multiple change-point model. J. Am. Stat. Assoc. 98: 427–437 CrossRefMathSciNetGoogle Scholar
  30. 30.
    Templeton A.R. (1996). Contingency tests of neutrality using intra/interspecific gene trees: the rejection of neutrality for the evolution of the mitochondrial cytochrome oxidase II gene in the hominoid primates. Genetics 144: 1263–1270 Google Scholar
  31. 31.
    Yang Z. (1994). Maximum likelihood phylogenetic estimation from DNA sequences with variable rates over sites: approximate methods. J. Mol. Evol. 39: 306–314 CrossRefGoogle Scholar
  32. 32.
    Yang Z. (1995). A space-time process model for the evolution of DNA sequences. Genetics 139: 993–1005 Google Scholar
  33. 33.
    Yang Z., Nielsen R., Goldman N. and Pedersen A.M.K. (2000). Codon-substitution models for heterogeneous selection pressure at amino acid sites. Genetics 155: 431–449 Google Scholar
  34. 34.
    Zheng Q. (2001). On the dispersion index of a Markovian molecular clock. Math. Biosci. 172: 115–128 MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  1. 1.Department of BiomathematicsDavid Geffen School of Medicine at UCLALos AngelesUSA
  2. 2.Department of StatisticsUniversity of WashingtonSeattleUSA
  3. 3.Department of BiostatisticsUCLA School of Public HealthLos AngelesUSA
  4. 4.Department of Human GeneticsDavid Geffen School of Medicine at UCLALos AngelesUSA

Personalised recommendations