Phase-Type Distribution Approximations of the Waiting Time Until Coordinated Mutations Get Fixed in a Population

  • Ola Hössjer
  • Günter Bechly
  • Ann Gauger
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 271)


In this paper we study the waiting time until a number of coordinated mutations occur in a population that reproduces according to a continuous time Markov process of Moran type. It is assumed that any individual can have one of \(m+1\) different types, numbered as \(0,1,\ldots ,m\), where initially all individuals have the same type 0. The waiting time is the time until all individuals in the population have acquired type m, under different scenarios for the rates at which forward mutations \(i\rightarrow i+1\) and backward mutations \(i\rightarrow i-1\) occur, and the selective fitness of the mutations. Although this waiting time is the time until the Markov process reaches its absorbing state, the state space of this process is huge for all but very small population sizes. The problem can be simplified though if all mutation rates are smaller than the inverse population size. The population then switches abruptly between different fixed states, where one type at a time dominates. Based on this, we show that phase-type distributions can be used to find closed form approximations for the waiting time law. Our results generalize work by Schweinsberg [60] and Durrett et al. [20], and they have numerous applications. This includes onset and growth of cancer for a cell population within a tissue, with type representing the severity of the cancer. Another application is temporal changes of gene expression among the individuals in a species, with type representing different binding sites that appear in regulatory sequences of DNA.


Coordinated mutations Fixed state population Moran model Phase-type distribution Waiting time 



The authors wish to thank an anonymous reviewer for several helpful suggestions that improved the clarity and presentation of the paper.


  1. 1.
    Asmussen, S., Nerman, O., Olsson, M.: Fitting phase-type distributions via the EM algorithm. Scand. J. Stat. 23, 419–441 (1996)zbMATHGoogle Scholar
  2. 2.
    Axe, D.D.: The limits of complex adaptation: an analysis based on a simple model of structured bacterial populations. BIO-Complex. 2010(4) (2010)Google Scholar
  3. 3.
    Barton, N.H.: The probability of fixation of a favoured allele in a subdivided population. Genet. Res. 62, 149–158 (1993)CrossRefGoogle Scholar
  4. 4.
    Beerenwinkel, N., Antal, T., Dingli, D., Traulsen, A., Kinzler, K.W., Velculescu, V.W., Vogelstein, B., Nowak, M.A.: Genetic progression and the waiting time to cancer. PLoS Comput. Biol. 3(11), e225 (2007)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Behe, M., Snoke, D.W.: Simulating evolution by gene duplication of protein features that require multiple amino acid residues. Protein Sci. 13, 2651–2664 (2004)CrossRefGoogle Scholar
  6. 6.
    Behe, M., Snoke, D.W.: A response to Michael Lynch. Protein Sci. 14, 2226–2227 (2005)CrossRefGoogle Scholar
  7. 7.
    Behrens, S., Vingron, M.: Studying evolution of promoter sequences: a waiting time problem. J. Comput. Biol. 17(12), 1591–1606 (2010)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Behrens, S., Nicaud, C., Nicodéme, P.: An automaton approach for waiting times in DNA evolution. J. Comput. Biol. 19(5), 550–562 (2012)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Bobbio, A., Horvath, Á., Scarpa, M., Telek, M.: A cyclic discrete phase type distributions: properties and a parameter estimation algorithm. Perform. Eval. 54, 1–32 (2003)CrossRefGoogle Scholar
  10. 10.
    Bodmer, W.F.: The evolutionary significance of recombination in prokaryotes. Symp. Soc. General Microbiol. 20, 279–294 (1970)Google Scholar
  11. 11.
    Carter, A.J.R., Wagner, G.P.: Evolution of functionally conserved enhancers can be accelerated in large populations: a population-genetic model. Proc. R. Soc. Lond. 269, 953–960 (2002)CrossRefGoogle Scholar
  12. 12.
    Cao, Y., et al.: Efficient step size selection for the tau-leaping simulation method. J. Chem. Phys. 124, 44109–44119 (2006)CrossRefGoogle Scholar
  13. 13.
    Chatterjee, K., Pavlogiannis, A., Adlam, B., Nowak, M.A.: The time scale of evolutionary innovation. PLOS Comput. Biol. 10(9), d1003818 (2014)CrossRefGoogle Scholar
  14. 14.
    Christiansen, F.B., Otto, S.P., Bergman, A., Feldman, M.W.: Waiting time with and without recombination: the time to production of a double mutant. Theor. Popul. Biol. 53, 199–215 (1998)CrossRefGoogle Scholar
  15. 15.
    Crow, J.F., Kimura, M.: An Introduction to Population Genetics Theory. The Blackburn Press, Caldwell (1970)zbMATHGoogle Scholar
  16. 16.
    Desai, M.M., Fisher, D.S.: Beneficial mutation-selection balance and the effect of linkage on positive selection. Genetics 176, 1759–1798 (2007)CrossRefGoogle Scholar
  17. 17.
    Durrett, R.: Probability Models for DNA Sequence Evolution. Springer, New York (2008)CrossRefGoogle Scholar
  18. 18.
    Durrett, R., Schmidt, D.: Waiting for regulatory sequences to appear. Ann. Appl. Probab. 17(1), 1–32 (2007)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Durrett, R., Schmidt, D.: Waiting for two mutations: with applications to regulatory sequence evolution and the limits of Darwinian evolution. Genetics 180, 1501–1509 (2008)CrossRefGoogle Scholar
  20. 20.
    Durrett, R., Schmidt, D., Schweinsberg, J.: A waiting time problem arising from the study of multi-stage carinogenesis. Ann. Appl. Probab. 19(2), 676–718 (2009)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Ewens, W.J.: Mathematical Population Genetics. I. Theoretical Introduction. Springer, New York (2004)CrossRefGoogle Scholar
  22. 22.
    Fisher, R.A.: On the dominance ratio. Proc. R. Soc. Edinb. 42, 321–341 (1922)CrossRefGoogle Scholar
  23. 23.
    Fisher, R.A.: The Genetical Theory of Natural Selection. Oxford University Press, Oxford (1930)CrossRefGoogle Scholar
  24. 24.
    Gerstung, M., Beerenwinkel, N.: Waiting time models of cancer progression. Math. Popul. Stud. 20(3), 115–135 (2010)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Gillespie, D.T.: Approximate accelerated simulation of chemically reacting systems. J. Chem. Phys. 115, 1716–1733 (2001)CrossRefGoogle Scholar
  26. 26.
    Gillespie, J.H.: Molecular evolution over the mutational landscape. Evolution 38(5), 1116–1129 (1984)CrossRefGoogle Scholar
  27. 27.
    Gillespie, J.H.: The role of population size in molecular evolution. Theor. Popul. Biol. 55, 145–156 (1999)CrossRefGoogle Scholar
  28. 28.
    Greven, A., Pfaffelhuber, C., Pokalyuk, A., Wakolbinger, A.: The fixation time of a strongly beneficial allele in a structured population. Electron. J. Probab. 21(61), 1–42 (2016)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Gut, A.: An Intermediate Course in Probability. Springer, New York (1995)CrossRefGoogle Scholar
  30. 30.
    Haldane, J.B.S.: A mathematical theory of natural and artificial selection. Part V: selection and mutation. Math. Proc. Camb. Philos. Soc. 23, 838–844 (1927)CrossRefGoogle Scholar
  31. 31.
    Hössjer, O., Tyvand, P.A., Miloh, T.: Exact Markov chain and approximate diffusion solution for haploid genetic drift with one-way mutation. Math. Biosci. 272, 100–112 (2016)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Iwasa, Y., Michor, F., Nowak, M.: Stochastic tunnels in evolutionary dynamics. Genetics 166, 1571–1579 (2004)CrossRefGoogle Scholar
  33. 33.
    Iwasa, Y., Michor, F., Komarova, N.L., Nowak, M.: Population genetics of tumor suppressor genes. J. Theor. Biol. 233, 15–23 (2005)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Kimura, M.: Some problems of stochastic processes in genetics. Ann. Math. Stat. 28, 882–901 (1957)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Kimura, M.: On the probability of fixation of mutant genes in a population. Genetics 47, 713–719 (1962)Google Scholar
  36. 36.
    Kimura, M.: Average time until fixation of a mutant allele in a finite population under continued mutation pressure: studies by analytical, numerical and pseudo-sampling methods. Proc. Natl. Acad. Sci. USA 77, 522–526 (1980)CrossRefGoogle Scholar
  37. 37.
    Kimura, M.: The role of compensatory neutral mutations in molecular evolution. J. Genet. 64(1), 7–19 (1985)CrossRefGoogle Scholar
  38. 38.
    Kimura, M., Ohta, T.: The average number of generations until fixation of a mutant gene in a finite population. Genetics 61, 763–771 (1969)Google Scholar
  39. 39.
    Knudson, A.G.: Two genetic hits (more or less) to cancer. Nat. Rev. Cancer 1, 157–162 (2001)CrossRefGoogle Scholar
  40. 40.
    Komarova, N.L., Sengupta, A., Nowak, M.: Mutation-selection networks of cancer initiation: tumor suppressor genes and chromosomal instability. J. Theor. Biol. 223, 433–450 (2003)MathSciNetCrossRefGoogle Scholar
  41. 41.
    Lambert, A.: Probability of fixation under weak selection: a branching process unifying approach. Theor. Popul. Biol. 69(4), 419–441 (2006)MathSciNetCrossRefGoogle Scholar
  42. 42.
    Li, T.: Analysis of explicit tau-leaping schemes for simulating chemically reacting systems. Multiscale Model. Simul. 6, 417–436 (2007)MathSciNetCrossRefGoogle Scholar
  43. 43.
    Lynch, M.: Simple evolutionary pathways to complex proteins. Protein Sci. 14, 2217–2225 (2005)CrossRefGoogle Scholar
  44. 44.
    Lynch, M., Abegg, A.: The rate of establishment of complex adaptations. Mol. Biol. Evol. 27(6), 1404–1414 (2010)CrossRefGoogle Scholar
  45. 45.
    MacArthur, S., Brockfield, J.F.Y.: Expected rates and modes of evolution of enhancer sequences. Mol. Biol. Evol. 21(6), 1064–1073 (2004)CrossRefGoogle Scholar
  46. 46.
    Maruyama, T.: On the fixation probability of mutant genes in a subdivided population. Genet. Res. 15, 221–225 (1970)CrossRefGoogle Scholar
  47. 47.
    Maruyama, T., Kimura, M.: Some methods for treating continuous stochastic processes in population genetics. Jpn. J. Genet. 46(6), 407–410 (1971)CrossRefGoogle Scholar
  48. 48.
    Maruyama, T., Kimura, M.: A note on the speed of gene frequency changes in reverse direction in a finite population. Evolution 28, 161–163 (1974)CrossRefGoogle Scholar
  49. 49.
    Moran, P.A.P.: Random processes in genetics. Proc. Camb. Philos. Soc. 54, 60–71 (1958)MathSciNetCrossRefGoogle Scholar
  50. 50.
    Neuts, M.F.: Matrix-Geometric Solutions in Stochastic Models: An Algorithmic Approach. John Hopkins University Press, Baltimore (1981)zbMATHGoogle Scholar
  51. 51.
    Nicodéme, P.: Revisiting waiting times in DNA evolution (2012). arXiv:1205.6420v1
  52. 52.
    Nowak, M.A.: Evolutionary Dynamics: Exploring the Equations of Life. Belknap Press, Cambridge (2006)zbMATHGoogle Scholar
  53. 53.
    Phillips, P.C.: Waiting for a compensatory mutation: phase zero of the shifting balance process. Genet. Res. 67, 271–283 (1996)CrossRefGoogle Scholar
  54. 54.
    Radmacher, M.D., Kelsoe, G., Kepler, T.B.: Predicted and inferred waiting times for key mutations in the germinal centre reaction: evidence for stochasticity in selection. Immunol. Cell Biol. 76, 373–381 (1998)CrossRefGoogle Scholar
  55. 55.
    Rupe, C.L., Sanford, J.C.: Using simulation to better understand fixation rates, and establishment of a new principle: Haldane’s Ratchet. In: Horstmeyer, M. (ed.) Proceedings of the Seventh International Conference of Creationism. Creation Science Fellowship, Pittsburgh, PA (2013)Google Scholar
  56. 56.
    Sanford, J., Baumgardner, J., Brewer, W., Gibson, P., Remine, W.: Mendel’s accountant: a biologically realistic forward-time population genetics program. Scalable Comput.: Pract. Exp. 8(2), 147–165 (2007)Google Scholar
  57. 57.
    Sanford, J., Brewer, W., Smith, F., Baumgardner, J.: The waiting time problem in a model hominin population. Theor. Biol. Med. Model. 12, 18 (2015)CrossRefGoogle Scholar
  58. 58.
    Schinazi, R.B.: A stochastic model of cancer risk. Genetics 174, 545–547 (2006)CrossRefGoogle Scholar
  59. 59.
    Schinazi, R.B.: The waiting time for a second mutation: an alternative to the Moran model. Phys. A. Stat. Mech. Appl. 401, 224–227 (2014)MathSciNetCrossRefGoogle Scholar
  60. 60.
    Schweinsberg, J.: The waiting time for \(m\) mutations. Electron. J. Probab. 13(52), 1442–1478 (2008)MathSciNetCrossRefGoogle Scholar
  61. 61.
    Slatkin, M.: Fixation probabilities and fixation times in a subdivided population. Evolution 35, 477–488 (1981)CrossRefGoogle Scholar
  62. 62.
    Stephan, W.: The rate of compensatory evolution. Genetics 144, 419–426 (1996)Google Scholar
  63. 63.
    Stone, J.R., Wray, G.A.: Rapid evolution of cis-regulatory sequences via local point mutations. Mol. Biol. Evol. 18, 1764–1770 (2001)CrossRefGoogle Scholar
  64. 64.
    Tuğrul, M., Paixão, T., Barton, N.H., Tkačik, G.: Dynamics of transcription factor analysis. PLOS Genet. 11(11), e1005639 (2015)CrossRefGoogle Scholar
  65. 65.
    Whitlock, M.C.: Fixation probability and time in subdivided populations. Genetics 164, 767–779 (2003)Google Scholar
  66. 66.
    Wodarz, D., Komarova, N.L.: Computational Biology of Cancer. Lecture Notes and Mathematical Modeling. World Scientific, New Jersey (2005)Google Scholar
  67. 67.
    Wright, S.: Evolution in Mendelian populations. Genetics 16, 97–159 (1931)Google Scholar
  68. 68.
    Wright, S.: The roles of mutation, inbreeding, crossbreeding and selection in evolution. In: Proceedings of the 6th International Congress on Genetics, vol. 1, pp. 356–366 (1932)Google Scholar
  69. 69.
    Wright, S.: Statistical genetics and evolution. Bull. Am. Math. Soc. 48, 223–246 (1942)MathSciNetCrossRefGoogle Scholar
  70. 70.
    Yona, A.H., Alm, E.J., Gore, J.: Random sequences rapidly evolve into de novo promoters (2017).,
  71. 71.
    Zhu, T., Hu, Y., Ma, Z.-M., Zhang, D.-X., Li, T.: Efficient simulation under a population genetics model of carcinogenesis. Bioinformatics 6(27), 837–843 (2011)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Department of MathematicsStockholm UniversityStockholmSweden
  2. 2.Biologic InstituteRedmondUSA

Personalised recommendations