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

## Abstract

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.

## Keywords

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

### Acknowledgements

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

## References

- 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.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.Barton, N.H.: The probability of fixation of a favoured allele in a subdivided population. Genet. Res.
**62**, 149–158 (1993)CrossRefGoogle Scholar - 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.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.Behe, M., Snoke, D.W.: A response to Michael Lynch. Protein Sci.
**14**, 2226–2227 (2005)CrossRefGoogle Scholar - 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.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.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.Bodmer, W.F.: The evolutionary significance of recombination in prokaryotes. Symp. Soc. General Microbiol.
**20**, 279–294 (1970)Google Scholar - 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.Cao, Y., et al.: Efficient step size selection for the tau-leaping simulation method. J. Chem. Phys.
**124**, 44109–44119 (2006)CrossRefGoogle Scholar - 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.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.Crow, J.F., Kimura, M.: An Introduction to Population Genetics Theory. The Blackburn Press, Caldwell (1970)zbMATHGoogle Scholar
- 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.Durrett, R.: Probability Models for DNA Sequence Evolution. Springer, New York (2008)CrossRefGoogle Scholar
- 18.Durrett, R., Schmidt, D.: Waiting for regulatory sequences to appear. Ann. Appl. Probab.
**17**(1), 1–32 (2007)MathSciNetCrossRefGoogle Scholar - 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.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.Ewens, W.J.: Mathematical Population Genetics. I. Theoretical Introduction. Springer, New York (2004)CrossRefGoogle Scholar
- 22.Fisher, R.A.: On the dominance ratio. Proc. R. Soc. Edinb.
**42**, 321–341 (1922)CrossRefGoogle Scholar - 23.Fisher, R.A.: The Genetical Theory of Natural Selection. Oxford University Press, Oxford (1930)CrossRefGoogle Scholar
- 24.Gerstung, M., Beerenwinkel, N.: Waiting time models of cancer progression. Math. Popul. Stud.
**20**(3), 115–135 (2010)MathSciNetCrossRefGoogle Scholar - 25.Gillespie, D.T.: Approximate accelerated simulation of chemically reacting systems. J. Chem. Phys.
**115**, 1716–1733 (2001)CrossRefGoogle Scholar - 26.Gillespie, J.H.: Molecular evolution over the mutational landscape. Evolution
**38**(5), 1116–1129 (1984)CrossRefGoogle Scholar - 27.Gillespie, J.H.: The role of population size in molecular evolution. Theor. Popul. Biol.
**55**, 145–156 (1999)CrossRefGoogle Scholar - 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.Gut, A.: An Intermediate Course in Probability. Springer, New York (1995)CrossRefGoogle Scholar
- 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.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.Iwasa, Y., Michor, F., Nowak, M.: Stochastic tunnels in evolutionary dynamics. Genetics
**166**, 1571–1579 (2004)CrossRefGoogle Scholar - 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.Kimura, M.: Some problems of stochastic processes in genetics. Ann. Math. Stat.
**28**, 882–901 (1957)MathSciNetCrossRefGoogle Scholar - 35.Kimura, M.: On the probability of fixation of mutant genes in a population. Genetics
**47**, 713–719 (1962)Google Scholar - 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.Kimura, M.: The role of compensatory neutral mutations in molecular evolution. J. Genet.
**64**(1), 7–19 (1985)CrossRefGoogle Scholar - 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.Knudson, A.G.: Two genetic hits (more or less) to cancer. Nat. Rev. Cancer
**1**, 157–162 (2001)CrossRefGoogle Scholar - 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.Lambert, A.: Probability of fixation under weak selection: a branching process unifying approach. Theor. Popul. Biol.
**69**(4), 419–441 (2006)MathSciNetCrossRefGoogle Scholar - 42.Li, T.: Analysis of explicit tau-leaping schemes for simulating chemically reacting systems. Multiscale Model. Simul.
**6**, 417–436 (2007)MathSciNetCrossRefGoogle Scholar - 43.Lynch, M.: Simple evolutionary pathways to complex proteins. Protein Sci.
**14**, 2217–2225 (2005)CrossRefGoogle Scholar - 44.Lynch, M., Abegg, A.: The rate of establishment of complex adaptations. Mol. Biol. Evol.
**27**(6), 1404–1414 (2010)CrossRefGoogle Scholar - 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.Maruyama, T.: On the fixation probability of mutant genes in a subdivided population. Genet. Res.
**15**, 221–225 (1970)CrossRefGoogle Scholar - 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.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.Moran, P.A.P.: Random processes in genetics. Proc. Camb. Philos. Soc.
**54**, 60–71 (1958)MathSciNetCrossRefGoogle Scholar - 50.Neuts, M.F.: Matrix-Geometric Solutions in Stochastic Models: An Algorithmic Approach. John Hopkins University Press, Baltimore (1981)zbMATHGoogle Scholar
- 51.Nicodéme, P.: Revisiting waiting times in DNA evolution (2012). arXiv:1205.6420v1
- 52.Nowak, M.A.: Evolutionary Dynamics: Exploring the Equations of Life. Belknap Press, Cambridge (2006)zbMATHGoogle Scholar
- 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.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.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.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.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.Schinazi, R.B.: A stochastic model of cancer risk. Genetics
**174**, 545–547 (2006)CrossRefGoogle Scholar - 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.Schweinsberg, J.: The waiting time for \(m\) mutations. Electron. J. Probab.
**13**(52), 1442–1478 (2008)MathSciNetCrossRefGoogle Scholar - 61.Slatkin, M.: Fixation probabilities and fixation times in a subdivided population. Evolution
**35**, 477–488 (1981)CrossRefGoogle Scholar - 62.Stephan, W.: The rate of compensatory evolution. Genetics
**144**, 419–426 (1996)Google Scholar - 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.Tuğrul, M., Paixão, T., Barton, N.H., Tkačik, G.: Dynamics of transcription factor analysis. PLOS Genet.
**11**(11), e1005639 (2015)CrossRefGoogle Scholar - 65.Whitlock, M.C.: Fixation probability and time in subdivided populations. Genetics
**164**, 767–779 (2003)Google Scholar - 66.Wodarz, D., Komarova, N.L.: Computational Biology of Cancer. Lecture Notes and Mathematical Modeling. World Scientific, New Jersey (2005)Google Scholar
- 67.Wright, S.: Evolution in Mendelian populations. Genetics
**16**, 97–159 (1931)Google Scholar - 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.Wright, S.: Statistical genetics and evolution. Bull. Am. Math. Soc.
**48**, 223–246 (1942)MathSciNetCrossRefGoogle Scholar - 70.Yona, A.H., Alm, E.J., Gore, J.: Random sequences rapidly evolve into
*de novo*promoters (2017). bioRxiv.org, https://doi.org/10.1101/111880 - 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