A basic model of mutations
ESAIM: Probability and Statistics, Tome 24 (2020), pp. 789-800.

We study a basic model for mutations. We derive exact formulae for the mean time needed to discover the master sequence, the mean returning time to the initial state, or to any Hamming class. These last two formulae are the same than the formulae obtained by Mark Kac for the Ehrenfest model.

DOI : 10.1051/ps/2020024
Classification : 60J10, 60J45, 92D10
Mots-clés : Markov chain, generating function, genetics, potential theory 
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     author = {Berger, Maxime and Cerf, Rapha\"el},
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     language = {en},
     url = {http://www.numdam.org/articles/10.1051/ps/2020024/}
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Berger, Maxime; Cerf, Raphaël. A basic model of mutations. ESAIM: Probability and Statistics, Tome 24 (2020), pp. 789-800. doi : 10.1051/ps/2020024. http://www.numdam.org/articles/10.1051/ps/2020024/

[1] R. Cerf and J. Dalmau, The quasispecies for the Wright–Fisher model. Evol. Biol. 45 (2018) 318–323. | DOI

[2] P. Diaconis and R. Griffiths, Exchangeable pairs of Bernoulli random variables, krawtchouck polynomials, and Ehrenfest urns. Aust. N. Z. J. Statist. 54 (2012) 81–101. | DOI | MR | Zbl

[3] E. Domingo and P. Schuster, Quasispecies: From Theory to Experimental Systems. Current Topics in Microbiology and Immunology. Springer International Publishing (2016).

[4] P.G. Doyle and J. Laurie Snell, Random Walks and Electric Networks, Vol. 22 of Carus Mathematical Monographs. Mathematical Association of America, Washington, DC (1984). | MR | Zbl

[5] M. Eigen, J. Mccaskill and P. Schuster, The molecular quasi-species. J. Phys. Chem. 92 (1988) 6881–6891. | DOI

[6] W. Feller, An Introduction to Probability Theory and its Applications, Vol. I. John Wiley & Sons Inc., New York (1968). | MR

[7] F.G. Hess, Alternative solution to the Ehrenfest problem. Am. Math. Monthly 61 (1954) 323–328. | DOI | MR | Zbl

[8] M. Kac, Random walk and the theory of Brownian motion. Am. Math. Monthly 54 (1947) 369–391. | DOI | MR | Zbl

[9] J. Neveu, Chaînes de Markov et théorie du potentiel. Ann. Fac. Sci. Univ. Clermont-Ferrand 24 (1964) 37–89. | Numdam | MR

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