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Distribution of the quasispecies for a Galton–Watson process on the sharp peak landscape

Part of: Markov processes

Published online by Cambridge University Press:  21 June 2016

Joseba Dalmau
Joseba Dalmau*
Affiliation:
Université Paris Sud and ENS Paris
*
* Postal address: DMA, École Normale Supérieure, 45, rue d'Ulm, 75230 Paris Cedex 05, France. Email address: [email protected]
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Abstract

We study a classical multitype Galton–Watson process with mutation and selection. The individuals are sequences of fixed length over a finite alphabet. On the sharp peak fitness landscape together with independent mutations per locus, we show that, as the length of the sequences goes to ∞ and the mutation probability goes to 0, the asymptotic relative frequency of the sequences differing on k digits from the master sequence approaches (σe-a - 1)(ak/k!)∑i≥ 1ik/σi, where σ is the selective advantage of the master sequence and a is the product of the length of the chains with the mutation probability. The probability distribution Q(σ, a) on the nonnegative integers given by the above equation is the quasispecies distribution with parameters σ and a.

Keywords

QuasispeciesGalton–Watson process

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 92D25: Population dynamics (general)
Type
Short Communications
Information
Journal of Applied Probability , Volume 53 , Issue 2 , June 2016 , pp. 606 - 613
Copyright
Copyright © Applied Probability Trust 2016 

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References

[1]Antoneli, F., Bosco, F., Castro, D. and Janini, L. M. (2013).Virus replication as a phenotypic version of polynucleotide evolution.Bull. Math. Biol. 75, 602–628.CrossRefGoogle ScholarPubMed
[2]Cerf, R. (2015).Critical population and error threshold on the sharp peak landscape for a Moran model.Mem. Amer. Math. Soc. 233, 1096.Google Scholar
[3]Cerf, R. (2015).Critical population and error threshold on the sharp peak landscape for the Wright–Fisher model.Ann. Appl. Prob. 25, 1936–1992.CrossRefGoogle Scholar
[4]Cerf, R. and Dalmau, J. (2016).The distribution of the quasispecies for a Moran model on the sharp peak landscape.Stoch. Process. Appl. 126, 1681–1709.Google Scholar
[5]Dalmau, J. (2015).The distribution of the quasispecies for a Wright–Fisher model on the sharp peak landscape.Stoch. Process. Appl. 125, 272–293.Google Scholar
[6]Demetrius, L., Schuster, P. and Sigmund, K. (1985).Polynucleotide evolution and branching processes.Bull. Math. Biol. 47, 239–262.CrossRefGoogle ScholarPubMed
[7]Eigen, M. (1971).Selforganization of matter and the evolution of biological macromolecules.Naturwissenschaften 58, 465–523.Google Scholar
[8]Harris, T. E. (1963).The Theory of Branching Processes.Springer, Berlin.CrossRefGoogle Scholar
[9]Kemeny, J. G. and Snell, J. L. (1960).Finite Markov Chains.Van Nostrand, Princeton, NJ.Google Scholar