Abstract
We exhibit a stochastic discrete time model that produces the Eigen model (Naturwissenschaften 58:465–523, 1971) in the deterministic and continuous time limits. The model is based on the competition among individuals differing in terms of fecundity but with the same viability. We explicitly write down the Markov matrix of the discrete time stochastic model in the two species case and compute the master sequence concentration numerically for various values of the total population. We also obtain the master equation of the model and perform a Van Kampen expansion. The results obtained in the two species case are compared with those coming from the Eigen model. Finally, we comment on the range of applicability of the various approaches described, when the number of species is larger than two.
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References
Alves, D., Fontanari, J.F., 1998. Error threshold in finite populations. Phys. Rev. E 57, 7008–7013.
Bonnaz, D., Koch, A.J., 1998. Stochastic model of evolving populations. J. Phys. A: Math. Gen. 31, 417–429.
Campos, P.R.A., Fontanari, J.F., 1999. Finite-size scaling of the error threshold transition in finite populations. J. Phys. A: Math. Gen. 32, L1–L7.
Demetrius, L., Schuster, P., Sigmund, K., 1985. Polynucleotide evolution and branching processes. Bull. Math. Biol. 47, 239–262.
Eigen, M., 1971. Selforganization of matter and the evolution of biological macromolecules. Naturwissenschaften 58, 465–523.
Grimstead, C.M., Snell, J.L., 1997. Introduction to Probability: Second Revised Edition. AMS, Providence.
Jones, B.L., Enns, R.H., Ragnekar, S.S., 1976. On the theory of selection of coupled macromolecular systems. Bull. Math. Biol. 38, 12–28.
Jones, B.L., Leung, H.K., 1981. Stochastic analysis of a nonlinear model for selection of biological macromolecules. Bull. Math. Biol. 43, 665–680.
McCaskill, J.S., 1984. A stochastic theory of macromolecular evolution. Biol. Cybern. 50, 63–73.
Moran, P.A.P., 1976. Global stability of genetic systems governed by mutation and selection. Math. Proc. Camb. Philos. Soc. 80, 331–336.
Nowak, M., Schuster, P., 1989. Error thresholds of replication in finite populations. Mutation frequencies and the onset of Muller’s ratchet. J. Theor. Biol. 137, 375–395.
Summers, J., Litwin, S., 2006. Examining the theory of error catastrophe. J. Virol. 80, 20–26.
Swetina, J., Schuster, P., 1982. Self-replication with errors. A model for polynucleotide replication. Biophys. Chem. 16, 329–345.
Thompson, C.J., McBride, J.L., 1974. On Eigen’s theory of self-organization of matter and the evolution of macromolecules. Math. Biosci. 21, 127–142.
Van Kampen, N.G., 2007. Stochastic Processes in Physics and Chemistry. North-Holland Personal Library, Amsterdam.
Zhang, Y.-C., 1997. Quasispecies evolution of finite populations. Phys. Rev. E 55, R3817–R3819.
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Musso, F. A Stochastic Version of the Eigen Model. Bull. Math. Biol. 73, 151–180 (2011). https://doi.org/10.1007/s11538-010-9525-4
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DOI: https://doi.org/10.1007/s11538-010-9525-4