Abstract
We study Eigen’s quasispecies model in the asymptotic regime where the length of the genotypes goes to \(\infty \) and the mutation probability goes to 0. We give several explicit formulas for the stationary solutions of the limiting system of differential equations.
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References
Bessho C, Kuroda N (1983) A note on a more general solution of Eigen’s rate equation for selection. Bull Math Biol 45(1):143–149
Bratus AS, Novozhilov AS, Semenov YS (2014) Linear algebra of the permutation invariant Crow-Kimura model of prebiotic evolution. Math Biosci 256:42–57
Carlitz L (1973) Permutations with prescribed pattern. Math Nachr 58:31–53
Cerf R (2015) Critical population and error threshold on the sharp peak landscape for a Moran model. Mem Am Math Soc 233(1096):vi+87
Cerf R, Dalmau J (2016) The distribution of the quasispecies for a moran model on the sharp peak landscape. Stoch Process Appl 126(6):1681–1709
Dalmau J (2014) Convergence of a moran model to eigen’s quasispecies model. arXiv:1404.2133
Eigen M (1971) Self-organization of matter and the evolution of biological macromolecules. Naturwissenschaften 58(10):465–523
Eigen M, McCaskill J, Schuster P (1989) The molecular quasi-species. Adv Chem Phys 75:149–263
Jones BL (1977) Analysis of Eigen’s equations for selection of biological molecules with fluctuating mutation rates. Bull Math Biol 39(3):311–316
Jones BL, Enns RH, Rangnekar SS (1976) On the theory of selection of coupled macromolecular systems. Bull Math Biol 38(1):15–28
Kingman JFC (1977) On the properties of bilinear models for the balance between genetic mutation and selection. Math Proc Camb Philos Soc 81(3):443–453
Moran PAP (1976) Global stability of genetic systems governed by mutation and selection. Math Proc Camb Philos Soc 80(2):331–336
Moran PAP (1977) Global stability of genetic systems governed by mutation and selection II. Math Proc Camb Philos Soc 81(3):435–441
Niven I (1968) A combinatorial problem of finite sequences. Nieuw Arch Wisk 3(16):116–123
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(4):375–395
Saakian DB (2007) A new method for the solution of models of biological evolution: derivation of exact steady-state distributions. J Stat Phys 128(3):781–798
Saakian DB, Biebricher CK, Chin-Kun H (2011) Lethal mutants and truncated selection together solve a paradox of the origin of life. PLoS ONE 6(7):1–12
Saakian DB, Hu C-K (2006) Exact solution of the eigen model with general fitness functions and degradation rates. Proc Natl Acad Sci USA 103(13):4935–4939
Seifert D, Di Giallonardo F, Metzner KJ, Günthard HF, Beerenwinkel N (2015) A framework for inferring fitness landscapes of patient-derived viruses using quasispecies theory. Genetics 199(1):192–203
Semenov YS, Bratus AS, Novozhilov AS (2014) On the behavior of the leading eigenvalue of Eigen’s evolutionary matrices. Math Biosci 258:134–147
Semenov YS, Novozhilov AS (2015) Exact solutions for the selection-mutation equilibrium in the crow-kimura evolutionary model. ArXiv preprint
Semenov YS, Novozhilov AS (2015) On eigen’s quasispecies model, two-valued fitness landscapes, and isometry groups acting on finite metric spaces. ArXiv preprint
Shevelev V (2012) Number of permutations with prescribed up-down structure as a function of two variables. Integers 12(4):529–569
Swetina J, Schuster P (1982) Self-replication with errors. A model for polynucleotide replication. Biophys Chem 16(4):329–345
Thompson CJ, McBride JL (1974) On Eigen’s theory of the self-organization of matter and the evolution of biological macromolecules. Math Biosci 21:127–142
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Cerf, R., Dalmau, J. Quasispecies on Class-Dependent Fitness Landscapes. Bull Math Biol 78, 1238–1258 (2016). https://doi.org/10.1007/s11538-016-0184-y
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DOI: https://doi.org/10.1007/s11538-016-0184-y