Skip to main content
Log in

Quasispecies on Class-Dependent Fitness Landscapes

  • Original Article
  • Published:
Bulletin of Mathematical Biology Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

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

    Article  MathSciNet  MATH  Google Scholar 

  • Bratus AS, Novozhilov AS, Semenov YS (2014) Linear algebra of the permutation invariant Crow-Kimura model of prebiotic evolution. Math Biosci 256:42–57

    Article  MathSciNet  MATH  Google Scholar 

  • Carlitz L (1973) Permutations with prescribed pattern. Math Nachr 58:31–53

    Article  MathSciNet  MATH  Google Scholar 

  • 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

    Article  MathSciNet  MATH  Google Scholar 

  • 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

    Article  Google Scholar 

  • Eigen M, McCaskill J, Schuster P (1989) The molecular quasi-species. Adv Chem Phys 75:149–263

    Google Scholar 

  • Jones BL (1977) Analysis of Eigen’s equations for selection of biological molecules with fluctuating mutation rates. Bull Math Biol 39(3):311–316

    MATH  Google Scholar 

  • Jones BL, Enns RH, Rangnekar SS (1976) On the theory of selection of coupled macromolecular systems. Bull Math Biol 38(1):15–28

    Article  MATH  Google Scholar 

  • 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

    Article  MathSciNet  MATH  Google Scholar 

  • Moran PAP (1976) Global stability of genetic systems governed by mutation and selection. Math Proc Camb Philos Soc 80(2):331–336

    Article  MathSciNet  MATH  Google Scholar 

  • Moran PAP (1977) Global stability of genetic systems governed by mutation and selection II. Math Proc Camb Philos Soc 81(3):435–441

    Article  MathSciNet  MATH  Google Scholar 

  • Niven I (1968) A combinatorial problem of finite sequences. Nieuw Arch Wisk 3(16):116–123

    MathSciNet  MATH  Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Article  MathSciNet  MATH  Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • Semenov YS, Bratus AS, Novozhilov AS (2014) On the behavior of the leading eigenvalue of Eigen’s evolutionary matrices. Math Biosci 258:134–147

    Article  MathSciNet  MATH  Google Scholar 

  • 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

    Article  MathSciNet  MATH  Google Scholar 

  • Swetina J, Schuster P (1982) Self-replication with errors. A model for polynucleotide replication. Biophys Chem 16(4):329–345

    Article  Google Scholar 

  • 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

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Joseba Dalmau.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

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

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11538-016-0184-y

Keywords

Navigation