The Quasispecies for the Wright–Fisher Model

Research Article
  • 5 Downloads

Abstract

We consider the classical Wright–Fisher model of population genetics. We prove the existence of an error threshold for the mutation probability per nucleotide, below which a quasispecies is formed. We show a new phenomenon, specific to a finite population model, namely the existence of a population threshold: to ensure the stability of the quasispecies, the population size has to be at least of the same order as the genome length. We derive an explicit formula describing the quasispecies.

Keywords

Quasispecies Error threshold Wright–Fsher: population threshold 

Notes

Compliance with Ethical Standards

Conflict of interest

The authors declare that they have no competing interest.

References

  1. Anderson, J. P., Daifuku, R., & Loeb, L. A. (2004). Viral error catastrophe by mutagenic nucleosides. Annual Review of Microbiology, 58(1), 183205.CrossRefGoogle Scholar
  2. Cerf, R. (2015). Critical population and error threshold on the sharp peak landscape for a Moran model. Memoirs of the American Mathematical Society, 233(1096), vi+87.CrossRefGoogle Scholar
  3. Cerf, R. (2015). Critical population and error threshold on the sharp peak landscape for the Wright–Fisher model. Annals of Applied Probability, 25(4), 1936–1992.CrossRefGoogle Scholar
  4. Cerf, R., & Dalmau, J. (2016). The distribution of the quasispecies for a Moran model on the sharp peak landscape. Stochastic Processes and Their Applications, 126(6), 1681–1709.CrossRefGoogle Scholar
  5. Crotty, S., Cameron, C. E., & Andino, R. (2001). RNA virus error catastrophe: Direct molecular test by using ribavirin. Proceedings of the National Academy of Sciences, 98(12), 68956900.CrossRefGoogle Scholar
  6. Dalmau, J. (2015). The distribution of the quasispecies for the Wright–Fisher model on the sharp peak landscape. Stochastic Processes and Their Applications, 125(1), 272–293.CrossRefGoogle Scholar
  7. Domingo, E. (2002). Quasispecies theory in virology. Journal of Virology, 76(1), 463–465.CrossRefPubMedCentralGoogle Scholar
  8. Domingo, E., Biebricher, C., Eigen, M., & Holland, J. J. (2001). Quasispecies and RNA virus evolution: Principles and consequences. Austin: Landes Bioscience.Google Scholar
  9. Eigen, M. (1971). Self-organization of matter and the evolution of biological macromolecules. Naturwissenschaften, 58(10), 465–523.CrossRefPubMedGoogle Scholar
  10. Eigen, M., McCaskill, J., & Schuster, P. (1989). The molecular quasi-species. Advances in Chemical Physics, 75, 149–263.Google Scholar
  11. Elena, S. F., Wilke, C. O., Ofria, C., & Lenski, R. E. (2007). Effects of population size and mutation rate on the evolution of mutational robustness. Evolution, 61(3), 666–674.CrossRefPubMedGoogle Scholar
  12. Kac, M. (1947). Random walk and the theory of Brownian motion. American Mathematical Monthly, 54(7), 369–391.CrossRefGoogle Scholar
  13. Kimura, M. (1985). The neutral theory of molecular evolution. Cambridge: Cambridge University Press.Google Scholar
  14. Nowak, M. A., & Schuster, P. (1989). Error thresholds of replication in finite populations. Mutation frequencies and the onset of Muller’s ratchet. Journal of Theoretical Biology, 137(4), 375–395.CrossRefPubMedGoogle Scholar
  15. Sumedha, Martin, O. C., & Peliti, L. (2007). Population size effects in evolutionary dynamics on neutral networks and toy landscapes. Journal of Statistical Mechanics: Theory and Experiment, 05, P05011.Google Scholar
  16. Tripathi, K., Balagam, R., Vishnoi, N. K., & Dixit, N. M. (2012). Stochastic simulations suggest that HIV-1 survives close to its error threshold. PLoS Computational Biology, 8(9), 1–14.CrossRefGoogle Scholar
  17. van Nimwegen, E., & Crutchfield, J. (2000). Metastable evolutionary dynamics: Crossing fitness barriers or escaping via neutral paths? Bulletin of Mathematical Biology, 62, 799–848.CrossRefPubMedGoogle Scholar
  18. Van Nimwegen, E., Crutchfield, J. P., & Huynen, M. (1999). Neutral evolution of mutational robustness. Proceedings of the National Academy of Sciences of the United States, 96, 9716–9720.CrossRefGoogle Scholar
  19. Wilke, C. (2005). Quasispecies theory in the context of population genetics. BMC Evolutionary Biology, 5, 1–8.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Département de mathématiques et applicationsÉcole normale supérieure, CNRS, PSL Research UniversityParisFrance
  2. 2.Laboratoire de Mathématiques d’OrsayUniv. Paris-Sud, CNRS, Université Paris-SaclayOrsayFrance
  3. 3.Centre de Mathématiques et ApplicationsÉcole PolytechniquePalaiseauFrance

Personalised recommendations