Skip to main content
SpringerLink
Log in

On Diffusive Stability of Eigen’s Quasispecies Model

  • Published:
Journal of Dynamical and Control Systems Aims and scope Submit manuscript

Abstract

Eigen’s quasispecies system with explicit space and global regulation is considered. Limit behavior and stability of the system in a functional space under perturbations of the diffusion matrix with non-negative spectrum are investigated. It is proven that if the diffusion matrix has only positive eigenvalues, then the solutions of the distributed system converge to the equilibrium solution of the corresponding local dynamical system. These results imply that many of the properties of the quasispecies model, including the critical mutation rates that specify the infamous error threshold, do not change if the spatial interactions under the principle of global regulation are taken into account.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

References

  1. Baake E, Gabriel W. Biological evolution through mutation, selection, and drift: an introductory review. In: Stauffer D, editor. Annual reviews of computational physics VII. Singapore: World Scientific; 1999. p. 203–264.

    Google Scholar 

  2. Baake E, Wagner H. Mutation—selection models solved exactly with methods of statistical mechanics. Genet Res. 2001;78(1): 93–117.

    Article  MathSciNet  Google Scholar 

  3. Berezovskaya FS, Novozhilov AS, Karev GP. Families of traveling impulses and fronts in some models with cross-diffusion. Nonlinear Anal Real World Appl. 2008;9(5): 1866–81.

    Article  MathSciNet  MATH  Google Scholar 

  4. Boerlijst MC, Hogeweg P. Spiral wave structure in pre-biotic evolution: hypercycles stable against parasites. Phys D. 1991;48(1): 17–28.

    Article  MATH  Google Scholar 

  5. Boerlijst MC, Lamers ME, Hogeweg P. Evolutionary consequences of spiral waves in a host-parasitoid system. Proc R Soc Lond B Biol Sci. 1993;253(1336): 15–8.

    Article  Google Scholar 

  6. Bratus AS, Novozhilov AS, Platonov AP. 2010. Dynamical systems and models in biology. Fizmatlit (in Russian).

  7. Bratus AS, Novozhilov AS, Semenov YS. 2013. Linear algebra of the permutation invariant Crow–Kimura model of prebiotic evolution. arXiv preprint arXiv:1306.0111.

  8. Bratus AS, Posvyanskii VP. Stationary solutions in a closed distributed Eigen–Schuster evolution system. Differ Equ. 2006;42(12): 1762–74.

    Article  MathSciNet  MATH  Google Scholar 

  9. Bratus AS, Posvyanskii VP, Novozhilov AS. Existence and stability of stationary solutions to spatially extended autocatalytic and hypercyclic systems under global regulation and with nonlinear growth rates. Nonlinear Anal: Real World Appl. 2010;11:1897–1917.

    Article  MathSciNet  MATH  Google Scholar 

  10. Bratus AS, Posvyanskii VP, Novozhilov AS. A note on the replicator equation with explicit space and global regulation. Math Biosci Eng. 2011;8(3): 659–76.

    Article  MathSciNet  MATH  Google Scholar 

  11. Bratus AS, Posvyanskii VP, Novozhilov AS. Replicator equations and space. Math Model of Nat Phen. 2014;9(3):47–67.

  12. Bull JJ, Meyers LA, Lachmann M. Quasispecies made simple. PLoS Comput Biol. 2005;1(6): e61.

    Article  Google Scholar 

  13. Cantrell RS, Cosner C. Spatial ecology via reaction-diffusion equations. New York: Wiley; 2003.

    MATH  Google Scholar 

  14. Chacón P, Nuńo JC. Spatial dynamics of a model for prebiotic evolution. Phys D. 1995;81(4): 398–410.

    Article  MATH  Google Scholar 

  15. Cronhjort MB. The interplay between reaction and diffusion. In: Dieckmann U, Law R, Metz JAJ, editors. The geometry of ecological interactions: simplifying spatial complexity. Cambridge: Cambridge University; 2000. p. 151–70.

    Google Scholar 

  16. Cronhjort MB, Blomberg C. Hypercycles versus parasites in a two dimensional partial differential equation model. J Theor Biol. 1994;169(1): 31–49.

    Article  Google Scholar 

  17. Czárán T, Szathmáry E. Coexistence of replicators in prebiotic evolution. In: Dieckmann U, Law R, Metz JAJ, editors. The geometry of ecological interactions: simplifying spatial complexity. Cambridge: Cambridge University; 2000. p. 116–34.

    Google Scholar 

  18. Dieckmann U., Law R., Metz JAJ. The geometry of ecological interactions: simplifying spatial complexity. Cambridge: Cambridge University Press; 2000.

    Book  Google Scholar 

  19. Eigen M. Selforganization of matter and the evolution of biological macromolecules. Naturwissenschaften. 1971;58(10): 465–523.

    Article  Google Scholar 

  20. Eigen M, McCascill J, Schuster P. The molecular quasi-species. Adv Chem Phys. 1989;75: 149–263.

    Google Scholar 

  21. Evans LC. 2010. Partial differential equations. 2nd ed. American Mathematical Society.

  22. Hadeler KP. Diffusion in Fisher’s population model. Rocky Mt J Math. 1981; 11: 39–45.

    Article  MathSciNet  MATH  Google Scholar 

  23. Hillen T, Painter KJ. A user’s guide to pde models for chemotaxis. J Math Biol. 2009;58(1): 183–217.

    Article  MathSciNet  MATH  Google Scholar 

  24. Hofbauer J.The selection mutation equation. J Math Biol. 1985;23(1): 41–53.

    Article  MathSciNet  MATH  Google Scholar 

  25. Jain K, Krug J. Adaptation in simple and complex fitness landscapes. In: Bastolla U, Porto M, Eduardo Roman H, Vendruscolo M, editors. Structural approaches to sequence evolution, chapter 14. Berlin: Springer; 2007. p. 299–339.

    Google Scholar 

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

    Article  MATH  Google Scholar 

  27. Karev GP, Novozhilov AS, Berezovskaya FS. On the asymptotic behavior of the solutions to the replicator equation. Math Med Biol. 2011;28(2): 89–110.

    Article  MathSciNet  MATH  Google Scholar 

  28. Novozhilov AS, Posvyanskii VP, Bratus AS. On the reaction–diffusion replicator systems: spatial patterns and asymptotic behaviour. Russ J Numer Anal Math Model. 2012;26(6): 555–64.

    Article  MathSciNet  Google Scholar 

  29. Okubo A, Levin SA. Diffusion and ecological problems. 2nd ed. Berlin: Springer; 2002.

    Google Scholar 

  30. Saakian DB, Biebricher CK, Hu CK. Lethal mutants and truncated selection together solve a paradox of the origin of life. PLoS One. 2011;6(7): e21904.

    Article  Google Scholar 

  31. Saakian DB, Hu CK. Exact solution of the Eigen model with general fitness functions and degradation rates. Proc Natl Acad Sci USA. 2006;103(13): 4935–9.

    Article  Google Scholar 

  32. Turing AM. The chemical basis of morphogenesis. Philos Trans R Soc B-Biol Sci. 1952;237(641): 37–72.

    Article  Google Scholar 

  33. Verhulst F. Nonlinear differential equations and dynamical systems. Berlin: Springer; 1996.

    Book  MATH  Google Scholar 

  34. Weinberger E.D. Spatial stability analysis of Eigen’s quasispecies model and the less than five membered hypercycle under global population regulation. Bull Math Biol. 1991;53(4): 623–38.

    Article  MathSciNet  MATH  Google Scholar 

  35. Wilke CO. Quasispecies theory in the context of population genetics. BMC Evol Biol. 2005;5(1): 44.

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgments

This research is supported in part by the Russian Foundation for Basic Research (RFBR) grant #10-01-00374 and joint grant between RFBR and Taiwan National Council #12-01-92004HHC-a. ASN’s research is supported in part by ND EPSCoR and NSF grant #EPS-0814442. CKH is supported in part by Taiwan-Russia collaborative research grant #101-2923-M-001-003-MY3 and NCTS (North).

Author information

Authors and Affiliations

  1. Faculty of Computational Mathematics and Cybernetics, Lomonosov Moscow State University, 119992, Moscow, Russia

    Alexander S. Bratus

  2. Applied Mathematics–1, Moscow State University of Railway Engineering, 127994, Moscow, Russia

    Chin-Kun Hu

  3. Institute of Physics, Academia Sinica, 11529, Nankang, Taipei, Taiwan

    Mikhail V. Safro

  4. Department of Mathematics, North Dakota State University, 58108, Fargo, ND, USA

    Artem S. Novozhilov

Authors
  1. Alexander S. Bratus
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Chin-Kun Hu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Mikhail V. Safro
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Artem S. Novozhilov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Artem S. Novozhilov.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bratus, A.S., Hu, CK., Safro, M.V. et al. On Diffusive Stability of Eigen’s Quasispecies Model. J Dyn Control Syst 22, 1–14 (2016). https://doi.org/10.1007/s10883-014-9237-4

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10883-014-9237-4

Keywords

Mathematics Subject Classifications (2010)

Search

Navigation