Journal of Statistical Physics

, Volume 48, Issue 1–2, pp 343–360 | Cite as

Statistical mechanics of Eigen's evolution model

  • Ira LeuthÄusser


The correspondence between Eigen's model of macromolecular evolution and the equilibrium statistical mechanics of an inhomogeneous Ising system is developed. The free energy landscape of random Ising systems with the Hopfield Hamiltonian as a special example is applied to the replication rate coefficient landscape. The coupling constants are scaled with 1/l, since the maxima of any landscape must not increase with the length of the macromolecules. The calculated error threshold relation then agrees with Eigen's expression, which was derived in a different way. It gives an explicit expression for the superiority parameter in terms of the parameters of the landscape. The dynamics of selection and evolution is discussed.

Key words

Macromolecular evolution inhomogeneous random Ising systems replication number landscape error threshold quasispecies 


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  1. 1.
    M. Eigen,Naturwissenschaften 58:465 (1971); M. Eigen and P. Schuster,Naturwissenschaften 64:541 (1977).Google Scholar
  2. 2.
    I. LeuthÄusser,J. Chem. Phys. 84:1884 (1986).Google Scholar
  3. 3.
    J. B. Kogut,Rev. Mod. Phys. 51:659 (1979).Google Scholar
  4. 4.
    W. Kinzel,Z. Phys. B 58:229 (1985); E. Domany and W. Kinzel,Phys. Rev. Lett. 53:311 (1984).Google Scholar
  5. 5.
    B. L. Jones, R. H. Enns, and S. S. Rangnekar,Bull. Math. Biol. 38:15 (1976).Google Scholar
  6. 6.
    L. Demetrius, P. Schuster, and K. Sigmund,Bull. Biol. 47:239 (1985).Google Scholar
  7. 7.
    P. W. Kasteleyn, inFundamental Problems in Statistical Mechanics, Vol. 3, E. G. D. Cohen, ed. (North-Holland/Elsevier, Amsterdam/New York, 1975); E. W. Montroll,J. Chem. Phys. 9:706 (1941).Google Scholar
  8. 8.
    R. J. Baxter,Exactly Solved Models in Statistical Mechanics (Academic Press, London, 1982).Google Scholar
  9. 9.
    S. F. Edwards and P. W. Anderson,J. Phys. F 5:965 (1965).Google Scholar
  10. 10.
    K. Binder, inPhase Transitions and Critical Phenomena, Vol. 8, C. Domb and J. L. Lebowitz, eds. (Academic Press, London, 1983).Google Scholar
  11. 11.
    U. LeuthÄusser, personal communication.Google Scholar
  12. 12.
    I. Rechenberg,Evolutionsstrategie: Optimierung technischer Systeme nach Prinzipien der biologischen Evolution (Frommann-Holzboog, Stuttgart, 1973); Q. Wang,Biol. Cybern. 56 (1987).Google Scholar
  13. 13.
    C. Biebricher, inEvolutionary Biology, Vol. 16, M. K. Hecht, B. Wallace, and G. T. Prance, eds. (Plenum Press, New York, 1983).Google Scholar
  14. 14.
    M. Eigen,Chem. Scripta 26B:13 (1986).Google Scholar
  15. 15.
    P. W. Anderson,Proc. Natl. Acad. Sci. USA 80:3386 (1983); D. S. Rokhsar, P. W. Anderson, and D. L. Stein,J. Mol. Evol. 23:119 (1986).Google Scholar
  16. 16.
    S. Kirkpatrick and D. Sherrington,Phys. Rev. B 17:4384 (1978).Google Scholar
  17. 17.
    J. J. Hopfield,Proc. Natl. Acad. Sci. USA 79:2554 (1982).Google Scholar
  18. 18.
    D. J. Amit, H. Gutfreund, and H. Sompolinsky,Phys. Rev. A 32:1007 (1985).Google Scholar
  19. 19.
    D. J. Amit, H. Gutfreund, and H. Sompolinsky,Phys. Rev. Lett. 55:1530 (1985).Google Scholar
  20. 20.
    G. A. Baker, Jr.,Phys. Rev. 130:1406 (1963).Google Scholar
  21. 21.
    T. Takeda and H. Fukuyama,J. Phys. Jpn. 40:925 (1976).Google Scholar
  22. 22.
    W. Ebeling, A. Engel, B. Esser, and R. Feistl,J. Stat. Phys. 37:369 (1984).Google Scholar

Copyright information

© Plenum Publishing Corporation 1987

Authors and Affiliations

  • Ira LeuthÄusser
    • 1
  1. 1.Max-Planck-Institut für biophysikalische ChemieGöttingenWest Germany

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