Skip to main content

An asymptotic maximum principle for essentially linear evolution models

Abstract.

Recent work on mutation-selection models has revealed that, under specific assumptions on the fitness function and the mutation rates, asymptotic estimates for the leading eigenvalue of the mutation-reproduction matrix may be obtained through a low-dimensional maximum principle in the limit N→∞ (where N, or Nd with d≥1, is proportional to the number of types). In order to extend this variational principle to a larger class of models, we consider here a family of reversible matrices of asymptotic dimension Nd and identify conditions under which the high-dimensional Rayleigh-Ritz variational problem may be reduced to a low-dimensional one that yields the leading eigenvalue up to an error term of order 1/N. For a large class of mutation-selection models, this implies estimates for the mean fitness, as well as a concentration result for the ancestral distribution of types.

This is a preview of subscription content, access via your institution.

References

  1. Akin, E.: The Geometry of Population Genetics. Lect. Notes Biomath., vol. 31, Springer, Berlin, 1979

  2. Athreya, K.B., Ney P.E.: Branching Processes. Springer, New York, 1972

  3. Baake, E., Baake, M., Wagner, H.: Ising quantum chain is equivalent to a model of biological evolution. Phys. Rev. Lett. 78, 559–562 (1997); Erratum: Phys. Rev. Lett. 79, 1782 (1997)

    Google Scholar 

  4. Baake, E., Baake, M., Wagner, H.: Quantum mechanics versus classical probability in biological evolution. Phys. Rev. E 57, 1191–1192 (1998)

    Article  Google Scholar 

  5. Baake, E., Wagner, H.: Mutation-selection models solved exactly with methods from statistical mechanics. Genet. Res. 78, 93–117 (2001)

    Article  Google Scholar 

  6. Ben-Arous, G., Bovier, A., Gayrard, V.: Glauber dynamics of the random energy model. 1. Metastable motion on the extreme states. Commun. Math. Phys. 235, 379–425 (2003)

    Google Scholar 

  7. Bovier, A., Eckhoff, M., Gayrard, V., Klein, M.: Metastability in stochastic dynamics of disordered mean-field models. Probab. Theor. Rel. Fields 119, 99–161 (2001); cond-mat/9811331

    MathSciNet  MATH  Google Scholar 

  8. Brémaud, P.: Markov Chains. Gibbs Fields, Monte Carlo Simulation, and Queues. Springer, New York, 1999

  9. de Bruijn, N.G.: Asymptotic Methods in Analysis. 3rd ed., corrected reprint, Dover, New York, 1981

  10. Bürger, R.: The Mathematical Theory of Selection, Recombination, and Mutation. Wiley, Chichester, 2000

  11. Bulmer, M.: Theoretical Evolutionary Ecology. Sinauer, Sunderland, 1994

  12. Burke, C., Rosenblatt, M.: A Markovian function of a Markov chain. Ann. Math. Statist. 29, 1112–1122 (1958)

    MATH  Google Scholar 

  13. Caswell, H.: Matrix Population Models: Construction, Analysis, and Interpretation. 2nd ed., Sinauer, Sunderland, 2000

  14. Charlesworth, B.: Mutation-selection balance and the evolutionary advantage of sex and recombination. Genet. Res. Camb. 55, 199–221 (1990)

    Google Scholar 

  15. Crow, J.F., Kimura, M.: An Introduction to Population Genetics Theory. Harper & Row, New York, 1970

  16. Durrett, R.: Probability models for DNA sequence evolution. Springer, New York, 2002

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

    Google Scholar 

  18. Ethier, S.N., Kurtz, T.G.: Markov Processes – Characterization and Convergence. Wiley, New York, 1986

  19. Ewens, W.: Mathematical Population Genetics. 2nd ed., Springer, New York, 2004

  20. Ewens, W., Grant, P.: Statistical Methods in Bioinformatics. Springer, New York, 2001

  21. Garske, T., Grimm, U.: A maximum principle for the mutation-selection equilibrium of nucleotide sequences. Bull. Math. Biol. 66, 397–421 (2004); physics/0303053

    Article  Google Scholar 

  22. Gayrard, V.: The thermodynamic limit of the q-state Potts–Hopfield model with infinitely many patterns. J. Stat. Phys. 68, 977–1011 (1992)

    MathSciNet  MATH  Google Scholar 

  23. Georgii, H.-O., Baake, E.: Multitype branching processes: the ancestral types of typical individuals. Adv. Appl. Prob. 35, 1090–1110 (2003); math.PR/0302049

    Article  MATH  Google Scholar 

  24. Gerland, U., Hwa, T.: On the selection and evolution of regulatory DNA motifs. J. Mol. Evol. 55, 386–400 (2002)

    Article  Google Scholar 

  25. Hadeler, K.P.: Stable polymorphisms in a selection model with mutation. SIAM J. Appl. Math. 41, 1–7 (1981)

    Google Scholar 

  26. Hermisson, J., Redner, O., Wagner, H., Baake, E.: Mutation-selection balance: Ancestry, load, and maximum principle. Theor. Pop. Biol. 62, 9–46 (2002); cond-mat/0202432

    Article  Google Scholar 

  27. Hermisson, J., Wagner, H., Baake, M.: Four-state quantum chain as a model of sequence evolution. J. Stat. Phys. 102, 315–343 (2001); cond-mat/0008123

    Article  MathSciNet  MATH  Google Scholar 

  28. Hofbauer, J.: The selection-mutation equation. J. Math. Biol. 23, 41–53 (1985)

    MathSciNet  MATH  Google Scholar 

  29. Jagers, P.: General branching processes as Markov fields. Stoch. Proc. Appl. 32, 183–242 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  30. Jagers, P.: Stabilities and instabilities in population dynamics. J. Appl. Prob. 29, 770–780 (1992)

    MathSciNet  MATH  Google Scholar 

  31. Karlin, K.S., Taylor, H.M.: A first course in stochastic processes. 2nd ed., Academic Press, San Diego, 1975

  32. Karlin, K.S., Taylor, H.M.: A Second Course in Stochastic Processes. Academic Press, San Diego, 1981

  33. Kato, T.: Perturbation Theory for Linear Operators. reprinted ed., Springer, New York, 1995

  34. Keilson, J.: Markov Chain Models – Rarity and Exponentiality. Springer, New York, 1979

  35. Kemeny, J.G., Snell, J.L.: Finite Markov Chains. Springer, New York, 1981

  36. Kondrashov, A.S.: Deleterious mutations and the evolution of sexual reproduction. Nature 336, 435–440 (1988)

    Article  Google Scholar 

  37. Kot, M.: Elements of Mathematical Ecology. Cambridge University Press, Cambridge, 2001

  38. Leuthäusser, I.: Statistical mechanics of Eigen’s evolution model. J. Stat. Phys. 48, 343–360 (1987)

    MathSciNet  Google Scholar 

  39. Mézard, M., Parisi, G., Virasoro, M.A.: Spin Glass Theory and Beyond. World Scientific, Singapore, 1987

  40. Nowak, M., Schuster, P.: Error thresholds of replication in finite populations. Mutation frequencies and the onset of Muller’s ratchet. J. Theor. Biol. 137, 375–395 (1989)

    Google Scholar 

  41. Prasolov, V.V.: Problems and Theorems in Linear Algebra. AMS, Providence, RI, 1994; corrected reprint, 1996

  42. Redner, O.: Discrete approximation of non-compact operators describing continuum-of-alleles models, Proc. Edinburgh Math. Soc. 47, 449–472 (2004); math.SP/0301024

    Article  Google Scholar 

  43. Rouzine, I.M., Wakeley, J., Coffin, J.M.: The solitary wave of asexual evolution. PNAS 100, 587–592 (2003)

    Article  Google Scholar 

  44. Swofford, D.L., Olsen, G.J., Waddell, P.J., Hillis, D.M.: Phylogenetic Inference. In: D.M. Hillis, C. Moritz, B.K. Mable (eds), Molecular Systematics, Sinauer, Sunderland, 1995, pp. 407–514

  45. Tarazona, P.: Error threshold for molecular quasispecies as phase transition: From simple landscapes to spin glass models. Phys. Rev. A 45, 6038–6050 (1992)

    Article  Google Scholar 

  46. Thompson, C.J., McBride, J.L.: On Eigen’s theory of the self-organization of matter and the evolution of biological macromolecules. Math. Biosci. 21, 127–142 (1974)

    Article  MATH  Google Scholar 

  47. Wagner, H., Baake, E., Gerisch, T.: Ising quantum chain and sequence evolution. J. Stat. Phys. 92, 1017–1052 (1998)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Ellen Baake.

Additional information

Acknowledgement Support and hospitality of the Erwin Schrödinger International Institute for Mathematical Physics in Vienna, where this work was initiated and partly carried out, is gratefully acknowledged. A.B. also thanks the German Research Council (DFG) for partial support in the Dutch-German Bilateral Research Group “Mathematics of random Spatial Models from Physics and Biology”. It is our pleasure to thank T. Garske and P. Blanchard for helpful discussions, and R. Bürger and an anonymous referee for reading the manuscript very carefully and providing valuable suggestions for improvement.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Baake, E., Baake, M., Bovier, A. et al. An asymptotic maximum principle for essentially linear evolution models. J. Math. Biol. 50, 83–114 (2005). https://doi.org/10.1007/s00285-004-0281-7

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00285-004-0281-7

Keywords

  • Asymptotics of leading eigenvalue
  • Reversibility
  • Mutation-selection models
  • Ancestral distribution
  • Lumping