How to Order

For AMS eBook frontlist subscriptions or backfile collection purchases:

   1a. To purchase any ebook backfile or to subscibe to the current year of Contemporary Mathematics, please download this required license agreement,

   1b. To subscribe to the current year of Memoirs of the AMS, please download this required license agreement.

   2. Complete and sign the license agreement.

   3. Email, fax, or send via postal mail to:

Customer Services
American Mathematical Society
201 Charles Street Providence, RI 02904-2213  USA
Phone: 1-800-321-4AMS (4267)
Fax: 1-401-455-4046

Visit the AMS Bookstore for individual volume purchases.

Browse the current eBook Collections price list

Powered by MathJax
  Remote Access

Critical population and error threshold on the sharp peak landscape for a Moran model

About this Title

Raphaël Cerf

Publication: Memoirs of the American Mathematical Society
Publication Year: 2015; Volume 233, Number 1096
ISBNs: 978-1-4704-0967-8 (print); 978-1-4704-1964-6 (online)
Published electronically: May 19, 2014
Keywords:Moran, quasispecies

View full volume PDF

View other years and numbers:

Table of Contents


  • Chapter 1. Introduction
  • Chapter 2. The Model
  • Chapter 3. Main Results
  • Chapter 4. Coupling
  • Chapter 5. Normalized Model
  • Chapter 6. Lumping
  • Chapter 7. Monotonicity
  • Chapter 8. Stochastic Bounds
  • Chapter 9. Birth and Death Processes
  • Chapter 10. The Neutral Phase
  • Chapter 11. Synthesis
  • Appendix A. Appendix on Markov Chains


The goal of this work is to propose a finite population counterpart to Eigen's model, which incorporates stochastic effects. We consider a Moran model describing the evolution of a population of size of chromosomes of length over an alphabet of cardinality . The mutation probability per locus is . We deal only with the sharp peak landscape: the replication rate is for the master sequence and for the other sequences. We study the equilibrium distribution of the process in the regime where

We obtain an equation in the parameter space separating the regime where the equilibrium population is totally random from the regime where a quasispecies is formed. We observe the existence of a critical population size necessary for a quasispecies to emerge and we recover the finite population counterpart of the error threshold. Moreover, in the limit of very small mutations, we obtain a lower bound on the population size allowing the emergence of a quasispecies: if then the equilibrium population is totally random, and a quasispecies can be formed only when . Finally, in the limit of very large populations, we recover an error catastrophe reminiscent of Eigen's model: if then the equilibrium population is totally random, and a quasispecies can be formed only when . These results are supported by computer simulations.

References [Enhancements On Off] (What's this?)

  • [1] Domingos Alves and Jose Fernando Fontanari, Error threshold in finite populations, Phys. Rev. E 57 (1998), 7008–7013.
  • [2] Jon P. Anderson, Richard Daifuku, and Lawrence A. Loeb, Viral error catastrophe by mutagenic nucleosides, Annual Review of Microbiology 58(1) (2004), 183–205.
  • [3] Ellen Baake and Wilfried Gabriel, Biological evolution through mutation, selection, and drift: An introductory review, Ann. Rev. Comp. Phys. VII (2000), 203–264.
  • [4] Michael Baake and Ellen Baake, An exactly solved model for mutation, recombination and selection, Canad. J. Math. 55 (2003), no. 1, 3–41. MR 1952324, 10.4153/CJM-2003-001-0
  • [5] N. H. Bingham, Fluctuation theory for the Ehrenfest urn, Adv. in Appl. Probab. 23 (1991), no. 3, 598–611. MR 1122877, 10.2307/1427624
  • [6] Leo Breiman, Probability, Addison-Wesley Publishing Company, Reading, Mass.-London-Don Mills, Ont., 1968. MR 0229267
  • [7] Leo Breiman, Probability, Classics in Applied Mathematics, vol. 7, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1992. Corrected reprint of the 1968 original. MR 1163370
  • [8] Shane Crotty, Craig E. Cameron, and Raul Andino, RNA virus error catastrophe: Direct molecular test by using ribavirin, Proceedings of the National Academy of Sciences 98(12) (2001), 6895–6900.
  • [9] Lloyd Demetrius, Peter Schuster, and Karl Sigmund, Polynucleotide evolution and branching processes, Bull. Math. Biol. 47 (1985), no. 2, 239–262. MR 803564, 10.1016/S0092-8240(85)90051-5
  • [10] Narendra M. Dixit, Piyush Srivastava, and Nisheeth K. Vishnoi, A finite population model of molecular evolution: theory and computation, J. Comput. Biol. 19 (2012), no. 10, 1176–1202. MR 2990752, 10.1089/cmb.2012.0064
  • [11] Esteban Domingo, Quasispecies theory in virology, Journal of Virology 76 (2002), no. 1, 463–465.
  • [12] Esteban Domingo, Christof Biebricher, Manfred Eigen, and John J. Holland, Quasispecies and rna virus evolution: principles and consequences, Landes Bioscience, Austin, Tex., 2001.
  • [13] Manfred Eigen, Self-organization of matter and the evolution of biological macromolecules, Naturwissenschaften 58 (1971), no. 10, 465–523.
  • [14] —, Natural selection: a phase transition?, Biophysical Chemistry 85 (2000), no. 2–3, 101–123.
  • [15] Manfred Eigen, John McCaskill, and Peter Schuster, The molecular quasi-species., Advances in Chemical Physics 75 (1989), 149–263.
  • [16] Santiago F. Elena, Claus O. Wilke, Charles Ofria, and Richard E. Lenski, Effects of population size and mutation rate on the evolution of mutational robustness, Evolution 61(3) (2007), 666–74.
  • [17] Warren J. Ewens, Mathematical population genetics. I, 2nd ed., Interdisciplinary Applied Mathematics, vol. 27, Springer-Verlag, New York, 2004. Theoretical introduction. MR 2026891
  • [18] William Feller, An introduction to probability theory and its applications. Vol. I and II, John Wiley & Sons Inc., New York, 1968 and 1971.
  • [19] Daniel T. Gillespie, A general method for numerically simulating the stochastic time evolution of coupled chemical reactions, J. Computational Phys. 22 (1976), no. 4, 403–434. MR 0503370
  • [20] Geoffrey Grimmett, Percolation, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 321, Springer-Verlag, Berlin, 1999. MR 1707339
  • [21] Mark Kac, Random walk and the theory of Brownian motion, Amer. Math. Monthly 54 (1947), 369–391. MR 0021262
  • [22] Samuel Karlin and Howard M. Taylor, A first course in stochastic processes, 2nd ed., Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975. MR 0356197
  • [23] John G. Kemeny and J. Laurie Snell, Finite Markov chains, Springer-Verlag, New York-Heidelberg, 1976. Reprinting of the 1960 original; Undergraduate Texts in Mathematics. MR 0410929
  • [24] Motoo Kimura, The neutral theory of molecular evolution, Cambridge University Press, 1985 (reprint).
  • [25] Ira Leuthäusser, Statistical mechanics of Eigen’s evolution model, J. Statist. Phys. 48 (1987), no. 1-2, 343–360. MR 914439, 10.1007/BF01010413
  • [26] Thomas M. Liggett, Interacting particle systems, Classics in Mathematics, Springer-Verlag, Berlin, 2005. Reprint of the 1985 original. MR 2108619
  • [27] John McCaskill, A stochastic theory of macromolecular evolution, Biological Cybernetics 50 (1984), 63–73.
  • [28] P. A. P. Moran, Random processes in genetics, Proc. Cambridge Philos. Soc. 54 (1958), 60–71. MR 0127989
  • [29] Fabio Musso, A stochastic version of the Eigen model, Bull. Math. Biol. 73 (2011), no. 1, 151–180. MR 2770281, 10.1007/s11538-010-9525-4
  • [30] Erik Van Nimwegen, James P. Crutchfield, and Martijn Huynen, Neutral evolution of mutational robustness, Proc. Natl . Acad. Sci . USA 96 (1999), 9716–9720.
  • [31] Martin A. Nowak, Evolutionary dynamics, The Belknap Press of Harvard University Press, Cambridge, MA, 2006. Exploring the equations of life. MR 2252879
  • [32] Martin A. Nowak and Peter Schuster, Error thresholds of replication in finite populations. Mutation frequencies and the onset of Muller's ratchet., Journal of theoretical Biology 137 (4) (1989), 375–395.
  • [33] Jeong-Man Park, Enrique Muñoz, and Michael W. Deem, Quasispecies theory for finite populations, Phys. Rev. E 81 (2010), 011902.
  • [34] David B. Saakian, Michael W. Deem, and Chin-Kun Hu, Finite population size effects in quasispecies models with single-peak fitness landscape, Europhysics Letters 98 (2012), no. 1, 18001.
  • [35] Roberto H. Schonmann, The pattern of escape from metastability of a stochastic Ising model, Comm. Math. Phys. 147 (1992), no. 2, 231–240. MR 1174411
  • [36] Ricard V. Solé and Thomas S. Deisboeck, An error catastrophe in cancer?, Journal of Theoretical Biology 228 (2004), 47–54.
  • [37] Sumedha, Olivier C Martin, and Luca Peliti, Population size effects in evolutionary dynamics on neutral networks and toy landscapes, Journal of Statistical Mechanics: Theory and Experiment 2007 (2007), no. 05, P05011.
  • [38] Kushal Tripathi, Rajesh Balagam, Nisheeth K. Vishnoi, and Narendra M. Dixit, Stochastic simulations suggest that HIV-1 survives close to its error threshold, PLoS Comput. Biol. 8 (2012), no. 9, e1002684, 14. MR 2993809, 10.1371/journal.pcbi.1002684
  • [39] Erik van Nimwegen and James Crutchfield, Metastable evolutionary dynamics: Crossing fitness barriers or escaping via neutral paths?, Bulletin of Mathematical Biology 62 (2000), 799–848.
  • [40] Edward D. Weinberger, A stochastic generalization of eigen's theory of natural selection, Ph.D. Dissertation, The Courant Institute of Mathematical Sciences, New York University, 1987.
  • [41] Claus Wilke, Quasispecies theory in the context of population genetics, BMC Evolutionary Biology 5 (2005), 1–8.
American Mathematical Society