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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)
DOI: http://dx.doi.org/10.1090/memo/1096
Published electronically: May 19, 2014
Keywords:Moran, quasispecies

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Table of Contents


Chapters

  • 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

Abstract


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.

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