# memo_has_moved_text();Critical population and error threshold on the sharp peak landscape for a Moran model

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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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 $m$ of chromosomes of length $\ell$ over an alphabet of cardinality $\kappa$. The mutation probability per locus is $q$. We deal only with the sharp peak landscape: the replication rate is $\sigma >1$ for the master sequence and $1$ for the other sequences. We study the equilibrium distribution of the process in the regime where

We obtain an equation $\alpha \,\phi (a)=\ln \kappa$ in the parameter space $(a,\alpha )$ 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 $\alpha < \ln \kappa /\ln \sigma$ then the equilibrium population is totally random, and a quasispecies can be formed only when $\alpha \geq \ln \kappa /\ln \sigma$. Finally, in the limit of very large populations, we recover an error catastrophe reminiscent of Eigen's model: if $\sigma \exp (-a)\leq 1$ then the equilibrium population is totally random, and a quasispecies can be formed only when $\sigma \exp (-a)>1$. These results are supported by computer simulations.

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