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 over an alphabet
of cardinality κ. The mutation probability per locus is q. We deal only with the
sharp peak landscape: the replication rate is σ 1 for the master sequence and
1 for the other sequences. We study the equilibrium distribution of the process in
the regime where
+∞ , m +∞ , q 0 ,
q a ∈]0, +∞[ ,
α [0, +∞] .
We obtain an equation α φ(a) = ln κ in the parameter space (a, α) 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 coun-
terpart 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 quasis-
pecies: if α ln κ/ ln σ then the equilibrium population is totally random, and a
quasispecies can be formed only when α ln κ/ ln σ. Finally, in the limit of very
large populations, we recover an error catastrophe reminiscent of Eigen’s model:
if σ exp(−a) 1 then the equilibrium population is totally random, and a quasis-
pecies can be formed only when σ exp(−a) 1. These results are supported by
computer simulations.
Received by the editor May 19, 2012, and, in revised form, November 26, 2012.
Article electronically published on May 19, 2014.
DOI: http://dx.doi.org/10.1090/memo/1096
2010 Mathematics Subject Classification. Primary 60F10, Secondary 92D25.
Key words and phrases. Moran, quasispecies, error threshold.
The author thanks an anonymous Referee for his careful reading and his remarks, which
helped to improve the presentation.
Affiliation at time of publication: Universit´ e Paris Sud, Math´ ematique, atiment 425, 91405
Orsay Cedex–France; email: rcerf@math.u-psud.fr.
2014 American Mathematical Society
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