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

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 $\ell, m\to +\infty$, $q\to 0$, $\ell q \to a$, $m/\ell\to\alpha$. 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. These results are supported by computer simulations.