Critical population and error threshold on the sharp peak landscape for the Wright–Fisher model

Abstract

We pursue the task of developing a finite population counterpart to Eigen’s model. We consider the classical Wright–Fisher 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 $q$. 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 \begin{eqnarray*}\ell&\to&+\infty,\qquad m\to+\infty,\qquad q\to0,\\{\ell q}&\to&a\in\,]0,+\infty[,\qquad\frac{m}{\ell}\to\alpha\in[0,+\infty].\end{eqnarray*} We obtain an equation $\alpha\psi(a)=\ln\kappa$ 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. The result is the twin brother of the corresponding result for the Moran model. The proof is more complex, and it relies on the Freidlin–Wentzell theory of random perturbations of dynamical systems.