CRITICAL POPULATION AND ERROR THRESHOLD ON THE SHARP PEAK LANDSCAPE FOR THE WRIGHT-FISHER MODEL

• 成果类型: Article
• 署名作者: Cerf, Raphael
• 署名单位: Universite Paris Saclay
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/14-AAP1039
• 发表日期: 2015
• 页码: 1936-1992
• 关键词: evolution

摘要

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 l 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 l -> +infinity, m -> +infinity, q -> 0, l(q) -> a is an element of]0, +infinity[, m/l -> alpha is an element of [0, +infinity]. We obtain an equation colf 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.