A localization threshold for macromolecular quasispecies from continuously distributed replication rates

Localization in the space of mutant sequences of an equilibrium distribution of independently self‐replicating macromolecules or quasispecies, as described by the deterministic evolution equations of Eigen (1971) is necessary for a dynamically stable, finite population and simple Darwinian selection of a particular genotype. In this work, the localization properties are investigated as a functional of the probability distribution (possibly continuous) of replication rates for the different mutants, including the neutral mutation extreme. The central result is the existence and evaluation of a threshold sequence length (for fixed monomer copying fidelity) below which the quasispecies is localized with unit probability around the particular mutant with maximum replication rate. This localization threshold differs in two respects from the error threshold of Eigen, which it confirms in the limit of identical superiority of the wild type over all other mutant replication rates: It is independent of mutant population variables and it predicts a localization threshold even in the presence of mutants arbitrarily close in exact replication rate to the maximum. General conclusions about the threshold are made on the basis of extreme value theory. Similarities of the problem with that of Anderson localization in disordered metals and quantum spin systems are exploited in the probability analysis and renormalization of the perturbation theory.