Finite-size scaling of the error threshold transition in finite population

TL;DR

This paper investigates how finite population size affects the error threshold transition in a molecular evolution model, revealing a specific scaling law for the time until the master sequence disappears.

Contribution

It introduces a finite-size scaling analysis of the error threshold transition in the stochastic quasispecies model, highlighting the scaling behavior of the master sequence extinction time.

Findings

The transition is first-order at a critical replication probability Q_c.

The master sequence disappearance time scales as N^{1/2} in the critical region.

A scaling function describes the transition dynamics near Q_c.

Abstract

The error threshold transition in a stochastic (i.e. finite population) version of the quasispecies model of molecular evolution is studied using finite-size scaling. For the single-sharp-peak replication landscape, the deterministic model exhibits a first-order transition at $Q=Q_c=1/a$, where $% Q$ is the probability of exact replication of a molecule of length $L \to \infty$, and $a$ is the selective advantage of the master string. For sufficiently large population size, $N$, we show that in the critical region the characteristic time for the vanishing of the master strings from the population is described very well by the scaling assumption $\tau = N^{1/2} f_a \left [ \left (Q - Q_c) N^{1/2} \right ] $, where $f_a$ is an $a$-dependent scaling function.