Non-Archimedean replicator dynamics and Eigen’s paradox

Abstract

We present a new non-Archimedean model of evolutionary dynamics, in which the genomes are represented by p-adic numbers. In this model the genomes have a variable length, not necessarily bounded, in contrast with the classical models where the length is fixed. The time evolution of the concentration of a given genome is controlled by a p-adic evolution equation. This equation depends on a fitness function f and on mutation measure Q. By choosing a mutation measure of Gibbs type, and by using a p-adic version of the Maynard Smith ansatz, we show the existence of threshold function Mc(f,Q), such that the long term survival of a genome requires that its length grows faster than Mc(f,Q). This implies that Eigen’s paradox does not occur if the complexity of genomes grows at the right pace. About twenty years ago, Scheuring and Poole, Jeffares and Penny proposed a hypothesis to explain Eigen’s paradox. Our mathematical model shows that this biological hypothesis is feasible, but it requires p-adic analysis instead of real analysis. More exactly, the Darwin–Eigen cycle proposed by Poole et al takes place if the length of the genomes exceeds Mc(f,Q).