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 M_{c}(f,Q), such that the long term survival of a genome requires that its length grows faster than M_{c}(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, 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 M_{c}(f,Q).