Eigen’s paradox and the quasispecies model in a non-Archimedean framework

In this article we present a new p-adic generalization of the Eigen–Schuster model where the genomes (sequences) are represented by words written in the alphabet 0,1,…,p−1, where p is a prime number, with a time variable length. The time evolution of the concentration of a given sequence is controlled by a p-adic evolution equation. The long term behavior of the concentration of a sequence depends on a fitness function f, a mutation measure Q, and an initial concentration distribution. The new model provides essentially two types of asymptotic scenarios for evolution. If the complexity of sequences grows at the right pace, then in the long term the survival is assured. This agrees with the fact that larger genome size improves the replication fidelity. In other case, the sequences cannot copy themselves with sufficiently fidelity, and in the long term they will not survive. Eigen’s paradox is one, among the infinitely many, possible scenarios in the long term. The mathematical formulation of this fact requires solving the Cauchy problem for the p-adic Eigen–Schuster model in a rigorous mathematical way. This requires imposing restrictions on the fitness function and on the mutation measure, among other conditions. The study of the mentioned initial value problem requires techniques of p-adic wavelets and p-adic heat kernels developed in the last 35 years.