An exactly solved model of biological evolution

Abstract

We reconsider the Eigen's quasi-species model for competing self-reproductive macromolecules in populations characterized by a single-peaked fitness landscape. The use of ideas and tools borrowed from polymers theory and statistical mechanics, allows us to exactly solve the model for generic DNA lengths d. The mathematical shape of the quasi-species confined around the master sequence is perturbatively found in powers of 1/d at large d. We rigorously prove the existence of the error-threshold phenomenon and study the quasi-species formation in the general context of critical phase transitions in physics. No sharp transitions exist at any finite d, and at $d\to \infty$ the transition is of first order. The typical r.m.s. amplitude of a quasi-species around the master sequence is found to diverge algebraically with exponent $ν_{\perp}=1$ at the transition to the delocalized phase in the limit $d\to \infty$.