Exact solution of the quasispecies model in a sharply peaked fitness landscape

We reconsider Eigen’s quasispecies model for competing self-reproductive macromolecules in populations characterized by a single-peaked fitness landscape. The use of ideas and tools borrowed from polymer theory and statistical mechanics allows us to exactly solve the model for generic DNA lengths $d$. The mathematical shape of the quasispecies 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 phenomena and study the quasispecies 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 rms amplitude of a quasispecies around the master sequence is found to diverge algebraically with exponent ${\nu }_{\perp }=1\mathrm{}$ at the transition to the delocalized phase in the limit $d\to \infty$.