Eigen's Quasispecies Research Articles

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 $d\ensuremath{\rightarrow}\ensuremath{\infty}$ the transition is of first order. The typical rms amplitude of a quasispecies around the master sequence is found to diverge algebraically with exponent ${\ensuremath{\nu}}_{\ensuremath{\perp}}=1$ at the transition to the delocalized phase in the limit $d\ensuremath{\rightarrow}\ensuremath{\infty}$.

An evolutionary version of the random energy model

We consider Eigen's quasispecies model (1971) of molecular evolution in the case in which the reproduction rates of different molecular species are quenched independent random variables. We show, by analogy with the random energy model, the existence of two phases: a 'neutral' phase, where no adaptation effects are exhibited, and a (pathological) 'adapted' phase, where the population is made of identical, optimally adapted molecules, and does not evolve.

Spatial stability analysis of Eigen's quasispecies model and the less than five membered hypercycle under global population regulation

Spatial stability analysis of Eigen's quasispecies model and the less than five membered hypercycle under global population regulation

Spatial stability analysis of Eigen's quasispecies model and the less than five membered hypercycle under global population regulation

A generalization of the “constant overall organization” constraint of Eigen's quasispecies and hypercycle models, called herein “global population regulation”, is shown to lead to mathematically tractable spatial generalizations of these two models. The spatially uniform steady state of Eigen's quasispecies model is shown to be stable and globally attracting for all possible values of the mutation and replication rates. In contrast, the spatially and temporally uniform solutions to the hypercycle with fewer than five members, the only ones insensitive to stochastic perturbations, are shown to be unstable, and a lower bound to the spatial inhomogeneities is obtained. The prospect that the spatially localized hypercycle might be immune to various instabilities cited in the literature is then briefly considered. Although spatial localization makes possible a much richer dynamical repertoire than previously considered, it is also more difficult to understand how Darwinian selection of hypercycles could result in a unique genetic code.