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

Abstract

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→ϱ the transition is of first order. The typical rms amplitude of a quasispecies around the master sequence is found to diverge algebraically with exponent Ќ ϭ1 at the transition to the delocalized phase in the limit d→ϱ.

Adaptation in Simple and Complex Fitness Landscapes

“…A variant of Eigen's model was solved exactly for any N in [43]. The model is defined in discrete time but the mutations are restricted to mutants within Hamming distance equal to one, as for the continuous time mutation rates (6).…”

Tempo and mode in quasispecies evolution

“…It was first observed by Leuthäusser [14] that the discrete time dynamics (3) can be interpreted as a transfer matrix of a two-dimensional Ising model, where the genotype sequences become one-dimensional spin configurations that are coupled in the time direction through the mutation matrix (1) [15]. A similar relation can be established between (3) and the transfer matrix of a polymer directed along the time axis [16,17]. In addition, Baake and coworkers have recently exploited the equivalence between quantum spin chains and a class of kinetic evolution equations closely related to the quasispecies model, in which mutation and selection occur in parallel [12,18].…”

“…For the simplest case of a single peak fitness landscape, where the master sequence replicates at rate W 0 and all other sequences replicate at rate W 1 < W 0 , the selective advantage is A = W 0 /W 1 , while for randomly distributed replication rates it is a functional of the rate distribution [19,20]. In terms of the physical analogies described above, the error threshold phenomenon is equivalent to the thermal phase transition in the Ising model [14,15,18,20,21] and to the thermal unbinding of a directed polymer bound to an attractive columnar defect along the time direction [16,17]. Much less appears to be known about the evolutionary dynamics of the model, that is, the approach to the final quasispecies distribution from an initial localized or delocalized state.…”

Selective advantage of topological disorder in biological evolution

“…Therefore, the first thing to try is the sharply-peaked landscape on the hypercube. This model was solved exactly by Galluccio et al [33,34].…”