  A quasispecies is a set of interrelated genotypes that have reached a
situation of equilibrium while evolving according to the usual Darwinian
principles of selection and mutation. Quasispecies studies invariably assume
that it is possible for any genotype to mutate into any other, but recent finds
indicate that this assumption is not necessarily true. Here we revisit the
traditional quasispecies theory by adopting a network structure to constrain
the occurrence of mutations. Such structure is governed by a random-graph
model, whose single parameter (a probability p) controls both the graph's
density and the dynamics of mutation. We contribute two further modifications
to the theory, one to account for the fact that different loci in a genotype
may be differently susceptible to the occurrence of mutations, the other to
allow for a more plausible description of the transition from adaptation to
degeneracy of the quasispecies as p is increased. We give analytical and
simulation results for the usual case of binary genotypes, assuming the fitness
landscape in which a genotype's fitness decays exponentially with its Hamming
distance to the wild type. These results support the theory's assertions
regarding the adaptation of the quasispecies to the fitness landscape and also
its possible demise as a function of p.
