In this paper, we study the equilibrium behavior of Eigen's quasispecies
equations for an arbitrary gene network. We consider a genome consisting of N genes, so that each gene sequence σ may be written as σ=σ1​σ2​...σN​. We assume a single fitness peak (SFP) model
for each gene, so that gene i has some ``master'' sequence σi,0​ for which it is functioning. The fitness landscape is then determined by
which genes in the genome are functioning, and which are not. The equilibrium
behavior of this model may be solved in the limit of infinite sequence length.
The central result is that, instead of a single error catastrophe, the model
exhibits a series of localization to delocalization transitions, which we term
an ``error cascade.'' As the mutation rate is increased, the selective
advantage for maintaining functional copies of certain genes in the network
disappears, and the population distribution delocalizes over the corresponding
sequence spaces. The network goes through a series of such transitions, as more
and more genes become inactivated, until eventually delocalization occurs over
the entire genome space, resulting in a final error catastrophe. This model
provides a criterion for determining the conditions under which certain genes
in a genome will lose functionality due to genetic drift. It also provides
insight into the response of gene networks to mutagens. In particular, it
suggests an approach for determining the relative importance of various genes
to the fitness of an organism, in a more accurate manner than the standard
``deletion set'' method. The results in this paper also have implications for
mutational robustness and what C.O. Wilke termed ``survival of the flattest.''Comment: 29 pages, 5 figures, to be submitted to Physical Review