Solution of the Quasispecies Model for an Arbitrary Gene Network

Abstract

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 $\sigma$ may be written as $\sigma ={\sigma }_1{\sigma }_2...{\sigma }_N$. We assume a single fitness peak (SFP) model for each gene, so that gene $i$ has some ``master'' sequence ${\sigma }_{i,0}$ 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.''

29 pages, 5 figures, to be submitted to Physical Review E

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