The Error and Repair Catastrophes: A Two-Dimensional Phase Diagram in the Quasispecies Model

This paper develops a two gene, single fitness peak model for determining the equilibrium distribution of genotypes in a unicellular population which is capable of genetic damage repair. The first gene, denoted by $ \sigma_{via} $, yields a viable organism with first order growth rate constant $ k > 1 $ if it is equal to some target ``master'' sequence $ \sigma_{via, 0} $. The second gene, denoted by $ \sigma_{rep} $, yields an organism capable of genetic repair if it is equal to some target ``master'' sequence $ \sigma_{rep, 0} $. This model is analytically solvable in the limit of infinite sequence length, and gives an equilibrium distribution which depends on $ \mu \equiv L\eps $, the product of sequence length and per base pair replication error probability, and $ \eps_r $, the probability of repair failure per base pair. The equilibrium distribution is shown to exist in one of three possible ``phases.'' In the first phase, the population is localized about the viability and repairing master sequences. As $ \eps_r $ exceeds the fraction of deleterious mutations, the population undergoes a ``repair'' catastrophe, in which the equilibrium distribution is still localized about the viability master sequence, but is spread ergodically over the sequence subspace defined by the repair gene. Below the repair catastrophe, the distribution undergoes the error catastrophe when $ \mu $ exceeds $ \ln k/\eps_r $, while above the repair catastrophe, the distribution undergoes the error catastrophe when $ \mu $ exceeds $ \ln k/f_{del} $, where $ f_{del} $ denotes the fraction of deleterious mutations.