Semiconservative Replication in the Quasispecies Model

Abstract

This paper extends Eigen's quasispecies equations to account for the semiconservative nature of DNA replication. We solve the equations in the limit of infinite sequence length for the simplest case of a static, sharply peaked fitness landscape. We show that the error catastrophe occurs when $ μ$, the product of sequence length and per base pair mismatch probability, exceeds $ 2 \ln \frac{2}{1 + 1/k} $, where $ k > 1 $ is the first order growth rate constant of the viable ``master'' sequence (with all other sequences having a first-order growth rate constant of $ 1 $). This is in contrast to the result of $ \ln k $ for conservative replication. In particular, as $ k \to \infty $, the error catastrophe is never reached for conservative replication, while for semiconservative replication the critical $ μ$ approaches $ 2 \ln 2 $. Semiconservative replication is therefore considerably less robust than conservative replication to the effect of replication errors. We also show that the mean equilibrium fitness of a semiconservatively replicating system is given by $ k (2 e^{-μ/2} - 1) $ below the error catastrophe, in contrast to the standard result of $ k e^{-μ} $ for conservative replication (derived by Kimura and Maruyama in 1966).