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 2ln[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 lnk for conservative replication. In particular, as k→∞, the error catastrophe is never reached for conservative replication, while for semiconservative replication the critical μ approaches 2ln2. 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(2e−μ∕2−1) below the error catastrophe, in contrast to the standard result of ke−μ for conservative replication (derived by Kimura and Maruyama in 1966). From this result it is readily shown that semiconservative replication is necessary to account for the observation that, at sufficiently high mutagen concentrations, faster replicating cells will die more quickly than more slowly replicating cells. Thus, in contrast to Eigen’s original model, the semiconservative quasispecies equations are able to provide a mathematical basis for explaining the efficacy of mutagens as chemotherapeutic agents.