Error-threshold exists in fitness landscapes with lethal mutants

Abstract

Background One of the important insights of quasi-species theory is an error-threshold. The error-threshold is the error rate of replication above which the sudden onset of the population delocalization from the fittest genotype occurs despite Darwinian selection; i.e., the break down of evolutionary optimization. However, a recent article by Wilke in this journal, after reviewing the previous studies on the error-threshold, concluded that the error-threshold does not exist if lethal mutants are taken into account in a fitness landscape. Since lethal mutants obviously exist in reality, this has a significant implication about biological evolution. However, the study of Wagner and Krall on which Wilke's conclusion was based considered mutation-selection dynamics in one-dimensional genotype space with the assumption that a genotype can mutate only to an adjoining genotype in the genotype space. In this article, we study whether the above conclusion holds in high-dimensional genotype space without the assumption of the adjacency of mutations, where the consequences of mutation-selection dynamics can be qualitatively different. Results To examine the effect of mutant lethality on the existence of the error-threshold, we extend the quasi-species equation by taking the lethality of mutants into account, assuming that lethal genotypes are uniformly distributed in the genotype space. First, with the simplification of neglecting back mutations, we calculate the error-threshold as the maximum allowable mutation rate for which the fittest genotype can survive. Second, with the full consideration of back mutations, we study the equilibrium population distribution and the ancestor distribution in the genotype space as a function of error rate with and without lethality in a multiplicative fitness landscape. The results show that a high lethality of mutants actually introduces an error-threshold in a multiplicative fitness landscape in sharp contrast to the conclusion of Wilke. Furthermore, irrespective of the lethality of mutants, the delocalization of the population from the fittest genotype occurs for an error rate much smaller than random replication. Finally, the results are shown to extend to a system of finite populations. Conclusion High lethality of mutants introduces an error-threshold in a multiplicative fitness landscape. Furthermore, irrespective of the lethality of mutants, the break down of evolutionary optimization happens for an error rate much smaller than random replication.