Phenotypic error threshold; additivity and epistasis in RNA evolution

Background: The error threshold puts a limit on the amount of information maintainable in Darwinian evolution. The error threshold was first formulated in terms of genotypes. However, if a genotype-phenotype map involves redundancy (mutational neutrality), the error threshold should be formulated in terms of phenotypes since there is no unique fittest genotype. A previous study formulated the error threshold in terms of phenotypes, and their results showed that a rather low degree of mutational neutrality can increase the error threshold unlimitedly. Results: We obtain an analytical formulation of the phenotypic error threshold by considering the additive assumption, in which base substitutions do not influence each other ( no epistasis). Our formulation shows that an increase of the error threshold due to mutational neutrality is limited. Computer simulations of RNA evolution are conducted to verify our formulation, and the results show a good agreement between the analytical prediction and the simulations. The comparison with the previous formulation illustrates that it is important for the prediction of the error threshold to consider that the number of base substitutions per replication is rather large near the error threshold. To examine the additive assumption, a detailed analysis of additivity and epistasis in RNA folding of a particular sequence is performed. The results show a high degree of epistasis in RNA folding; furthermore, the analysis also elucidates the reason of the success of the additive assumption. Conclusions: We conclude that an increase of the error threshold by mutational neutrality is limited, and that the additive assumption achieves a good prediction of the error threshold in spite of a high degree of epistasis in RNA folding because the average number of base substitutions of sequences retaining the phenotype per replication is sufficiently small to avoid of the effect of epistasis.