Overdispersed molecular evolution in constant environments

According to recent data analysis of DNA sequences, the dispersion index, defined as the variance-to-mean ratio of the number of base substitutions in a lineage, is often much larger than unity, which is in conflict with simple Poisson processes assumed in the molecular clock hypothesis. In this paper, it will be shown that the dispersion index can be much larger than unity in a model in which the fitness of DNA sequences, mutation rate, and population size are all constant with time, and that moderately deleterious sequences are more abundant than the best fit sequences. Since the fixation probability of novel mutations depends on their fitness relative to the current sequence, the neutral mutation rate is enhanced once a deleterious mutation is fixed. In a simple case with two fitness classes, a large dispersion index can be produced by moderately deleterious mutations (3 < 4Ns < 6), but neither by nearly neutral (4Ns < 2) nor by strongly deleterious (4Ns > 7) mutations. Analysis of the case with 100 fitness classes shows that the dispersion index is insensitive to the population size, but greatly changes with the fitness distribution of DNA sequences. The model can explain why non-synonymous substitutions often have a larger dispersion index than synonymous substitutions.