Molecular evolution modeled as a fractal renewal point process in agreement with the dispersion of substitutions in mammalian genes

A fractal renewal point process (FRPP) is used to model molecular evolution in agreement with the relationship between the variance and the mean numbers of nonsynonymous and synonymous substitutions in mammals. Like other episodic models such as the doubly stochastic Poisson process, this model accounts for the large variances observed in amino acid substitution rates, but unlike certain other episodic models, it also accounts for the increase in the index of dispersion with the mean number of substitutions in Ohta's (1995) data. We find that this correlation is significant for nonsynonymous substitutions at the 1% level and for synonymous substitutions at the 10% level, even after removing lineage effects and when using Bulmer's (1989) unbiased estimator of the index of dispersion. This model is simpler than most other overdispersed models of evolution in the sense that it is fully specified by a single interevent probability distribution. Interpretations in terms of chaotic dynamics and in terms of chance and selection are discussed.