Molecular evolution modeled as a fractal statistical process

Modeling the rate of nucleotide substitutions in DNA as a dichotomous stochastic process with an inverse power-law correlation function describes evolution by a fractal stochastic process (FSP). This FSP model agrees with recent findings on the relationship between the variance and mean number of synonymous and nonsynonymous substitutions in 49 different genes in mammals, that being a power-law increase in the ratio of the variance to the mean, the index of dispersion, with the number of substitutions in a protein. The probability of a given number of substitutions occuring in a time t is determined by a fractional diffusion equation whose solution is a truncated Lévy distribution implying that evolution is a Lévy process in time and yields the same functional behavior for the variance in the number of substitutions as does the FSP model. In addition to obtaining these relationships, the FSP model implies lognormal statistics for the index of dispersion as a function of the mean number of substitutions in a protein, which is confirmed in the regression of the FSP model to data. Lognormal statistics suggest that molecular evolution can be viewed as a multiplicative stochastic process, rather than the linear additive processes of Darwinian selection and drift.