Implications of fluctuations in substitution rates: Impact on the uncertainty of branch lengths and on relative-rate tests

Many tests of the lineage dependence of substitution rates, computations of the error of evolutionary distances, and simulations of molecular evolution assume that the rate of evolution is constant in time within each lineage descended from a common ancestor. However, estimates of the index of dispersion of numbers of mammalian substitutions suggest that the rate has time-dependent variations consistent with a fractal-Gaussian-rate Poisson process, which assumes common descent without assuming rate constancy. While this model does not affect certain relative-rate tests, it substantially increases the uncertainty of branch lengths. Thus, fluctuations in the rate of substitution cannot be neglected in calculations that rely on evolutionary distances, such as the confidence intervals of divergence times and certain phylogenetic reconstructions. The fractal-Gaussian-rate Poisson process is compared and contrasted with previous models of molecular evolution, including other Poisson processes, the fractal renewal process, a Levy-stable process, a fractional-difference process, and a log-Brownian process. The fractal models are more compatible with mammalian data than the nonfractal models considered, and they may also be better supported by Darwinian theory. Although the fractal-Gaussian-rate Poisson process has not been proven to have better agreement with data or theory than the other fractal models, its Gaussian nature simplifies the exploration of its impact on evolutionary distance errors and relative-rate tests.