Multiplicative and fractal processes in DNA evolution

Darwin's theory of evolution by natural selection revolutionized science in the nineteenth century. Not only did it provide a new paradigm for biology, the theory formed the basis for analogous interpretations of complex systems studied by other disciplines, such as sociology and psychology. With the subsequent linking of macroscopic phenomena to microscopic processes, the Darwinian interpretation was adopted to patterns observed in molecular evolution by assuming that natural selection operates fundamentally at the level of DNA. Thus, patterns of molecular evolution have important implications in many fields of science. Although the evolution rate of a given gene seems to be of approximately the same order of magnitude in all species, genes appear to differ in rate from one another by orders of magnitude, a fact which standard theory does not adequately explain. An understanding of the statistics of rates across different genes may shed light on this problem. The evolution rates of mammalian DNA, based on recent estimates of numbers of nonsynonymous substitutions in 49 genes of humans, rodents, and artiodactyls, are studied. We find that the rate variations are better described by lognormal statistics, as would be the case for a multiplicative process, than by Gaussian statistics, which would correspond to a linear, additive process. Thus, we introduce a multiplicative evolution statistical hypothesis (MESH), in which the theoretical explanation of these statistics requires the evolution of different substitution rates in different genes to be a multiplicative process in that each rate results from the interaction of a number of interdependent contingency processes. Lognormal statistics lend support to fractal process models of DNA substitutions, including anomalous diffusion processes and fractal stochastic point processes, such as the fractal renewal process and the fractal doubly-stochastic Poisson process. The realization of a fractal process is a random self- similar time series with a power-law autocorrelation function, spectral density, and Fano factor over many time scales.