MULTIPLICATIVE AND FRACTAL PROCESSES IN DNA EVOLUTION

Darwin's theory of evolution by natural selection revolutionized scien ce in the nineteenth century. Not only did it provide a new paradigm f or 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 m icroscopic processes, the Darwinian interpretation was adopted to patt erns observed in molecular evolution by assuming that natural selectio n operates fundamentally at the level of DNA. Thus, patterns of molecu lar evolution have important implications in many fields of science. A lthough the evolution rate of a given gene seems to be of approximatel y the same order of magnitude in all species, genes appear to differ i n 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 e volution rates of mammalian DNA, based on recent estimates of numbers of nonsynonymous substitutions in 49 genes of humans, rodents, and art iodactyls, are studied. We find that the rate variations are better de scribed by lognormal statistics, as would be the case for a multiplica tive process, than by Gaussian statistics, which would correspond to a linear, additive process. Thus, we introduce a multiplicative evoluti on 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 r ate results from the interaction of a number of interdependent conting ency processes. Lognormal statistics lend support to fractal process m odels of DNA substitutions, including anomalous diffusion processes an d fractal stochastic point processes, such as the fractal renewal proc ess and the fractal doubly-stochastic Poisson process. The realization of a fractal process is a random self- similar time series with a pow er-law autocorrelation function, spectral density, and Fano factor ove r many time scales.