A Mathematical Formulation of Evolution and Innovation I. Unicellular Organisms

Most types of evolution indicated on unicellular organisms are shown to be derived mathematically from the time change equation of organisms that self-reproduce, occasionally mutate, and die. Darwinian evolution corresponds to the approximate solution with respect to the first order of mutation term, showing that the mutant with the higher selfreproducing rate and/or slower death rate increases its number. Such a mutant is mainly due to the nucleotide base substitutions. However, the scaffold, under which Darwinian evolution takes place, has been sometimes changed during the longer time. The well-known examples are the gene duplication to generate new genes and the gene transfer from the endosymbiont to the host genome of eukaryotes after the acceptance of endosymbionts. The mutants having experienced such drastic change in genome first decline to the minor members in the population but some of them recover the increase rate as a new style of organisms by Darwinian evolution under the reconstructed genome. The evolutionary process containing such innovation is formulated by successively solving the time change equations of satellite variants up to the higher order of mutation terms until the appearance of new style organisms. This mathematical formulation applied to the self-reproducing proto-cells in the RNA-protein world also provides a possibility to explain the origin of protein genes as well as of translation apparatus and the transition of the proto-cells to the DNA-RNA-protein world.