Gene-mating dynamic evolution theory: fundamental assumptions, exactly solvable models and analytic solutions

摘要

Fundamental properties of macroscopic gene-mating dynamic evolutionarysystems are investigated. A model is proposed to describe a large class ofsystems within population genetics. We focus on a single locus, arbitrarynumber alleles in a two-gender dioecious population. Our governing equationsare time-dependent continuous differential equations labeled by a set ofgenotype frequencies. The full parameter space consists of all allowed genotypefrequencies. Our equations are uniquely derived from four fundamentalassumptions within any population: (1) a closed system; (2) average-and-randommating process (mean-field behavior); (3) Mendelian inheritance; (4)exponential growth and exponential death. Even though our equations arenon-linear with time evolutionary dynamics, we have an exactly solvable model.Our findings are summarized from phenomenological and mathematical viewpoints.From the phenomenological viewpoint, any initial genotype frequency of a closedsystem will eventually approach a stable fixed point. Under time evolution, weshow (1) the monotonic behavior of genotype frequencies, (2) any genotype orallele that appears in the population will never become extinct, (3) theHardy-Weinberg law, and (4) the global stability without chaos in the parameterspace. To demonstrate the experimental evidence, as an example, we show amapping from the blood type genotype frequencies of world ethnic groups to ourstable fixed-point solutions. From the mathematical viewpoint, the equilibriumsolutions consist of a base manifold as a global stable attractor, attractingany initial point in a Euclidean fiber bundle to the fixed point where thefiber is attached. We can define the genetic distance of two populations astheir geodesic distance on the equilibrium manifold. In addition, themodification of our theory under the process of natural selection and mutationis addressed.