# General Models of Evolution

## Mathematical models of population genetics.

The mathematical theory of population genetics was grounded by R.A.Fisher, J.B.S. Haldane, and S.Wright in the 1910-1930s. Population genetics or the synthetic theory of evolution is based on the Darwinian concept of natural selection and Mendelian genetics. Numerous experiments on a small fruit fly, Drosophila, played an important role in finding an agreement between the Darwinian assumption of gradual, continuous evolutionary improvements and the discrete character of evolution in Mendelian genetics. According to population genetics, the main mechanism of progressive evolution is the selection of organisms with advantageous mutations.

The mathematical theory of population genetics describes quantitatively the gene distribution dynamics in evolving populations. The theory includes two main types of models: deterministic models (implying an infinitely large population size) and stochastic ones (finite population size). See Mathematical methods of population genetics for details.

Population genetics was intensively developed until the 1960s, when the difficulties of genetics became clear from molecular investigations.

## The neutral theory of molecular evolution.

The revolution in molecular genetics occurred in the 1950-1960s. The structure of DNA was established (F.H.C.Crick, J.D.Watson, 1953), the scheme of protein synthesis (according to information coded in DNA) became known, and the genetic code was deciphered.

As to the evolution aspects, the evolutionary rate of amino-acids substitutions as well as the protein polymorphism were estimated. In order to explain these experimental results, Motoo Kimura proposed the neutral theory of molecular evolution [1,2]. The main assumption of Kimura's theory is: the mutations at the molecular level (amino- and nuclear-acid substitutions) are mostly neutral or slightly disadvantageous (essentially disadvantageous mutations are also possible, but they are eliminated effectively from populations by selection). This assumption agrees with the mutational molecular substitution rate observed experimentally and with the fact that the rate of the substitutions for the less biologically important part of macromolecules is greater than for the active macromolecule centers.

Using mathematical methods of population genetics, M. Kimura deduced a lot of the neutral theory consequences, which are in rather good agreement with molecular genetics data [2].

The mathematical models of the neutral theory are essentially stochastic, that is, a relatively small population size plays an important role in the fixation of the neutral mutations.

If molecular substitutions are neutral, then why is progressive evolution possible? To answer this question, M.Kimura uses the concept of gene duplication developed by S.Ohno [3]. According to M.Kimura, gene duplications create unnecessary, surplus DNA sequences, which in turn drift further because of random mutations, providing the raw material for a creation of new, biologically significant genes.

The evolutionary concepts of the neutral theory came from interpretations of biological experiments; this theory was strongly empirically inspired. The other type of theory, a more abstract one, was proposed by Stuart A. Kauffman: NK automata or Boolean networks.

Theoretical population genetics and the neutral theory of molecular evolution describe the general features of genetic evolution. Nevertheless, these theories don’t consider the cybernetic properties of biological organisms. The theory of NK automata by S.A.Kauffman is a very interesting step towards understanding the evolution of the "program-like, computational" abilities of biological systems. This theory is mainly illustrative, however, it provides "a challenging scenario" (well developed mathematically) of the cybernetic evolution of the living cells.

References:

1. M. Kimura. Nature. London, 1968.V.217. PP.624.

2. M. Kimura. "The neutral theory of molecular evolution". Cambridge Un-ty Press. 1983.

3. S. Ohno. "Evolution by gene duplication". Berlin, Springer-Verlag, 1970.