Mathematical models of population genetics.
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
The neutral theory of molecular evolution.
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
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
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 .
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 .
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
dont 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.
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.