ISSN 1022-7954, Russian Journal of Genetics, 2016, Vol. 52, No. 9, pp. 985–992. © Pleiades Publishing, Inc., 2016.
Mathematical Models in Genetics
M. Traykov and Iv. Trenchev
Cеnter for Advanced Bioinformatics Research, South-West University “Neofit Rilski”, Blagoevgrad, 2700 Bulgaria
Received October 1, 2015
Abstract—In this study, we present some of the basic ideas of population genetics. The founders of population
genetics are R.A. Fisher, S. Wright, and J. B.S. Haldane. They, not only developed almost all the basic theory
associated with genetics, but they also initiated multiple experiments in support of their theories. One of the
first significant insights, which are a result of the Hardy–Weinberg law, is Mendelian inheritance preserves
genetic variation on which the natural selection acts. We will limit to simple models formulated in terms of
differential equations. Some of those differential equations are nonlinear and thus emphasize issues such as
the stability of the fixed points and time scales on which those equations operate. First, we consider the classic
case when selection acts on diploid locus at which wу can get arbitrary number of alleles. Then, we consider
summaries that include recombination and selection at multiple loci. Also, we discuss the evolution of quan-
titative traits. In this case, the theory is formulated in respect of directly measurable quantities. Special cases
of this theory have been successfully used for many decades in plants and animals breeding.
Keywords: genetics, mathematical models, mathematical genetics, bioinformatics
The theoretical population genetics and mathe-
matical genetics represent the study of the temporal
and spatial variations of species frequencies (e.g.,
genes, genotypes, gametes, etc.) in the populations,
formed by various ecological and genetic influences.
In natural populations operate two general opposite
trends: 1—a tendency of adaptability and resistance to
specific events, favorable to environment and 2—the
necessity on the population to maintain the potential
for change to deal with the changing environment. The
use of mathematics in study of genetic systems is as old
as genetics itself. From the rediscovery of Mendel’s
work did not take long to put the Hardy–Weinberg low
(1908) for constancy of gene frequency over a certain
time interval. Between 1915 and 1950 mathematical
genetics has pioneered and dominated by the names of
R.A. Fisher, S. Wright, and J.B.S. Haldane.
In series of articles in the 20th years of the 19th cen-
tury, Haldane shows variety of simple mathematical
analyzes related with the way of natural selection. In
particular, he points out how evolutionary forces such
as a viable selection, mutation and migration can be
quantified and shown by mathematical models [1–3].
Fisher and Wright also participated in the drafting
of these theories. In small populations, Wright further
found that the evolution theory should take into
account the effect of samples, involved in producing of
one generation from previous. He called this effect
“random drift”. This aspect of population genetics has
had significant consequences for Feller’s mathemati-
cal studies within the theory of diffusion processes.
Also, Wright and Fisher were pioneers in the theory
of systems for mating between relatives, used by plant
and animal breeders. The result was the theory of del-
eterious inbreeding which leads to intriguing algebraic
and analytical structures. Statistical theory probably
originated from the R. A. Fisher’s attempts to design
and analysis of experiments, whose targets were
mostly to solve problems in genetics [1–3].
The aim of this article is to acquaint the students in
mathematics with several classic mathematical genetic
models. Attention is paid main on the formulation of
models accompanied by a brief analysis and appropri-
ate references. We apply some interpretations and
conclusions of the results described in the literature.
We will try to highlight a few important genetic factors
and concepts by presenting models involving different
models for mating, selective forces, migrations and
mutations, and recombination mechanism etc.
Classical genetic definition of our interests pro-
ceeds the modern molecular era. First, the genes occur
under certain sets or loci along a chromosome. Every
locus can be occupied by one of several different genes,
called alleles. Most human cells contain 46 chromo-
somes. Two of them are sex chromosomes – two
paired, X’s for a female and X and Y for a male. The
other 22 homologous pairs of chromosomes are called
The article is published in the original.