Nonlinear Genetics Inbreeding and Genetic LoadVolobuev Andrey Nikolaevich Romanc

Abstract

Comparison of concepts of occurrence and accumulation of the genetic load in result of inbreeding for a family tree and a population is given. It is shown that for the population the analysis of the genetic load concerns to area of nonlinear genetics. The nonlinear differential equation of the second order being the special case of the Hardy-Weinberg law at presence inbreeding for the population is found. The numerical decision of this equation allowed to find the law of increase of recessive allele frequency in the population is given. Some aspects of the traditional theory of the genetic load are considered.

Key words： Genetic load； Inbreeding； Hardy-weinberg law； Nonlinear genetics

Nikolaevich， V. A.， Ivanovich， R. P.， & Kirikovich， M. V. （2014）. Nonlinear Genetics Inbreeding and Genetic Load. Advances in Natural Science， 7（1）， -0. Available from： http：///index.php/ans/article/view/4164

DOI： http：///10.3968/4164

INTRODUCTION

In （Volobuev， Romanchuk， & Malishev， 2013） it is shown that variation in during time of a population life of the recessive gene frequency linked to the Х-chromosome in condition of the population balance is described by the linear differential equation of the second order. This equation is Hardy-Weinberg law for the panmixed population. Also the linear differential equations of the second order describe， for example， the mutational process resulting in hemophilia（Volobuev， Romanchuk & Malishev， 2013） or mutational process under action of radiation （Volobuev & Petrov， 2011）. Similar processes can be attributed to processes of linear genetics.

However in genetics there are the processes resulting in the nonlinear differential equations. These processes concern to nonlinear genetics. As an example of similar process we shall consider development of the population at presence of the inbreeding.

1. THE GENETIC LOAD IN RESULT OF INBREEDING IN A FAMILY TREE

More detailed data are available on the population of Japan. In different prefectures the ratio 15.2（5）. Negative value means that children’s death rate in consanguineous marriages was lower than in non consanguineous marriages （Vogel & Motulsky， 1990）. Average sizes on the country A=0.1036， B=0.67 and =6.7.

It is possible to assume （Vogel & Motulsky， 1990） that there is some threshold of recessive homozygous genes allowable frequency compatible with a survival of individual.

Taking into account values A and B， and also inbreeding factor for Japan F=0.004 （Vogel & Motulsky， 1990） under the Equation （22） we shall find the fraction of the survived zygotes S=0.9. Fraction of the perish zygotes there is 1S=0.1. The elementary estimation under the formula 1S=q2f =0.1 gives the fraction of lethal （parameter of selection is equal 1） at association in homozygous recessive alleles qf ≈0.316 （Ayala & Kiger， 1984）. Hence the fraction of not mutant genes equal to threshold frequency of recessive alleles is qf max=1qf=0.684.

On Figure 1 dotted line shows the threshold frequency. For Japan this frequency is attained approximately in 167 generation.

The received result shows that accumulation of the genetic load it is process very slow. Achievement of threshold frequency for Japan to be carried out only in 167 generations or 4175 years at time of one generation change T=25 years.

In （Vogel & Motulsky， 1990） it is noticed that from the beginning of the population development with factor inbreeding F=0.0030.005 up to achievement of equilibrium frequency which is established at action of selection should pass about 4500 years. In calculation the formula of linear increase of recessive alleles frequency was used at mutation speed μ=105. Increase of the mutant alleles number was compensated by selection against recessive homozygous with parameter of selection 0.5.

Therefore most likely that limiting values of the genetic load in a human population of many countries especially taking into account small sizes of inbreeding factor （Canada F=45・105， France F=23・105， Germany F=19・105， Italy F=16・105 （Vogel & Motulsky， 1990）， etc.） even at presence of selection till now are not achieved.

In our opinion with help of the offered method it is possible to examine not only harmful for the population genetic load related with inbreeding but also useful and indifferent genetic load （or attributes）.

At early stages of mankind development the role of inbreeding was more in connection with smaller number of people and isolation of separate groups. Therefore accumulation of the genetic load was more intensively （the inbreeding factor was much more）. Primeval people after the going out from Africa and moving in northern Europe began to live in conditions of smaller light exposure including in the ultra-violet part of spectrum. In result the skin with originally dark pigmentation started to have white color. It promoted occurrence of vitamin D which deficiency results in rickets （Vogel & Motulsky， 1990）. Process of dark pigmentation reduction of skin can be examined as accumulation of the recessive useful genetic load. Natural selection fixed given phenotypic attribute at people in region.

To the indifferent genetic load in population it is possible to ascribe occurrence such recessive attribute as blond hair， for example， in northern part of Europe where apparently at early stages the inbreeding has been especially widespread. Color of hair does not play a role in survival of population. Fixed of this attribute was carried out by the sexual selection i.e. marriage preferences of people in region.

However it is traditionally accepted to examine accumulation of the genetic load in the population are determined by harmful mutations. Some fears of H. J. Muller are connected to this process about danger of the future biological degeneration of mankind （Muller， 1950）.

CONCLUSION

Among processes of population genetics it is possible to allocate linear processes， for example， development of the panmixed populations， the mutational process resulting in hemophilia， the mutational process under action of radiation which described by the linear differential equations of the second order. For nonlinear genetics processes which described by the nonlinear differential equations of the second order are typical， for example， development of a population at presence of inbreeding.

The concept of genetic load allows lead the analysis of the population with inbreeding， in particular to find the law of the recessive allele frequency increase in the population.

The existing theory of genetic load occurrence and accumulation is actually developed only for a family tree as small component of population.

Using the technique of transition offered by the authors from the analysis of a family tree to the analysis of a population the theory of the genetic load arising for the account inbreeding in a population is developed.

REFERENCES

Ayala， F.， & Kiger， J. J. （1984）. Modern genetics （Vol.3， pp.153， 171）. California： The Benjamin/Cummings Publishing Company， Inc.

Morton， N. E.， Crow， J. F.， & Muller， H. J. （1956）. An estimate of the mutational damage in man from data on consanguineous marriages. Proc. Natl. Acad. Sci. USA， 42， 855-863.

Muller， H. J. （1950）. Our load of mutation. Am. J. Hum. Genet.， 2， 111-176.

Vogel， F.， & Motulsky， A. （1990）. Human genetics （Vol.2， p.346， 349， 351， 352， 353， 354， 358； Vol.3， p.37）. Berlin： Springer-Verlag.

Volobuev， A. N.， & Petrov， E. S. （2011）. Modelling of the populating development of the genome in the radiation of the environment. Natural Science， 3（12）， 1029-1033.

Volobuev， A. N.， Romanchuk， P. I.， & Malishev， V. K. （2013）. Population and mutagenesis or about hardy and weinberg one methodical mistake. Advances in Natural Science， 6（4）， 55-63.