Andrey N. Volobuev; Peter I. Romanchuk; Vladimir K. Malishev

Abstract

The existing discrete form of the Hardy-Weinberg genetic law is applicable for a family tree. For population it is necessary to use a continuous time scale. The differential form of the Hardy-Weinberg law is offered. On the basis of this form of the law the demographic problem is considered where oncological diseases connected with the action of the stochastic mutagen factor. Geneticmathematical aspects of hemophilia are considered in the assumption of the equivalent constant mutagen factor action.

Key words： Hardy-Weinberg law； Family tree； Population； Mutations； Selection

INTRODUCTION

The Hardy-Weinberg law which was found by English mathematician Hardy and German doctor Weinberg in 1908 plays key role in the mathematical analysis of genetic processes. In the elementary kind the essence of this law will consist in the following.

In the elementary kind of two alleles of autosomal genes relative frequencies of genotypes in generations correspond to terms of binomial expansion （p+q）2so p+q=1 where p and q is alleles frequencies. Relative frequencies of genotypes remain constant from generation to generation in case of the ideal population （number of species is very great， exist panmixia， there is no selection， mutations， migrations of species， etc.）. Since founders of the law Hardy and Weinberg it is supposed that in such kind the law describes the processes in population （Vogel& Motulsky， 1990； Ayala & Kiger， 1984； Li， 1976； Weir， 1990； Volobuev， 2005； Brown & Rothery， 1994）.

However founders of the law and the subsequent authors at use of the law make essential methodical mistake. The matter is that the population will consist of set of family trees which periodically contact among themselves. On Figure 1 the principle of population formation from separate family trees is shown. There the square means a male individual， a circle - female.

For example， three family trees are shown. For the family tree it is possible to consider the time of one generation change approximately T≈25-30 years. For the population this time can be any t1，t2， etc. For the population continuous alternation of generations is characteristic and hence the form of the Hardy-Weinberg law should not have discrete character. In it there is essence of the basic methodical mistake of application of the Hardy-Weinberg law for the population. In used form the Hardy-Weinberg law is written down for the family tree.

As it was already specified the Hardy-Weinberg law in the kind considered above there is concerns to separate family tree. Implicitly this law includes time since alternation of generations occurs through certain time T. Thus， HardyWeinberg law in form （1） has the expressed discrete character on time. The population will consist of family trees crossed among themselves and lives in continuous time. Alternation of generations of set of family trees results to the generations vary actually according to continuous time scale.

Extremely important topic of mathematical genetics there are mutations. The mutations explain genetically dependent hereditary diseases. Mutations allow explain process of the evolution of organisms. Mutations underlie of animals and plants breeding.

Mutations are spontaneous and induced （Vogel & Motulsky， 1990）.

During induced mutations there is interaction of the mutagen factor and individual exposed mutation.

Process of the mutation has stochastic character. The individuals under action of the mutagen factor with some probability can be subjected to mutations， and can not be subjected.

During mutation the mutagen factor has the important role. It is possible classification mutagen factors into two groups： determined and stochastic.

The determined mutagen factors can be constant or functionally time-dependent.

For example， process of mutagenesis under action of the determined mutagen factor in due course reducing the intensity according to exponential law of disintegration of radioactive elements in the environment is considered（Volobuev， 2005）.

3.1 Action of the Stochastic Mutagen Factor on the Population

Lets analyze influence of the stochastic mutagen factor by the example of the occurrence malignant newgrowths.

Among other kinds of diseases the occurrence of malignant newgrowths has some features. First of all， it is the big variability of the newgrowths site. It can practically arise in any place of the organism. Besides for oncological diseases there are typically a variety of the mutagen （cancerigenic） factors： poor-quality food， polluted environment， mode of life and professional work， smoking and many other factors.

All these cancerogenic factors finally affect on the mitogenetic function of a cell causing its malignant transformation.

Is generalized we shall consider that set of the reasons resulting to occurrence of the malignant newgrowths it is the influence on the organism of some stochastic mutagen factor.

Despite of the stochastic character of influence it is difficult to assume the situation at which the given stochastic mutagen factor completely would be absent. It concerns even completely isolated primitive societies. Especially such factor in any kind always is present at a modern civilized society.

As investigated model we shall consider homogeneous and stable in the demographic attitude a human society of very much advanced country with a high level of development of the medicine accessible to all population. In such countries death rate of the population basically should be caused by oncological diseases which start to play the role of the natural factor of inevitable alternation of generations. We shall name such countries demographic stationary.

It is possible to assume also that in similar societies the genetic-mathematical laws determining death rate of the population from oncological diseases should operate， i.e. as result of action of the stochastic mutagen factor.

To understand how it is possible to take into account action of the stochastic mutagen factor we will address to other well investigated physical phenomenon—to the Brownian motion （Matveev， 1981）. Brownian motion of a particle in a liquid at first sight should not exist. Really， on Brownian particle， for example， flower pollen impacts the molecules of the liquid which operation are counterbalanced from different directions. Therefore， the most probable condition of the particle is motionless. The particle should shiver only but should not have some constant displacement from a point of supervision. Einstein and Smoluchowski have shown that physically the Brownian motion is consequence of statistical properties of the second law of thermodynamics. If the researcher has relative small number of the molecules the essential deviation from the most probable state of system should be observed in this case the motionless state of the Brownian particles.

Lets note the main similarity of two phenomena： the Brownian motion and existence of the population in conditions of the stochastic mutagen factor action.

At the Brownian motion on the determined system- particle in the liquid – stochastic force acts from the molecules of a liquid.

In the researched case on the determined system -reproductive genome─some stochastic mutagen factor acts.

At the Brownian motion the equation of movement of the particle looks like：

where m there is mass of particle， S - displacement of the particle from initial position， r - factor of medium resistance to movement of the particle， t - time， F -stochastic force acting on the particle from the molecules of liquid. We shall note absence in the equation （28） the elastic forces which is determined returned the particle in initial position causing its oscillation around of the balance point.

The equation （27） is similar to the equation （28） for F=0.

If there is some stochastic mutagen factor D（n） randomly time-dependent lives of the population the equation （27） by analogy with （28） it is necessary to copy as：

p . Alleles O has in this case frequency at men mq at women fq . The ratio （1） for blood system АВO is not frequency distribution of blood genotypes but the genotype frequency aa （or a genotype OO）， and also phenotype frequency corresponding to a blood group I it the ratio reflects truly.

The basic demonstration of existence Х-linked recessive inheritance for the blood system АВO consists that the destruction at disease of blood， for example， hemophilia are men and daughters phenotypic are healthy（Vogel & Motulsky， 1990）.

For the first time the mathematical genetics laws has applied Haldane to a problem of hemophilia on basis of Danforth idea about the equilibration of frequency of mutations and selection. Occurrence of hemophilia there is usually concern to spontaneous mutations. However formally meaning balance of mutations and selection， and also constancy of the population mutation occurrence（otherwise illness quickly would disappear） it is possible to calculate the problem of hemophilia assuming action on the blood system of some equivalent constant mutagen factor. Action of selection will be appreciated further.

The analysis we shall make on the basis of HardyWeinberg law written down as：

where C4there is constant of integration.

In connection with that the basic results for genetic research of hemophilia have been earlier received at use of discrete alternation generations principle at the given of analysis stage， for use of the previous researchers results， it is convenient to return to the discrete scale of generations.

Change of alleles a frequency for one generation is equal：

For the analysis of the selection action on the population， with the purpose of the previous researches use， we shall return to the separate family tree. In the family tree where the hemophilia is observed the selection resulting in decrease of genic frequencies in particular of allele O blood operates.

Action of selection is intensive enough. For example， the life duration of the men who were ill by hemophilia makes 1/3 from the life duration of healthy people （Vogel& Motulsky， 1990）， male fertility i.e. chances to have posterity in comparison with healthy men is reduced. Therefore not all men of the given family tree participate in reception of posterity and damaged alleles O eliminated from the family tree.

Lets consider selection against homozygotes аа （orОО）.

Genotypes before selection， for example， in generation n-2 are distributed according to （1）.

We accept fitnesses of genotypes （Vogel & Motulsky， 1990）：

CONCLUSION At the first record of the Hardy-Weinberg law the methodical mistake has been admitted connected by that this law was applied to the analysis of a population in the form correct only to the family tree. For the analysis of population it is necessary to use continuous time scale and the mathematical section of the differential equations.

Exception of the given methodical mistake allows to expand the numbers of the problems solved with the help of the Hardy-Weinberg law in particular in the demographic problem it permit to clear geneticmathematical laws of generations alternation in the countries with a high level of medicine development and the homogeneous population also more clearly to understand the interrelation of mutagenesis and selection， for example， at hemophilia.

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