Vol.3, No.12, 1029-1033 (2011) Natural Science
http://dx.doi.org/10.4236/ns.2011.312128
Modelling of the populating development of the genome
in the radiation condition of the environment
Andrey N. Volobuev*, Eugene S. Petrov
Samara State Medical University, Samara, Russia; *Corresponding Author:
Received 8 November 2011; revised 30 November 2011; accepted 12 December 2011.
ABSTRACT
The existence of the genome population in
condition of radiation environment has been
considered. The differences between the laws of
the allele frequencies for autosomal genes and
genes linked to sex are described. Radiation
conditions were found at maintenance of the
balance of the Hardy-Weinberg genotype in the
population, as well as conditions of complete
elimination of the targeted allele by ionizing ra-
diation. Conclusions about the nature of radia-
tion resistance of the population are drawn.
Keywords: Population; Ionizing Radiation;
Genotypes; Frequency of Allele; Recessive Allele;
Heterozygote; Hardy-Weinberg Equilibrium;
Inbreeding
1. INTRODUCTION
Intensive production, extracting and use of radioactive
elements, the lack of protection system and so on, as
well as accidents at nuclear power plants - all these rea-
sons lead to the fact that elevated background radiation
is a source of mutations affecting the gene pool of the
human population, which may cause irreversible conse-
quences not only for living people, but also for future
generations [1-4]. It is therefore interesting to consider
the existence and development of the population in ex-
posure to mutagenic ionizing radiation.
2. HARDY-WEINBERG EQUILIBRIUM
When crossed opposite-sex individuals in one genera-
tion, equilibrium frequency of alleles is established for
the autosomal genome, the law of which was found by
the English mathematician Hardy and German physician
Weinberg in 1908—the so-called Hardy-Weinberg equi-
librium.
Under this law, the genotypes AA, Aa, and aa have the
following frequency relation:




22
:2:
A
ApAapqaa q (1)
where p—is the dominant frequency of allele A, q—the
frequency of the recessive allele a, (p + q = 1) so that


:
A
paq (2)
Ratio (1) remains unchanged from generation to gen-
eration in an ideal population (number of individuals is
very large, there is pan-mixing, there is no selection,
mutation, migration of individuals, etc.) [5]. Violation
ratio (1) due to any external influences: mutation, migra-
tion, and so on—leads only to a change in the next gen-
eration of gene frequency ratio (2) and recovery ratio
(1).
One of the most significant factors of mutagenesis is
ionizing radiation. Violations of the genetic apparatus,
arisen with it, are generally incompatible with the nor-
mal functioning of the organism further. Mutations in
genes in sex chromosomes can lead either to the disap-
pearance of reproductive function in individuals, or to
the appearance in the offspring of serious violation.
We will analyze the existence of a population in a ra-
diation environment, and mathematical modeling of
changes in the gene pool taking as an example changes
in both sex-linked allele frequencies and inherited auto-
somally.
This problem can be solved analytically in complete
form.
Assume that during the impact on the population, the
frequencies of genotypes changed in the following pro-
portions:








2
2
:21
:
AA pFpqAapqF
aa qFpq



  

 (3)
that breaks the Hardy-Weinberg relation (1) (or taking
into account to inbreeding—Wright). In (3) condition of
pan-mixing weakened. The opportunity of closely-re-
lated marriages with inbreeding coefficient F there are.
This ratio characterizes the decrease in the proportion of
individuals in heterozygote (1 – F) times compared with

A. N. Volobuev et al. / Natural Science 3 (2011) 1029-1033 1030
pan-mixed population.
In reality, the impact on the population may be related
to migration, the emergence of subpopulations, radiation
exposure, negative social phenomena, etc.
In the next generation Hardy-Weinberg equilibrium is
restored with the new correlation of allele frequencies.
To obtain the dominant frequency of allele A in the new
balance must sum up the frequency of homozygote AA
and half the frequency of heterozygote Aa from (3).
Similarly we will do and for the allele a. As a result, we
obtain



2: 2Ap aq    (4)
Thus, the ratio of the frequencies of genotypes in the
next generation is





2
2
2
:2 22
:2
AA p
Aa pq
aa q

 


 

(5)
Let’s note that in (5) and (4) is not an inbreeding co-
efficient F.
Let us consider some consequences of the resulting
frequency ratios (4) and (5).
In the absence of influences on the population, i.e.
0

 , ratio (5) is identical to (1), i.e. we have a
well-known position to maintain the frequency of geno-
types in a set of generations.
When exposed to mutagenic agents on the population
of a differently nature Hardy-Weinberg equilibrium is
restored, but the population does not compensate for the
impact—the allele frequencies changed. This gives an
answer to the question of what type of equilibrium takes
place: stable, unstable and indifferent. Hardy-Weinberg
equilibrium is indifferent.
The mechanical analogy of the three possible types of
equilibrium: stable-1, unstable-2, indifferent-3, shown in
Figure 1.
3. INFLUENCE ON A POPULATION OF
RADIOACTIVE RADIATION
The question of the long and gradual reduction of the
1 2 3
Figure 1. Possible types of equilibrium (1: stable, 2: unstable,
3: indifferent). Mechanical analogy of the Hardy-Weinberg equi-
librium.
radiation exposure in which the population evolves, is of
a great interest.
Let’s consider a model situation where the existence
of a population, in the region it is only one element of
the radioactive half-life T. A was established [6] that the
frequency of mutations (usually lethal) is proportional to
the power dose of radiation 1
pkP, where 1 k is a
constant. As the power dose of radiation is proportional
to the activity of radioactive elements, scattered in the
environment, it is possible to record the frequency of
mutations, using the fundamental law of radioactive de-
cay, the ratio 1
~2
tn
T
p





 , where t ~ 30 years is the
approximate time of one generation [6], n is a number of
the considered generation the existing in conditions of
radiation.
Accepting 1
2
t
T
R





 and considering the influence
of radiation on recessive homozygote only aa, i.e.
2p



, we obtain:

22
n
kR   (6)
where the coefficient 1/2 is taken for the convenience of
further transformations. The coefficient k characterizes
the initial activity of the radioactive elements that affect
the population. In accordance with [7] 0
~kNT,
where 0
N is the total initial quantity of a radioactive
substance that acts on a population.
For example, there are recessive genes responsible for
the content of catalase in the blood. One can cite as an
example of reducing the frequency of recessive alleles of
these genes at 15.5 times for about two generations of
people living in Hiroshima and Nagasaki in comparison
with the general population of Japan. In this case, the
frequency of dominant alleles has not changed [8].
Based on (4), the recursive formula for changing the
frequency of allele a for the generations of the popula-
tion is as follows:

11
22
n
nn n
qqq kR 

 (7)
where 0, 1kR

 (at 1 ,2 tTR

).
The solution of this equation is not difficult and it is
the sum of a geometric progression:

0
1
21
n
n
RR
k
qq R

  (8)
where q0—is the frequency of allele a in the initial gen-
eration.
If the number of generations tends to infinity n,
frequency of allele a, tends to the value:

A. N. Volobuev et al. / Natural Science 3 (2011) 1029-1033
1031031
0
ˆ21
kR
qq R  . (9)
The possibility of establishing equilibrium is deter-
mined by the condition

021
kR
qR . Taking 00q

,
we can solve Eq.8 with respect to n a number of genera-
tions for which will complete elimination of the allele a
that reduces the variability of the population’s gene pool.
Figure 2 is a graph of the dependence frequency n q
of allele a the number of generations n, constructed in
accordance with the Formula (8) for the half-life of ra-
dioactive elements in the region, respectively, T = 176;
93.8 and 22.6 years. For example, half-life of radioactive
substances is such as 55Cs137—30.2 years, 38Sr90—28.7
years, and 27Co60—5.3 years [9]. But usually a half-life
isotope is considerably smaller.
As can be seen from the graphs, for the half-life 93.8
years the allele a will be disappears in the eighth genera-
tion. For the half-life T = 22.6 years after a fifth-genera-
tion set a new equilibrium frequency of allele a ˆ
q = 0.17.
In the calculations there are for allele a used the initial
frequency 00.9q; 50
kT  .
The critical value of the half-life, above which the
Hardy-Weinberg equilibrium is not established as a re-
sult of complete elimination of the allele a, T = 37.4
years.
Figure 2 shows that for small half-life, curve 3, the
frequency of alleles a at first drops very quickly, because
high activity radioactive elements. But then equilibrium
is established, because rapid decrease in the number of
highly active substances reduces the effects of radiation.
For long half-life the frequency of allele a decreases
more slowly, but there are completely eliminates the
allele at the affect of long radiation exposure, curves 1
and 2.
Suppose that A and a alleles and linked to the X
chromosome. The frequency of allele A in men and
women is pm and pf. For allele a they will be qm and qf.
When crossed in the first generation there are arise
genotype correlation in women:
   
::.
f
mmffmmf
A
AppAa pqpqaaqq (10)
In men, there is a frequency ratio gemizygote deter-
mined that the X chromosome in female in crossing goes
to the male offspring:
 
:
f
f
A
paq (11)
Suppose, as in the case of autosomal inheritance, on
the population act external factor, changing the ratio of
genotype frequencies in women:
Figure 2. The relationship between the frequency qn affected
by the radiation of the alleles of autosomal genes and the
number of generations n populations for different half-life (1: T
= 176 years, 2: T = 93.8 years, 3: T = 22.6 years).



  


2
:1
:2
fmmffm
mf fm
fmmf fm
AAppp qp qF
Aap qpqF
aaq qp qp qF





 







 


(12)
Assume also that the gemizygote state men (11) is
unaffected. Then, using the previous method of calcu-
lating allele frequencies, we can find that the change in
frequency of alleles a generations of women is
 
11
12.
2
fn fn mn
qqq 




(13)
As in the case of autosomal inheritance, the inbreed-
ing coefficient of F in relation (13) is not included.
Take into account that the X-chromosome male inher-
its from his mother: (1) (2)mnf n
qq


.
As in the case of autosomal inheritance, we consider
the effects of ionizing radiation on the recessive homo-
zygote aa.
 

12
12
2
n
fn fn fn
qqq kR




(14)
Solution of this equation is the sum of the general so-
lution of the homogeneous Eq.15 and a particular solu-
tion of inhomogeneous Eq.14 [10]:
 
12
1
2
fn fn fn
qqq







(15)
so
f
nfnfn
qqq


. (16)
The general solution of homogeneous Equation (15)
f
n
q
 will be sought in the form n
fn
qa
. Substituting
this expression into (15), we find:
12
1
2
nnn
aaa





 (17)

A. N. Volobuev et al. / Natural Science 3 (2011) 1029-1033 1032
Dividing both sides by 2n a

, we obtain a quadratic
characteristic equation:
2
210aa (18)
the solution of which are the two roots 12
1
1; 2 aa.
So, the general solution of (15) has the form:
12
1
2
n
fn
qCC




 (19)
where 1 C and 2 C—are constants. Solution (19), with
the initial conditions, describes the change of frequency
allele a women in the generations in the absence of ex-
ternal factors on the population.
A particular solution of Eq.14 will be sought in the
form n
fn
qLR
, where L—is yet unknown. Substituting
this formula into (14) we find:
212
2nn n LR LRLRkR

  (20)
Dividing both sides by 2n R

, we find the value of L:

.
21 12
n
kR
LRR  (21)
Thus, the solution of Eq.14 in accordance with (16) is:

2
12
1
221 12
nn
fn
kR
qCCRR


 
 
 (22)
Constants 1 C and 2 C we find the basis of the initial
conditions at 0
0,
f
nf
nqq and at 1n according
to (13) and (6)



100
00
22
22
fn fmf
mf
qq qq
qq kR

  
 
Consequently, the solution of Eq.14 is:



00
00
2
2
331
1
36122
21 12
fm
fn
n
fm
n
qq kR
qR
qq kR
R
kR
RR




 

 






(23)
If the number of generations in women tends to infin-
ity (n), there is an equilibrium frequency of allele
a (if it is possible):

00
2
ˆ331
fm
fn
qq kR
qR


 (24)
Similarly to the case of autosomal inheritance at
0
fn
q the number of generations, which would com-
pletely eliminate allele a in women can be found.
Figure 3 is a graph of the dependence of the fre-
quency
f
n
q of allele a in women and the number of
generations n, constructed by the Formula (23) for the
same half-life as for autosomal inheritance.
As can be seen from the graphs, for the half-life T =
93.8 years—the allele a will be disappears in the ninth
generation, the curve 2. For the half-life T = 22.6 years
since the sixth generation a new equilibrium is estab-
lished with the frequency of allele a in women
ˆ0.14
fn
q

, curve 3. Curve 1 is designed for the half-life
of T = 176 years. At the calculations used the initial fre-
quency of allele a women 00.9
f
q; men 00.1
m
q

.
The critical value of the half-life, above which the
Hardy-Weinberg equilibrium is not established as a re-
sult of complete elimination of the allele a, in the case of
inheritance of genes linked to the X chromosome, T =
44.7 years.
Figure 4 is a graph of the number of generations k n
of the complete elimination of the allele a and half-life
of radioactive element T for autosomal genes. 1 and the
genes of sex-linked, 2. The graphs are based on the solu-
tions of Eqs.8 and 23 at initial conditions, respectively,
0
n
q

and 0
fn
q

.
The graphs show that there are more dangerous ra-
dioactive elements to the population. For the given
model calculation they are the elements with half-life T
= 52.2 years for autosomal genes and T = 60.2 years for
genes linked to sex. In the first case of complete elimi-
nation of the allele a it will occur after 6
k
n genera-
tions, and in the second case—after 8
k
n generations.
Thus, genes, sex-linked, are more resistant to radiation
than autosomal.
4. CONCLUSIONS
As a result of calculations we can draw the following
Figure 3. The relationship between the frequency qfn affected
by the radiation of the alleles of the gene linked with X-chro-
mosome in females and the number of generations n popula-
tions for different half-life (1: T = 176 years, 2: T = 93.8 years,
3: T = 22.6 years).

A. N. Volobuev et al. / Natural Science 3 (2011) 1029-1033
1031033
Figure 4. The relationship between the limiting number of
generations nk population which still retained its gene pool and
installed the Hardy-Weinberg equilibrium for the targeted of
affected radiation gene and half-life of radioactive element T
(1: for autosomal genes, 2: for genes linked with X-chromo-
some).
conclusions:
1) When there is the action of radiation on the popula-
tion 2 outcomes are possible depending on the half-life
of radioactive elements in the environment:
 Establishment of the Hardy-Weinberg equilibrium at
maintaining the existing in the population genome,
but at a lower level of the targeted allele frequencies;
 The disappearance of the targeted allele, and conse-
quently the impossibility of preserving the Hardy-
Weinberg equilibrium.
2) There is a boundary value of the half-life of radio-
active substances in the environment, below which the
population maintains its genome. Consequently, the
outcome of the population depends on the nature of the
radiation environment (disaster at nuclear power plants,
nuclear explosion, accident at work, a natural phenome-
non, etc.).
3) Genes that are sex-linked, have a somewhat higher
resistance to radiation compared with the autosomal
ones.
4) Inbreeding does not affect the radiation resistance
of the population.
The obtained results allow us to be more cautious, at
presence the choice, to use of radioactive elements with
specific half-lives, which have a strong mutagenic effect
on autosomal genes and genes linked to the X-chromo-
some.
Considering in the evolutionary terms that genes
linked to the X-chromosome have greater resistance, we
can recall that genes are located in the X-chromosome,
mutations of which affect the vital functions of the hu-
man organism (vision, blood clotting, skin [11]). There-
fore, these genes are phylogenetically were better pro-
tected against the action of mutagens such as radiation.
Greater resistance to radiation of alleles, which are lo-
calized in 23 pairs of chromosomes, again suggests that
the female organism is genetically more stable. It plays
an important role in maintaining the size of the popula-
tion.
The obtained results allow us to determine the radia-
tion conditions of the maintaining of the balance of the
Hardy-Weinberg in the population, as well as conditions
of complete elimination of the targeted allele by ionizing
radiation.
The above method of analysis of the effects of radia-
tion on the genome can be applied to other types of in-
fluences. Thus it is necessary only to replace the funda-
mental law of radioactive decay in (6) and (14) with an-
other one for the investigated impact.
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