Vol.5, No.9, 979-986 (2013) Natural Science
http://dx.doi.org/10.4236/ns.2013.59119
Family tree and population: Distinction and similarity
of the analysis on example of hemophilia
Andrey N. Volobuev, Peter I. Romanchuk, Vladimir K. Malishev
Samara State Medical University, Samara, Russia;
Received 24 June 2013; revised 24 July 2013; accepted 1 August 2013
Copyright © 2013 Andrey N. Volobuev et al. This is an open access article distributed under the Creative Commons Attribution Li-
cense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
ABSTRACT
On the basis of Hardy-Weinberg law the distinc-
tion in the philosophy of the description of a
family tree and a population is considered. With
the help of introduction of the equivalent con-
stant mutagen factor both in a family tree and in
a population, the analysis of hemophilia is led.
Influence of selection at presence of hemophilia
is considered. On the basis of idea of Danforth
and Haldane about the balance mutagenesis and
selection at presence of hemophilia, the inter-
relation of selection parameter and the equiva-
lent constant mutagen factor is found. It is shown,
that dynamics of development of hemophilia in a
family tree and in population are similar.
Keywords: Alleles; Equivalent Constant Mutagen
Factor; Family Tree; Population; Hemop hilia;
Balance of Selection and Mutagenesis
1. INTRODUCTION
Mathematical genetics is a completely special section
of biophysics. It enables to understand those or other phe-
nomena in human community, to explain and predict de-
velopment of this community as from the point of view
of health of people, and social processes in community. It
allows us to understand the purposes and problems of
human community for improvement of the life.
The mathematical genetics is based on Hardy-Weinberg
law which was found by English mathematician Hardy
and German doctor Weinberg in 1908.
This law in the elementary kind of two alleles of a
gene determines that relative frequencies of genotypes in
generations correspond to terms of binomial expansion
, so where p and q there are alleles
frequencies in a population or that is more correct in a
family tree. Relative frequencies of genotypes remain
constant from generation to generation in case of an ideal

2
qp 
population (number of individuals is very large, there is
pan-mixing, there is no selection, mutation, migration of
individuals, etc.) [1].
The population will consist of set of family trees which
periodically contact among themselves.
The exponent 2 specifies that two different-sex species
are necessary for reception of posterity. And the persons
are the man and the woman.
Hardy-Weinberg law in genetics has the same funda-
mental role as, for example, 2-nd Newton law in me-
chanics. But in the relation to world around these two
laws of genetics and mechanics there is an important dis-
tinction. The exponent 2 at the Hardy-Weinberg law re-
flects concrete conditions of life occurrence in the Earth. It
is possible to assume that on other manned planets it
exponent can have other value. If three different-sex spe-
cies, for example, are necessary for reception of posterity
at the Hardy-Weinberg law there should be an exponent 3.
The second Newton law undoubtedly has the same kind
everywhere in Universe.
Extremely important topic of mathematical genetics is
that there are mutations. The mutations explain geneti-
cally dependent hereditary diseases. Mutations allow to
explain process of evolution of organisms. Mutations un-
derlie animals and plants breeding.
Mutations are spontaneous and induced [2].
During induced mutations there is an interaction of the
mutagen factor and a species with an exposed mutation.
Process of a mutation has stochastic character. The
species under action of the mutagen factor with some
probability can be subjected to mutations, and can not be
subjected.
During the mutation the mutagen factor has the im-
portant role. It is possible to classify mutagen factors into
two groups: determined and stochastic.
 1 qp The determined mutagen factors can be constant or
functionally time-dependent.
There is pertinently to present the following mechani-
cal analogy.

A. N. Volobuev et al. / Natural Science 5 (2013) 979-986
980
In a room there is an air ball-analogue of the mutagen
factor. Molecules of air in a room play a role of species.
The model mutation is an impact of a molecule with a
ball.
The probability of a mutation depends on a state of a
molecule, in this case distances from a ball. With a ball it
can cooperate, and the distant molecule and a near mole-
cule can depart from a ball during thermal movement not
having heat around it.
But the probability of a mutation depends on behavior
of a ball.
The ball can not change the volume—the constant
mutagen factor.
It can increase at blow-up and reduce at blow-away.
The probability of mutations is changing. For example,
at full blow-away a ball probability of mutations become
equal to zero. The ball can periodically or aperiodically
change the volume. All these processes with a ball are
model of the determined mutagen factors [3].
The ball also can change the volume completely sto-
chastic and it is the model of stochastic mutagen factor
[4].
2. ACTION OF THE CONSTANT
MUTAGEN FACTOR ON A COUPLED
WITH THE Х-CHROMOSOME
GENOME
The intensive use of the mobile communication, insuf-
ficient protection against electromagnetic radiation of
computers on a workplace result to that there is a con-
stant mutagen electromagnetic background which is a
source of the changes touching a genofund of a human
population. Therefore is of interest to consider existence
and development genome in conditions of influence of
some constant mutagen factor.
At action of the constant mutagen factor there is proba-
bly occurrence of the selection resulting in change of
genic frequencies in one direction [2].
Let’s consider action of the constant mutagen factor on
two-alleles genome coupled with the Х-chromosome. For
this purpose we shall assume, that alleles A and a are
coupled with the Х-chromosome. The frequency of domi-
nant alleles A we shall designate at men m and at
women
p
f
p. For recessive alleles a it is accordingly
and
m
q
f
q.
At crossing in the first generation there is a ratio of
genotypes at women, according to product (pf + qf) (pm +
qm). Thus:
   
::
f
mmffmmf
A
AppAa pqpqaaqq. (1)
Men have gemizygote on genes in the Х-chromosome
the frequency ratio determined by that the Х-chromo-
some of the woman at crossing passes to the man’s de-
scendant:


:
f
f
A
paq. (2)
Distribution (1) can be used also for the description of
blood system АВO. In spite of the fact that to this system
corresponds three-alleles ensemble of the genes the two
alleles A and B are dominant and their general frequency
can be designated at men pm and at women pf. Alleles O
has in this case frequency at men m at women q
f
q.
The ratio (1) for blood system АВO is not frequency
distribution of genotypes of blood but the genotype fre-
quency aa (or a genotype OO), and also phenotype fre-
quency corresponding to a blood group I it the ratio re-
flects truly. For example, frequency of alleles O at men
and women in Berlin is equal , [2]. 0,6057 qq
fm

The basic demonstration of existence Х-coupled reces-
sive inheritance for a blood system АВO consists that the
destruction at disease of blood, for example, hemophilia
are men and daughters phenotypic are healthy.
For the first time the mathematical genetics laws has
applied Haldane to a problem of hemophilia on basis of
Danforth idea about an equilibration of frequency of mu-
tations and selection. Occurrence of hemophilia there is
usually concern to spontaneous mutations. However, for-
mally meaning balance of mutations and selection, and
also a constancy of occurrence of a family tree mutation
(otherwise illness quickly would disappear) it is possible
to calculate a task of a hemophilia assuming action of
some equivalent constant mutagen factor. Action of se-
lection will be appreciated further.
For a finding of recessive alleles a frequency in a new
generation it is necessary to add frequency of homozy-
gote aa and half of frequency of heterozygote Aa from a
ratio (1). It is similarly necessary to act and for alleles A.
Using this way of calculation of allele frequencies we
find that change of alleles a frequency in generations of
women equally:
 
1
1
2
fn fn mn
qqq
1




. (3)
Let’s assume:
 
1
1
mnf n
qq
b2


, (4)
where


1
2
mn
fn
q
bq


 it is some constant factor which we
shall name the constant mutagen factor. This factor
specifies that alleles frequency a at the woman of the
previous generation not in accuracy is equal to alleles
frequency at the man of the following generation. Such
situation develops if alleles frequency a in a female part
of a population during to live before the reproductive
period is reduced due to the arisen mutations resulting in

A. N. Volobuev et al. / Natural Science 5 (2013) 979-986 981
impossibility to posterity. [2].
According to (4) Eq.3 becomes:
 
1
1
2
fn fn fn
qqbq




2


. (5)
For the solution of the Eq.5 we shall search as qfn = an
where a there is constant. Substituting this formula in (5),
we shall find:
1
1
2
nnn
aaba




2


. (6)
Having divided the both parts of the equation on 2n a

,
we shall receive some characteristic equation:
210
22
b
аа
, (7)
which solutions are two roots

1
11
4
a


 and

2
11
4
a


 where 18b   we shall name the
reduced constant mutagen factor. At b = 1 or 3


the
expression (5) describes genome equilibrium condition at
absence mutagenesis.
Hence the general solution of the Eq.5 looks like:
 
12
11
11
44
nn
fn
qC C 








, (8)
where 1 and 2 there are constants. The solution (8)
with account of initial conditions describes change of
alleles a frequency at women in generations with account
of the mutagen factor action.
C C
Constants 1 and 2 we shall find proceeding from
initial conditions: at n = 0, qfn = qf0 and at n = 1 accord-
C C
ing to (3) 00
12
mf
fn f
qq
qq   .
We believe that in zero generation of men and women
action of the mutagen factor is absent. Thus:

00
1
21
2
mf
qq
C



,


00
2
21
2
mf
qq
C



 .
Substituting constants 1 and 2 in the formula (8),
we receive the solution of the Eq.5 as:
C C

 

 
0
11
0
121 1
24
11
nn
fn m
n
nn
f
qq
q







, (9)
where for the further transformations it is convenient to
use the function of integer argument:
 
11
4
nn
n
Fn


. (10)
In this case, the formula (9) will be transformed to a
kind:
  
00
12
fn mf
qqFnqFn

1. (11)
If number of generations there is n then at 
11
4
 

a limit 1
lim 0
4
n
 



 if n is integer. In
this case frequency . On Figure 1, the depend-
ence plotted under the formula (11) of alleles a frequency
at women from number of generation n at various values
of the mutagen factor b is shown. Initial alleles a fre-
quency at women was accepted 0 at men
0
0
fn
q
0.7
f
q
0.3
m
q

. Curve АВO is plotted for frequency of blood
alleles O at women and initial frequencies of it alleles
00
0.6057
fm
qq

. The equivalent constant mutagen
factor for curve АВO was accepted equal . 0.b8
From Figure 1, it is visible that at absence of the muta-
gen factor influence 1b

the alleles a frequency at
women gradually drawing near to equilibrium frequency
00
20.57
3
mf
f
qq
q

 that coincides with [5]. At oc-
currence of the mutagen factor there is a decrease in al-
leles a frequency and if all over again are observed
jumps of frequency then there is a smooth decrease.
Let’s find change of alleles a frequency in generation
of women:
 

  

11
00 00
1221
fn
fn fn
mf fm
qqq
qqFnq Fnq Fn



1


.
(12)
Figure 1. Dependence of alleles a (or O) frequency at women
from number of generation n at various sizes of the mutagen
factor b.

A. N. Volobuev et al. / Natural Science 5 (2013) 979-986
982
On Figure 2, the calculation of change of alleles a fre-
quency used under the formula (12) is shown depending
on generation of women n at various values of the mu-
tagen factor b.
Apparently from Figure 2, the approach to equilib-
rium state of genome

1 is observed only at
absence of the mutagen factor influence, i.e. at
0
fn
q 
1b

.
Otherwise there is a constant decrease in alleles a fre-
quency in generations. And approximately to the seventh
generation the constancy of such decrease is established:
in 10-th generation at size , at
b = 0.8 size .
0.b
0.021

10. fn
q 014

1fn
On Figure 3, change of alleles blood O frequency is
shown in generations of women at the equivalent con-
stant mutagen factor b = 0.8.
q 
Figure 2. Change Δqf(n−1) of alleles a frequency depending
on number of generation of women at various values of the
mutagen factor b.
Figure 3. Change Δqf(n−1) of alleles blood O frequency in
generations of women at the equivalent constant mutagen
factor b = 0.8.
The mutagen factor acting on alleles O can result, for
example, in hemophilia. In 10-th generation, size Δqf(n−1)
= −0.02. It specifies, that there is a gradual reduction of
healthy alleles O frequency at women and during too
time increase at them destructive alleles O frequency.
According to the formula (11), see Figure 1, at the sec-
ond generation healthy alleles O frequency at women to
become equal 20.492
f
q

.
Hence, frequency destructive alleles has increased at
them by size 0.6057 0.4920.113

. Also increase fre-
quency destructive alleles O at men of the third genera-
tion since it is transferred the man. At men destructive
alleles O phenotypic it can be shown as hemophilia.
On Figure 4 according to [2], the family tree of the
European royal houses in which men frequently were ill
hemophilia is shown. Distribution of the Х-chromosome
in a family tree is shown also. The circle leads round the
Х-chromosome having destructive alleles O which caus-
es hemophilia. Black squares there are designate men
which are sick of hemophilia. From this family tree the
gradual increase in number of men which are sick of
hemophilia is well visible. The shown family tree not full
since on it is possible to conclude that in the third gen-
eration all men are sick of hemophilia. Actually it is far
from being so. And now in the given royal dynasty the
share of sick hemophilia men are not too great. Accord-
ing to the submitted family tree in the first generation
from four men one is sick of hemophilia, and in the sec-
ond generation from seven men three are sick of hemo-
philia. Rough calculation of increase in frequency sick of
hemophilia in a considered dynasty in the first and sec-
ond generation at men has size 310.178
74
 .
Corresponding reduction of a share of genotypic healthy
women in the zero and first generation, counted up under
the formula (12) is equal 0.106, see Figure 3. If should
be more full family tree results apparently should be
more close.
3. CONNECTION BETWEEN STANDARD
PARAMETER OF SELECTION AND
THE CONSTANT MUTAGEN FACTOR
In a family tree where the hemophilia is observed, there
is selection resulting in decrease of genic frequencies in
particular of alleles O to one direction.
Action of selection there is intensive enough. For ex-
ample, the life period of the men which are sick of he-
mophilia makes 1/3 from life period of healthy people
[2]. Ability to leave offspring at men in comparison with
healthy men is reduced. Therefore, not all men of a fam-
ily tree take part in reception of offspring and destructive
alleles O eliminate from a family tree. However is possi-
ble the balance of mutagenesis and selection resulting to

A. N. Volobuev et al. / Natural Science 5 (2013) 979-986
983
Figure 4. Family tree of the European royal houses in which men frequently were sick of hemophilia.
Further, using a standard rule of a finding of alleles
frequency in the following generation (half of heterozy-
gotes frequency plus of homozygotes frequency) and the
formula (14) we calculate the frequency of recessive
alleles a at women in generation n it is similar [2] where
such calculation is made for autosomal genome:
preservation of Hardy-Weinberg equilibrium on some
level i.e. ratio of genotypes frequencies (1).
The balance at hemophilia mutagenesis and selection
allows find connection between the equivalent constant
mutagen factor b and standard parameter of selection s
[2].
Let’s consider selection against homozygotes aa. In [5],
there is concrete example of such selection—the phe-
nomenon of industrial melanism, i.e. change of painting
of butterfly Biston betularia in industrial regions of Eng-
land is resulted at the end of XIX century.
 




11 1111
11
11
2
1
fn
mnf nf nmnmnf n
mnf n
q
pq pqqqs
qqs
 




.
(15)
Genotypes before selection, for example, in generation
are distributed according to (1). 2nLet’s transform the formula (15), using
 
11
1
mn mn
pq



 and :
 
11
1
fn fn
pq


We accept fitness of genotypes [2]:

1:1: 1

s
, (13)
 




11 1
11
1
2
1
f nmnmnf n
fn
mnf n
q qqqs
qqqs
 



1
2
. (16)
where s there is reduction of homozygotes fraction of
recessive alleles as a result of selection (selection pa-
rameter).
Genotypes after selection we shall write down for the
following generation (n − 1):

 








11 11
11 11
:
:1
f nmnmnf n
f nmnmnf n
AA ppAapq
pq aaqqs
 
 

. (14)
At selection all changes in a population occur due to
size s therefore we believe that alleles a frequency at the
man is equal to alleles a frequency at the woman of the
previous generation
 
1mnf n
qq


. Hence:
Taking into account, that
 

 

 

1111 11
111 11 1
1
fnmnmnfnfn mn
mn fnfnfnmnmn
pppq pq
qqp qpq
 
  

 



,





12 2
21
1
2
1
fnfnfn fn
fn
fn fn
qq qq
qqqs
 



1
s
. (17)
As it is underlined in [2], Eq.17 has no general solu-
tion even for autosomal genome. Especially it is not pre-
sent for genome linked with the Х-chromosome. It is
essentially complicates the mathematical analysis of the
Eq.17. First of all does not allow is proved that the size s
re- mains constant from generation to generation.
we find the sum of frequencies of genotypes:
.

11
1mnfn qqs



A. N. Volobuev et al. / Natural Science 5 (2013) 979-986
984
Mutagenesis and selection act in one direction. If arisen
for the account mutagenesis destructive alleles will be
eliminated from the family tree with the same speed due
to selection the balance to be kept. A quantity of the de-
structive genes is kept in a family tree but this quantity
will not increase.
We assume in process of mutagenesis in generation n
at women the quantity of the destructive alleles O has
increased in comparison with generation

1n


. We
assume generation n where the constancy

1fn
q


q


was
already established, Figure 3. Frequency of destructive
alleles has increased on
 
11
fn
fnfn . These
alleles due to selection eliminate from a family tree. The
balance of mutagenesis and selection will arise in case of
equality of frequency
qq
f
n in mutagenesis and selection
since frequency

fn up to a mutation is supposed in
both cases identical.
q

1
q
Taking into account balance mutagenesis and selection,
we equate (5) and (17). After simple transformations, we
find:
 


11
1
12
12
fnfn fn
fn
fn
bqq bq
qps


 


2
s
3
. (18)
In a case with hemophilia at the mutagen factor b = 0.8
and identical genic frequencies at men and the women
equal 0.6057 standard parameter of selection designed
under the formula (18) , and fitness of a geno-
type . The estimation of fitness is enough
challenge. One of possible ways to calculate fitness to
find it is as the ratio of an average of survived children of
the falling one parent which is sick of hemophilia to an
average of survived children on one healthy parent. For
example, in [6] the opportunity of calculation is offered
(though and with reserves) to fitness at hemophilia
0.27s
10.7s
11.752.50.s 7.
Thus, the opportunity of the accuracy solution of the
equation describing change of frequencies in generations
with the help of the constant mutagen factor b as against
use of the s-dependent Eq.17 apparently specifies pref-
erable of value b.
At the analysis of the accuracy solution of process of
mutagenesis, for example, in a case with hemophilia,
there is an opportunity of more reliable calculation of
constant mutagen factor size if there are authentic data
on a full family tree with destruct of corresponding al-
lele.
4. ACTION OF THE CONSTANT
MUTAGEN FACTOR ON A
POPULATION
In the previous section, action of the constant mutagen
factor on a genome of a family tree has been considered.
Besides, it has been marked that spontaneous mutagene-
sis at balance of mutations and selection (in particular at
hemophilia), it is possible to present as action of the
equivalent constant mutagen factor.
In the present section, we will analyze action of the
constant mutagen factor (mobile communication, radia-
tion of office equipment, etc.) on a human population.
The Hardy-Weinberg law in the kind considered above
there is concerns to a separate family tree. Implicitly this
law includes time since alternation of generations occurs
through certain period. Average time of a life of one
generation is 25 30Т

 years. Thus Hardy-Weinberg
law has definitely discrete character on time. The popu-
lation will consist of family trees crossed among them-
selves and lives in continuous time. Alternation of gen-
erations into family trees set results to that generations
vary actually according to a continuous time scale.
Therefore we shall pass to a continuous scale of gen-
erations n. Under size n in this case we mean time of a
life of the population normalized on average on a popu-
lation time of a life of one generation, i.e. actually con-
tinuous dimensionless time.
The analysis we shall make on the basis of Hardy-
Weinberg law written down as [4]:

2
11
2
dd
ln 2d
d
fn fn
qq
n
n



, (19)
where the value

there is characterizes some constant
mutagen factor.
The Eq.19 can be integrated once:


1
1
1
dln 2
d
fn
fn
qqn
n


C

, (20)
where there is a constant of integration.
1
The Eq.20 is integrated in quadratures. The general
solution looks like:
С


ln 2
1
2 12e
ln 2ln 2
ln 2
n
fn
C n
qС


 , (21)
where there is a constant of integration.
2
Change of alleles a frequency for one generation is
equal:
С
 




11
1ln2
1
2
2
ln 2
1
2
2
ln 2
2
1e
ln 2ln 2
ln 2
e
ln 2ln 2
ln 2
1e
ln 22
fn
fn fn
n
n
n
qqq
nCС
C nС
С












  



. (22)
At increase in number of generations (or time) n→∞
the change of alleles frequency ln 2
f
q


 .

A. N. Volobuev et al. / Natural Science 5 (2013) 979-986 985
Let’s address now to the analysis of decrease of alleles
blood O frequency at there is hemophilia. As it was
specified earlier a spontaneous mutation at hemophilia it
is possible to present as action of some constant equiva-
lent mutagen factor, in this case

.
Frequency of mutations at hemophilia in different coun-
tries (on a population) changes from 5 4.4 10

 (Swit-
zerland), up to (Denmark), i.e. a gene of a
hemophilia have from 44 up to 64 women on one million
[2]. Frequency of mutations it is ratio of number of cases
of anomaly display to the double number of the exam-
ined individuals the corrected sizes of frequencies of
mutations therefore are used.
5
6.410 
Let at the size is . I.e. 60
girls which birth from one million have a gene of a he-
mophilia. In this case the equivalent constant mutagen
factor .
n
lnq
5
610
f
q


5
10  2 4.16   
f
The basic uncertain size in dynamics of change of al-
leles frequencies (22) is the constant C2. According to (22),
we shall find the law of decrease in genic frequency:

ln 2
2
1
1e
ln 22
n
fn fn
qq С



 . (23)
We use the initial condition: at according to (3) 1n
00
12
mf
fn f
qq
qq   .
Hence:
0
10 2
1
ln 242
mf
ff
qq
qq С


 0 . (24)
From (24) we find constant :
2
С

200
4
2ln 2
fm
Сqq

. (25)
Substituting (25) in (22), we shall find:


ln 2
00
1
2e
ln 2ln 2
n
fm
fn
qqq







. (26)
Believing as well as in case of a family tree 00
f
m
qq

,
we shall receive:




1
ln 2
112e 12
ln 2ln2
n
n
fn
q




 

. (27)
At n = 0 we find initial change of alleles O frequency
which is equal

1ln 2 f
q


. Taking into account

1ln2 f
q


 we find . We shall

5
1610 f
q


note that the value

1f has rated character. Change
of genic frequency is real at mutagenesis begins from
time coordinate n = 1 at which according to (27)
.
q 
0
 
10fn f
On Figure 5, the dependence of change of genic al-
leles O frequency
 1fn at women from time of the
population life plotted under the formula (27) is shown.
qq


q 
Figure 5. Dependence of change of genic frequency Δqf(n−1)
of blood allele O at women from time of a population life.
Comparison about Figure 3 for change of genic alleles
O frequency (under condition of the equivalent constant
mutagen factor action) plotted for a separate family tree
shows that as whole dependences are similar. Smooth-
ness of a curve on Figure 5 is defined by a continuity of
a scale alternation of generations or dimensionless time
for a population as against discrete time of a separate
family tree.
The analysis of Figure 5 allows to assume that under
action of the equivalent constant mutagen factor

acting on a population, there is average on species of a
population the mutation resulting in average on a popula-
tion to continuous reduction of frequency of alleles O
healthy at women and during too time to increase at them
of frequency of alleles O destructive i.e. to a hemophilia.
Action of selection, for example, decrease in life period
of men with hemophilia on 1/3 and decrease in their birth
rate are compensates reduction of destructive alleles fre-
quency [2]. In this the dynamics of development of he-
mophilia in a population as a whole does not differ from
those in a family tree.
5. ACTION OF SELECTION ON A
POPULATION
Let’s consider action of selection on a population. Us-
ing (17) the change of alleles a frequency in a family tree
it is possible to calculate under the formula:




 


  

11
12 21
1
21
2
1221 2
21
1
2
1
1
2
1
fn
fn fn
fnfnfn fn
fn
fn fn
fnfnfn fnfn fn
fn fn
qqq
qq qqs
q
qqs
qq qqsqq
qqs

 


 





 

1
s

.(28)

A. N. Volobuev et al. / Natural Science 5 (2013) 979-986
986
At transition to a continuous time scale we believe dif-
ference in genic frequencies of two generations follow-
ing one after another infinitesimal, i.e.

21fn fn
 
qq

.
Hence, the formula (28) will be transformed to a kind
similar autosomal genome [2]:

 


2
1
12
1
1
1
fn fn
fn
fn
qsq
qqs






1
1
. (29)
At small frequencies of alleles a can be approximated
change of frequency (actually the differential) by expres-
sion [2]:
 

 
22
111 1
1
fnfnfnfn fn
qqsq qsp
 
 , (30)
where
  11fn fn
there is frequency of domi-
nant alleles A at women. For blood system ABO
1pq

1fn
p

there is total frequency of blood dominant alleles.
If process of selection is stationary it is possible to use
the frequencies of alleles at : n
2
fff
qqp

 s

. (31)
Actually, this formula accordance with Figure 5 it is
valid already at . 8n
Thus, the parameter of selection s in a population can
be calculated under the formula:




ff
f
pq
q
s2. (32)
Let’s note, that parameter of selection s it is size posi-
tive since . The alleles a frequency is reduced. 0qf
Using the formula (27) at we find n
2ln

 f q. Substituting this expression in (32) we
shall find connection between parameter of selection s
and the characteristic of the constant equivalent mutagen
factor

in a population:
2ln
2
 ffpsq  . (33)
Apparently from (33), there is a linear connection be-
tween parameters s and

that confirms of Danforth
and Haldane ideas about an opportunity of mutagenesis
and selection balance at presence of hemophilia.
6. CONCLUSION
Hereditary disease hemophilia can be analyzed as one
example of separate family tree, and all populations as
whole. For a quantitative estimation on the basis of
Hardy-Weinberg law, allele blood O frequency changes can
be used as the equivalent constant mutagen factor and it
is also an idea of Danforth and Haldane about balance of
mutagenesis and selection at presence of hemophilia. As
the whole the analysis of a family tree on example of the
European royal houses and populations leads to the same
basic results, change of allele frequencies after some
lives of a population (or generations in a family tree) be-
comes constant. Selection compensates the given change
of allele frequencies.
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