Family tree and population: Distinction and similarity of the analysis on example of hemophilia

doi:10.4236/ns.2013.59119

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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; [email protected]

Received 24 June 2013; revised 24 July 2013; accepted 1 August 2013

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



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. For recessive alleles a it is accordingly

and

At crossing in the first generation there is a ratio of

genotypes at women, according to product (pf + qf) (pm +

qm). Thus:

 



mmffmmf

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:





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

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



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:

 

fn fn mn

qqq

1









. (3)

Let’s assume:

 

mnf n



, (4)

where





 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:

 

fn fn fn

qqbq













. (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:

nnn

aaba













. (6)

Having divided the both parts of the equation on 2n



we shall receive some characteristic equation:

210

аа

, (7)

which solutions are two roots







 and







 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:

 

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

fn f

qq 

 .

We believe that in zero generation of men and women

action of the mutagen factor is absent. Thus:







,









 .

Substituting constants 1 and 2 in the formula (8),

we receive the solution of the Eq.5 as:

C C













121 1

fn m















, (9)

where for the further transformations it is convenient to

use the function of integer argument:

 





. (10)

In this case, the formula (9) will be transformed to a

kind:

 



fn mf

qqFnqFn



1. (11)

If number of generations there is n then at









a limit 1

lim 0











 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

q

0.7

q

0.3



. Curve АВO is plotted for frequency of blood

alleles O at women and initial frequencies of it alleles

0.6057



. 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

20.57

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:

 



  



00 00

1221

fn fn

mf fm

qqq

qqFnq Fnq Fn











(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

q





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.

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



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

 .

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

mnf nf nmnmnf n

mnf n

pq pqqqs

qqs

 









(15)

Genotypes before selection, for example, in generation

are distributed according to (1).

2nLet’s transform the formula (15), using

 

mn mn





 and :

 

fn fn





We accept fitness of genotypes [2]:



1:1: 1



, (13)

 







11 1

f nmnmnf n

mnf n

q qqqs

qqqs

 







. (16)

where s there is reduction of homozygotes fraction of

recessive alleles as a result of selection (selection pa-

rameter).

OPEN ACCESS

Genotypes after selection we shall write down for the

following generation (n − 1):



 

















11 11

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



. Hence:

Taking into account, that

 



 



 



1111 11

111 11 1

fnmnmnfnfn mn

mn fnfnfnmnmn

pppq pq

qqp qpq

 

  



 













12 2

fnfnfn fn

fn fn

qq qq

qqqs

 







. (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:



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







. We

assume generation n where the constancy



1fn









was

already established, Figure 3. Frequency of destructive

alleles has increased on

 

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

n in mutagenesis and selection

since frequency



fn up to a mutation is supposed in

both cases identical.



q

Taking into account balance mutagenesis and selection,

we equate (5) and (17). After simple transformations, we

find:

 





fnfn fn

bqq bq

qps





 





. (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]:



ln 2d

fn fn







, (19)

where the value



there is characterizes some constant

mutagen factor.

The Eq.19 can be integrated once:



dln 2

qqn





C



, (20)

where there is a constant of integration.

The Eq.20 is integrated in quadratures. The general

solution looks like:



ln 2

12e

ln 2ln 2

ln 2

qС





 , (21)

where there is a constant of integration.

Change of alleles a frequency for one generation is

equal:

 



1ln2

ln 2

ln 2ln 2

ln 2

ln 2ln 2

ln 2

ln 22

fn fn

qqq

nCС

nС























 







. (22)

At increase in number of generations (or time) n→∞

the change of alleles frequency ln 2





 .

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.

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

610

q





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

ln 22

fn fn

qq С





 . (23)

We use the initial condition: at according to (3)

1n

fn f

qq 

 .

Hence:

10 2

ln 242

qq С





 0

. (24)

From (24) we find constant :



200

2ln 2

Сqq



. (25)

Substituting (25) in (22), we shall find:



ln 2

ln 2ln 2

qqq













. (26)

Believing as well as in case of a family tree 00



we shall receive:





ln 2

112e 12

ln 2ln2







 



. (27)

At n = 0 we find initial change of alleles O frequency

which is equal



1ln 2





. Taking into account



1ln2





 we find . We shall



1610

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.





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:







 





  



12 21

1221 2

fn fn

fnfnfn fn

fn fn

fnfnfn fnfn fn

fn fn

qqq

qq qqs

qqs

qq qqsqq

qqs



 





 











 





.(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

 



Hence, the formula (28) will be transformed to a kind

similar autosomal genome [2]:

OPEN ACCESS



 



fn fn

qsq

qqs







1

. (29)

At small frequencies of alleles a can be approximated

change of frequency (actually the differential) by expres-

sion [2]:

 



 

111 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



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

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:









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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