## IV.6.1 The models of population genetics make it possible to calculate the progress of changes in the frequency of a dominant, recessive or superdominant allele.

Study of the changes in the frequency of alleles in a certain gene can be based on the following model:We have a population of randomly cross-breeding diploidal organisms.The population is sufficiently large so that no random processes are important and thus there are no random fluctuations in the size of the population and frequencies of the individual alleles, reproduction of organisms and selection processes are synchronous and the individual generations are discrete, i.e. do not overlap.The studied gene has only two alleles, **A _{1}**and

**A**, whose frequencies are

_{2}*X*

_{1}a

*X*

_{2}, where

*X*

_{1 }= 1 –

*X*

_{2}.The

*fitnesses*of the three possible genotypes

**A**,

_{1}A_{1}**A**and

_{1}A_{2 }**A**are in the order

_{2}A_{2 }*W*

_{11},

*W*

_{12}and

*W*

_{22 }and their frequencies prior to the action of selection are and, i.e. they are subject to the

*Hardy-Weinberger equilibrium.*The frequency

*X*

_{1}of the following generation is given by the equation:

(1)

where is the average fitness of the population

(2)

A change in frequency *X*_{1}in a generation is given by the equation

** **(3)

which, following substitution from equation (2), yields

(4)

As _{ }depends on the relative and not the absolute values of *W*_{11}, *W*_{12}, *W*_{22}, these quantities can be set equal in the series 1, 1 – *s*_{1}, 1 – *s*_{2}.Symbols *s*_{1 }and *s*_{2} denote *selection coefficients* expressing the degree of selection pressure against the individual genotypes.

It holds for the semidominant allele **A _{1}**, where

*W*

_{11 }= 1,

*W*

_{12 }= 1 –

*s*/2,

*W*

_{22 }= 1 –

*s*(the

*heterozygote*exhibits the relevant trait to a smaller degree, in the analyzed case to half the degree of the

*homozygote*

**A**), that

_{2}A_{2}

(5)

for allele **A _{1}** with

**complete dominance**, where

*W*

_{11 }=

*W*

_{12 }= 1,

*W*

_{22 }= 1 –

*s*(

*the heterozygote*has the same phenotype and thus the same fitness as the

*homozygote*

**A**) it holds that

_{1}A_{1}

(6)

for the **recessive allele** **A _{1}**, where

*W*

_{11 }= 1,

*W*

_{12 }=

*W*

_{22 }= 1 –

*s*(

*the heterozygote*has the same phenotype as the

*homozygote*

**A**), it holds that

_{1}A_{1}

(7)

and, for the case of **superdominance**, where *W*_{11 }= 1 – *s*_{1}, *W*_{12 }= 1, *W*_{22 }= 1 – *s*_{2 }(the *hereozygote* has greater fitness), it holds that

(8)

Equations (5) – (8) are nonlinear, so that it is easier to solve them numerically.In equations (5) – (7), the frequency of allele **A _{1}**gradually increases to complete suppression of allele

**A**.For the cases of

_{2}*dominance*and

*semidominance*, the increase is relatively rapid; in the case of

*recessivity*, selection occurs very slowly and accelerates substantially only after an increase in the frequency above a certainvalue (Fig.IV.6).In case of

*superdominance*,

_{ }is positive while

If *X*_{1 }increases above this value, then *x*_{1}is negative.This thus means that there exists an equilibrium frequency of allele **A _{1}**(

*R*), for which it holds that

R = s_{2} /( s_{1} + s_{2} ), (9)

i.e., a frequency that the population sooner or later attains and to which it returns after any fluctuations.

If the *selection coefficient *is low, then fitness is approximately equal to 1 and is very small, so that equation (3) can be replaced by the equation

(10)

where *X = X*_{1}and *t* is time expressed in the number of generation periods.It holds for the *semidominant allele* that

(11)

and, after integration

(12)

where *X*_{0 }and *X _{t }*are the frequencies of allele

**A**at times

_{1}*0*and

*t.*For the dominant allele we similarly obtain the equation

(13)

and, for the *recessive allele,*

(14)

*Fig. IV.6. Selection in favour of advantageous dominant, semi-dominant and recessive alleles. The action of selection leads to the fastest increase in the content of dominant alleles and the slowest in the content of recessive alleles. However, semi-dominant alleles become fixed soonest.*

Equations (12), (13) and (14) permit determination of the number of generations necessary to change the frequency of allele **A _{1}**from the initial frequency

*X*

_{0 }to the final frequency

*X*.

_{t}