Chapter 6. Basic Principle 

of Population Genetics

To predict the phenotypic and genotypic frequencies of progeny resulting from matings among the total population the probabilities of matings between individuals of various genotypes are needed. That is, for a population of individuals having genotypes PP, Pp, and pp, the frequency of matings between, for example, PP males and pp females must be known to predict the frequencies in their progeny.

Population genetics(집단유전학) is the study of gene and genotypic frequencies within a population and the prediction of these frequencies in subsequent generations. This chapter describes concepts of population genetics as they apply to qualitative traits(질적형질).

6- 1 Gene and Genotypic Frequencies

For the polled or horned condition in cattle there are two alleles involved at an autosomal locus(상염색체좌위). The frequency of each allele and of each genotype(유전자형) must be known to describe a population relative to this locus. A frequency(빈도) is denoted as f( ); hence, f(P) represents the frequency of the P allele and f(Pp) the frequency of the heterozygote(이형접합체) in the population(집단). Because each individual of a diploid(2배체) species carries two genes(유전자) at a particular autosomal locus, in a population of n individuals there are 2n total genes at that locus.

Genotypic frequency

The frequency of a genotype is defined as the proportion of the n individuals in the 

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population with a particular genotype. If the number of individuals in a population of polled and horned cattle with each genotype is represented by n subscripted with the genotype, then 

The genotypic frequencies can then be represented

The sum of frequencies of all possible events must equal 1:

f(PP) + f(Pp) + f(pp) = 1

Gene frequencies

The number of P alleles is twice the number of homozygous(동형접합체) polled animals (they each carry two P alleles) plus the number of heterozygotes, (each of which carries one P allele); that is, 

Similarly, np is the number of heterozygotes plus twice the number of horned animals:

Gene frequencies may also be calculated from genotypic frequencies. If f(PP), f(Pp), and f(pp) are the frequencies of the three genotypes, then

f(P) = f(PP) + (1/2)f(Pp) and

f(p) = (1/2)f(Pp)+ f(pp)

Example 6- 1 illustrates the calculation of gene and genotypic frequencies of coat color in a sample of Shorthorn cattle.

Example 6- 1

In Shorthorn cattle, three coat colors are red, roan, and  white. In a sample of 1000 Shorthorns assume the number of animals with each coat color is 

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The genotypic frequencies are

f(red) = f(RR) = 360/1000 = .36 

f(roan) = f(Rr) = 480/1000 = .48 

f(white) = f(rr) = 160/1000 = .16

Because of codominance(공우성) these are also the phenotypic frequencies. In this sample, there are a total of 2000 genes:

= 720 + 480 = 1200

= 480 + 320 = 800

Therefore,

Note also that the same results are obtained using genotypic frequencies to calculate gene frequencies:

f(R) = f(RR) + (1/2)f(Rr) 

= .36 + (1/2)(.48) = .6 and

f(r) = (1/2)f(Rr) + f(rr) 

= (1/2)(.48) + .16 = .4

Frequencies are equivalent to probabilities. For example, the probability of randomly drawing a PP individual from a population is equal to f(PP). Likewise, the probability of randomly drawing a gamete(배우체) carrying the P gene from the polled of gametes is f(P). Equating frequencies and probabilities becomes important when determining the chance of a mating between two individuals of given genotypes,

Random mating

Random mating(임의교배), also called panmixia(잡혼번식), is a mating system in 

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which each individual has an equal opportunity to mate with any individual of the opposite sex. When the impact of alternative mating strategies on gene and genotypic frequencies is considered, random mating is often used as the basis of comparison. That is, how do the gene and genotypic frequencies obtained from a particular mating strategy differ from those that would have resulted had the population mated at random?

Probability of mating

The probability of random mating between two animals of given genotypes is the product of the frequencies of the two genotypes in the population. For a particular autosomal locus with two alleles, for example, B and b, and three genotypes, BB, Bb, and bb, there are nine possible mating combinations with the following frequencies:

An assumption in determining frequencies of these matings is that genotypic frequencies are the same in males as in females. In some cases, for example, with sex- linked alleles(성연관대립유전자), this assumption may not be valid and the frequencies of genotypes must be established for each sex. Nevertheless, with random mating, the probability of a particular mating is always the product of the genotypic frequencies of the mates.

6- 2 The Hardy- Weinberg Law

With the rediscovery in 1900 of Mendel's work, scientists began to reconcile(조정하다) evolutionary theory with the particulate theory. One question was, what happens to gene and genotypic frequencies from one generation to the next? In 1908, G. H. Hardy and W. Weinberg, working independently, developed the fundamental relationship between gene and genotypic frequencies over many generations under 

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certain assumptions. The Hardy- Weinberg law(법칙) may be stated as follows:

For a large random mating population, in the absence of forces that change gene frequencies (mutation, migration, and selection), the gene and genotypic frequencies remain constant from one generation to the next.

When gene and genotypic frequencies remain constant a population is said to be in equilibrium(평형). The necessary conditions for equilibrium are

1. a large population,

2. random mating, and

3. the absence of forces acting to change gene frequency.

Chapter 7 discusses the consequences when these conditions are not met.

Beginning with the following discussion of gene and genotypic frequencies in populations in Hardy- Weinberg equilibrium(평형), a shorthand notation for gene frequencies will be used. For an autosomal locus with two alleles, a common notation is to represent the frequency of one allele, say B, with the letter p and the frequency of the other allele, b, with the letter q. That is, 

f(B) = p

f(b) = q

Gene and genotypic frequencies at Hardy- Weinberg equilibrium

A population in Hardy- Weinberg equilibrium for a single autosomal locus with two alleles exhibits the existence of a distinct relationship between gene and genotypic frequencies. Assume a population at equilibrium with f(B) = p and f(b) = q. The genotypic frequencies are

This means the frequency of genotype BB is equal to the square of the frequency of the B allele, the frequency of the heterozygote Bb is two times the product of the frequency of B and b, and the frequency of genotype bb is the square of the frequency of the b allele.

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The sum of the probabilities of drawing B or b is 1, or

p + q = 1

Similarly, the sum of the frequencies of all possible events must equal 1; thus

That the gene and genotypic frequencies remain constant from one generation to the next can be easily demonstrated. The frequency of the B allele, in both the male and female pools of gametes, is p and of the b allele, q. The union(결합:combination) of these gametes is random. The frequency of zygotes(접합체) from P(the union of two gametes) for four types of unions is as follows:

f(B) = f(BB) + (1/2)f(Bb)

which algebraically reduces to

= p(p + q)

= p (recall that p + q = 1) and

f(b) = 1/2 f(Bb) + f(bb)

= q(p + q)

= q

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Example 6- 2

The distribution of coat color in a sample of 1000 Angus cattle was determined to be

If the population is in equilibrium, then the estimate of q is

In Example 6- 1 exact gene frequencies were obtained for the sample by gene counting, which was possible since the heterozygote had a distinct phenotype. When one allele is dominant the gene frequencies can only be estimated because the true distribution of homozygotes(동형접합체) and heterozygotes cannot be known. 

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Example 6- 3

Assume that the following frequencies of phenotypes were observed in a population of Shorthorn cattle for both males and females:

For this population

f(R) = f(RR) + f(Rr)

= .6 + .2 =.8 and

f(r) = .2

The population is obviously not in equilibrium because no white cattle were observed despite the fact that f(r) = .2. 

This population is now randomly mated. The probabilities of unions between particular gametes are 

Hence, the expected genotypic frequencies in the progeny are

f(RR) = .64

f(Rr) = .32

f(rr) = .04

The progeny population is at equilibrium; the gene and genotypic frequencies will remain constant for all future generations under the assumptions of the Hardy- Weinberg law.

6- 3 Sex- linked Loci

The calculation of genotypic frequencies for a sex- linked locus(성염색체연관좌위) with two alleles is similar to that for autosomal loci with the obvious difference of the 

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heterogametic state of one of the sexes (males in mammals, females in birds). Assume that f(B) = p and f(b) = q for two alleles at a sex- linked locus. At equilibrium, the genotypic frequencies within each sex are

ln the homogametic sex, the frequency of gametes carrying the B aIlele is p and of those carrying the b aIlele is q. The heterogametic sex, however, produces three types of gametes -  one- half carrying either the B or b aIlele and the other half carrying the Y chromosome. The frequency of genotypes in the progeny is the probability of union of male and female gametes of each type under random mating. For mammals the genotypes and frequencies are

The frequency of males in the progeny is

f(males) = (1/2)p + (1/2)q

= (1/2) (p + q)

= 1/2

and of females is

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= 1/2

This is the same frequency as in females of the parent population. Similar calculations provide the expected frequencies of the other genotypes in the progeny. Therefore, under the conditions of the Hardy- Weinberg law, the gene and genotypic frequencies for sex- linked loci also are constant from one generation to the next. If a population is in equilibrium, estimates of gene frequencies can be obtained directly from the phenotypic frequencies within the heterogametic sex:

For autosomal loci, a population not in Hardy- Weinberg equilibrium reaches equilibrium after one generation of random mating if the gene frequencies are the same in males and females. The same is true for sex- linked loci. If the initial gene frequencies are not equal in both sexes for autosomal loci, two generations of random mating are required to bring the population to equilibrium. For sex- linked loci, however, more than two generations of random mating are required to reach equilibrium. Assume a mammalian population with the following genotypic frequencies:

gives the expected frequencies of genotypes in the progeny:

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The frequencies of the alIeles in the male progeny are exactly the same as the frequencies of the alleles in the female parents because males receive the B or b allele only from their mothers. The frequencies of alleles in the female progeny are the averages of the gene frequencies of the parents because females receive half their genes from each parent:

which is the same as 

= .3 + .25 = .55

In the second generation of progeny from random mating, f(B) in males will be .55, and f(B) in females the average of frequencies of their parents, or ½(.6 + .55) = .575. The frequencies for three generations are 

The absolute difference in gene frequency between sexes is halved from one generation to the next. The frequencies approach equality in both sexes when

f(B in equilibrium) = (1/3)f(B in the initial population of males)

+ (2/3)f(B in the initial population of females)

For the given example,

f(B at equilibrium) = (1/3)(.5) + (2/3)(.6)

= .5667

Intuitively, this equilibrium frequency makes sense. For sex- linked loci, three X chromosomes are involved: in mammals the male has one and the female has two. The contribution of genes from the male parents is one of every three and from the female parents two of every three. The frequencies of the alleles at equilibrium are expected 

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to be a weighted average of the frequencies within sex for the initial generation, the weights being 1/3 for males and 2/3 for females.

6- 4 Multiple Alleles at a Single Locus

At loci with more than two alleles, the consequences of random mating are the same as for two alleles. Using a three- allele system as an example, assume that 

with the subscripts designating alleles at the A locus. At equilibrium the genotypic frequencies can be derived from the trinomial expansion(3항전개)

which is

As in the case of two alleles, the sum of gene frequencies must be 1:

p + q + r = 1

Under the assumptions of the Hardy- Weinberg law, the gene and genotypic frequencies for multiple alleles remain constant over generations. If the gene frequencies are the same for both sexes, a population not in equilibrium will reach equilibrium after one generation of random mating.

For sex- linked loci, the genotypic frequencies at equilibrium for males are

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The frequencies for females are the same as for autosomal loci. Calculating the gene frequencies for multiple alleles for situations where each genotype has a distinct phenotype is the same as for the two- allele system that is, count genes and divide by the total number of genes. If dominance exists between pairs of alleles, the approach becomes more complex, as shown in Example 6- 4.

Example 6- 4

(q + .4)(q -  .2) = 0 

so that

q = - .4 or.2

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6- 5 Multiple Loci

A population with equal gene frequencies in males and females, which is not in equilibrium for autosomal loci, requires only one generation of random mating to reach equilibrium. Intuitively it would seem that, if two different loci independently reach equilibrium, they should also be jointly at equilibrium. This assumption may not be true.

f(AB)f(ab) = f(Ab)f(aB) and that

f(AB)f(ab) -  f(Ab)f(aB) = 0

If the loci are not in joint equilibrium then

and the difference will be denoted as d. Therefore, d represents the difference in production of gametes in the coupling state(상인상태) versus the repulsion state(상반상태). Two different situations must be considered to demonstrate disequilibrium(불평형):

1. the loci are on separate chromosomes, or are far enough apart on one chromosome so that they appear to segregate independently; and

2. the loci are linked.

Independent loci

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Example 6- 5

Assume the following frequencies of joint genotypes:

For this population 

f(A) = .7

f(a) = .3

f(B) = .6

f(b) = .4

Now 

= (.48)(.18) -  (.22)(.12)

= .06

If gametes combine randomly, the genotypic frequencies are

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The new genotypic frequencies become

While each locus is itself in equilibrium, the two loci are not jointly in equilibrium. The gametes produced by this generation have the following frequencies:

f(AB) = .45

f(Ab) = .25

f(aB) = .15

f(ab) = .15

Linked loci

Loci that are linked approach equilibrium more slowly than do loci segregating independently, and the closer the linkage(연관), the slower the approach to equilibrium.

Linkage between loci influences the proportion of gametic types produced, favoring the parental type gametes over the recombinant type gametes (see Section 4.2). The recombinant gametes reflect crossing- over(교차) between the homologous(상동) pair of chromosomes. Frequencies of gametes produced from a particular 

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population are calculated just as for independent loci in Example 6- 5, except for the treatment of contributions from the dihybrid Aa Bb. From Example 6- 5 it is evident that the dihybrid(두좌위잡종) is the only genotype in which crossing- over rates(교차율) affect the frequency of gametes produced. That is, crossing- over does not influence gamete frequencies produced by genotypes homozygous for at least one locus. When obtaining the frequencies of gametes produced by the dihybrid, the two possible types of dihybrids must be considered: AB/ab and Ab/aB. Example 6- 6 shows how these two different dihybrids contribute to the gametic pool.

Example 6- 6

Assume in Example 6- 5 that the original dihybrids were equally represented in the population such that the frequencies of joint genotypes are

As in Example 6- 5,

f(A) = .7

f(a) = .3

f(B) = .6

f(b) = .4

and the population is not in equilibrium at either locus or jointly. The gametes produced by the genotypes in this population are the same for all genotypes except the dihybrids (Aa Bb) as those in Example 6- 5:

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(see page 15)

The contributions from the dihybrids, which must take into account crossing- over, are

where r is the proportion of recombinants(재조합) and cannot exceed 0.5. Gametic frequencies in progeny are calculated exactly as in Example 6- 5 except that the dihybrids are identified.

6- 6 Summary 

Population genetics deals with gene and genotypic frequencies in populations and the prediction of these frequencies in subsequent generations. Gene frequency is the proportion of the total number of genes represented by a particular allele. If no allele is dominant, gene frequencies may be determined by simply counting alleles. Genotypic frequency is the proportion of total animals with a particular genotype.

The Hardy- Weinberg law states that gene and genotypic frequencies remain constant from one generation to the next if the following conditions are met:

1. large population,

2. random mating, and

3. the absence of any force (mutation, migration, or selection) acting to change gene frequency.

A population that meets these assumptions is in equilibrium. If f(B) = p and f(b) = q, 

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then the genotypic frequencies can be determined from the binomial expansion

For multiple alleles at a single locus, the multinomial expansion is used to obtain genotypic frequencies at equilibrium. 

Random mating is a mating system in which each individual has an equal opportunity to mate with any individual of the opposite sex. 

For populations in equilibrium, the frequency of the recessive allele, f(b), may be estimated as

A population not in equilibrium for an autosomal locus will reach equilibrium in one generation of random mating if the gene frequencies are equal in both sexes; if the gene frequencies are not equal, two generations of random mating are required.

At equilibrium, gene frequencies at sex- Iinked loci can be estimated directly from phenotypes of the heterogametic sex. If the gene frequencies are the same in both sexes, populations not in equilibrium will reach equilibrium in one generation of random mating. If the gene frequencies are different between sexes, the population needs more generations of random mating to reach equilibrium than with autosomal loci. With sex linkage in mammals the frequency of genes in male progeny is equal to the frequency in their mothers, and the frequency in female progeny is the average of frequencies in both parents. 

For two loci considered jointly, populations not in equilibrium do not reach joint equilibrium in one generation of random mating. A measure of disequilibrium when considering two loci jointly is 

d = f(AB)f(ab) -  f(Ab)f(aB)

At equilibrium, d = 0. For independent loci, d in the nth generation is

where do is the measure of disequilibrium in the initial generation. For linked loci

where r is the measure of linkage, measured in map units

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