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If two individuals mate that are heterozygous (e.g., **Bb**) for a trait, we find that

- 25% of their offspring are homozygous for the dominant allele (
**BB**) - 50% are heterozygous like their parents (
**Bb**) - 25% are homozygous for the recessive allele (
**bb**) and thus, unlike their parents, express the recessive phenotype.

This is what Mendel found when he crossed monohybrids. It occurs because meiosis separates the two alleles of each heterozygous parent so that 50% of the gametes will carry one allele and 50% the other and when the gametes are brought together at random, each **B** (or **b**)-carrying egg will have a 1 in 2 probability of being fertilized by a sperm carrying **B** (or **b**). (Left table)

Results of random union of the two gametes produced by two individuals, each heterozygous for a given trait. As a result of meiosis, half the gametes produced by each parent with carry allele B; the other half allele b. | Results of random union of the gametes produced by an entire population with a gene pool containing 80% B and 20% b. | ||||
---|---|---|---|---|---|

0.5 B | 0.5 b | 0.8 B | 0.2 b | ||

0.5 B | 0.25 BB | 0.25 Bb | 0.8 B | 0.64 BB | 0.16 Bb |

0.5 b | 0.25 Bb | 0.25 bb | 0.2 b | 0.16 Bb | 0.04 bb |

However, the frequency of two alleles in an **entire population** of organisms is unlikely to be exactly the same. Let us take as a hypothetical case, a population of hamsters in which 80% of all the gametes in the population carry a dominant allele for black coat (**B**) and 20% carry the recessive allele for gray coat (**b**).

Random union of these gametes (right table) will produce a generation:

- 64% homozygous for
**BB**(0.8 x 0.8 = 0.64) - 32%
**Bb**heterozygotes (0.8 x 0.2 x 2 = 0.32) - 4% homozygous (
**bb**) for gray coat (0.2 x 0.2 = 0.04)

So 96% of this generation will have black coats; only 4% gray coats.

Will gray coated hamsters eventually disappear? No. Let's see why not.

- All the gametes formed by
**BB**hamsters will contain allele**B**as will one-half the gametes formed by heterozygous (**Bb**) hamsters. - So, 80% (0.64 + .5*0.32) of the pool of gametes formed by this generation with contain
**B**. - All the gametes of the gray (
**bb**) hamsters (4%) will contain**b**but one-half of the gametes of the heterozygous hamsters will as well. - So 20% (0.04 + .5*0.32) of the gametes will contain
**b**.

So we have duplicated the initial situation exactly. The proportion of allele **b** in the population has remained the same. The heterozygous hamsters ensure that each generation will contain 4% gray hamsters. Now let us look at an algebraic analysis of the same problem using the expansion of the binomial (**p**+**q**)^{2}.

[(p+q)^2 = p^2 + 2pq + q^2]

The total number of genes in a population is its *gene pool*.

- Let (p) represent the frequency of one gene in the pool and (q) the frequency of its single allele.
- So, (p + q = 1)
- (p^2) = the fraction of the population homozygous for (p)
- (q^2) = the fraction homozygous for (q)
- (2pq) = the fraction of heterozygotes

- In our example,
**p**= 0.8,**q**= 0.2, and thus [(0.8 + 0.2)^2 = (0.8)^2 + 2(0.8)(0.2) + (0.2)^2 = 064 + 0.32 + 0.04]

The algebraic method enables us to work backward as well as forward. In fact, because we chose to make **B** fully dominant, the only way that the frequency of **B** and **b** in the gene pool could be known is by determining the frequency of the recessive phenotype (gray) and computing from it the value of **q**.

**q ^{2}** = 0.04, so

**q**= 0.2, the frequency of the

**b**allele in the gene pool. Since

**p**+

**q**= 1,

**p**= 0.8 and allele

**B**makes up 80% of the gene pool. Because

**B**is completely dominant over

**b**, we cannot distinguish the

**Bb**hamsters from the

**BB**ones by their phenotype. But substituting in the middle term (

**2pq**) of the expansion gives the percentage of heterozygous hamsters.

**2pq**= (2)(0.8)(0.2) = 0.32

So, recessive genes do not tend to be lost from a population no matter how small their representation.

Hardy-Weinberg law

So long as certain conditions are met (discussed below), **gene frequencies** and **genotype ratios** in a randomly-breeding population remain constant from generation to generation. This is known as the Hardy-Weinberg law.

The Hardy-Weinberg law is named in honor of the two men who first realized the significance of the binomial expansion to population genetics and hence to evolution. Evolution involves changes in the gene pool. A population in Hardy-Weinberg equilibrium shows no change. What the law tells us is that populations are able to maintain a reservoir of variability so that if future conditions require it, the gene pool can change. If recessive alleles were continually tending to disappear, the population would soon become homozygous. Under Hardy-Weinberg conditions, genes that have no present selective value will nonetheless be retained.

## When the Hardy-Weinberg Law Fails

To see what forces lead to evolutionary change, we must examine the circumstances in which the Hardy-Weinberg law may fail to apply. There are five:

- mutation
- gene flow
- genetic drift
- nonrandom mating
- natural selection

### Mutation

The frequency of gene **B** and its allele **b** will not remain in Hardy-Weinberg equilibrium if the rate of mutation of **B** -> **b** (or vice versa) changes. By itself, this type of mutation probably plays only a minor role in evolution; the rates are simply too low. However, gene (and whole genome) duplication - a form of mutation - probably has played a major role in evolution. In any case, evolution absolutely depends on mutations because this is the only way that new alleles are created. After being shuffled in various combinations with the rest of the gene pool, these provide the raw material on which natural selection can act.

### Gene Flow

Many species are made up of local populations whose members tend to breed within the group. Each local population can develop a gene pool distinct from that of other local populations. However, members of one population may breed with occasional immigrants from an adjacent population of the same species. This can introduce new genes or alter existing gene frequencies in the residents.

In many plants and some animals, gene flow can occur not only between subpopulations of the same species but also between different (but still related) species. This is called **hybridization**. If the hybrids later breed with one of the parental types, new genes are passed into the gene pool of that parent population. This process is called **introgression**. It is simply gene flow between species rather than within them.

Comparison of the genomes of contemporary humans with the genome recovered from Neanderthal remains shows that from 1–3% of our genes were acquired by introgression following mating between members of the two populations tens of thousands of years ago.

Whether within a species or between species, gene flow increases the variability of the gene pool.

### Genetic Drift

As we have seen, interbreeding often is limited to the members of local populations. If the population is small, Hardy-Weinberg may be violated. Chance alone may eliminate certain members out of proportion to their numbers in the population. In such cases, the frequency of an allele may begin to drift toward higher or lower values. Ultimately, the allele may represent 100% of the gene pool or, just as likely, disappear from it.

Drift produces evolutionary change, but there is no guarantee that the new population will be more fit than the original one. Evolution by drift is aimless, not adaptive.

### Nonrandom Mating

One of the cornerstones of the Hardy-Weinberg equilibrium is that mating in the population must be random. If individuals (usually females) are choosy in their selection of mates, the gene frequencies may become altered. Darwin called this **sexual selection**.

Nonrandom mating seems to be quite common. Breeding territories, courtship displays, "pecking orders" can all lead to it. In each case certain individuals do not get to make their proportionate contribution to the next generation.

#### Assortative mating

Humans seldom mate at random preferring phenotypes like themselves (e.g., size, age, ethnicity). This is called *assortative mating*. Marriage between close relatives is a special case of assortative mating. The closer the kinship, the more alleles shared and the greater the degree of **inbreeding**. Inbreeding can alter the gene pool. This is because it predisposes to **homozygosity**. Potentially harmful recessive alleles — invisible in the parents — become exposed to the forces of natural selection in the children.

It turns out that many species - plants as well as animals - have mechanisms be which they avoid inbreeding. Examples:

- Link to discussion of self-incompatibility in plants.
- Male mice use olfactory cues to discriminate against close relatives when selecting mates. The preference is learned in infancy - an example of imprinting. The distinguishing odors are controlled by the MHC alleles of the mice and are detected by the vomeronasal organ (VNO).

### Natural Selection

If individuals having certain genes are better able to produce mature offspring than those without them, the frequency of those genes will increase. This is simply expressing Darwin's natural selection in terms of alterations in the gene pool. (Darwin knew nothing of genes.) Natural selection results from *differential mortality* and/or *differential fecundity*.

#### Mortality Selection

Certain genotypes are less successful than others in surviving through to the end of their reproductive period. The evolutionary impact of mortality selection can be felt anytime from the formation of a new zygote to the end (if there is one) of the organism's period of fertility. Mortality selection is simply another way of describing Darwin's criteria of fitness: **survival**.

#### Fecundity Selection

Certain phenotypes (thus genotypes) may make a disproportionate contribution to the gene pool of the next generation by producing a disproportionate number of young. Such fecundity selection is another way of describing another criterion of fitness described by Darwin: **family size**. In each of these examples of natural selection, certain phenotypes are better able than others to contribute their genes to the next generation. Thus, by Darwin's standards, they are more **fit**. The outcome is a gradual change in the gene frequencies in that population.

## Calculating the Effect of Natural Selection on Gene Frequencies

The effect of natural selection on gene frequencies can be quantified. Let us assume a population containing

- 36% homozygous dominants (
**AA**) - 48% heterozygotes (
**Aa**) and - 16% homozygous recessives (
**aa**)

The gene frequencies in this population are (p = 0.6) and (q = 0.4). The heterozygotes are just as successful at reproducing themselves as the homozygous dominants, but the homozygous recessives are only 80% as successful. That is, for every 100 **AA** (or **Aa**) individuals that reproduce successfully only 80 of the **aa** individuals succeed in doing so. The *fitness *((w)) of the recessive phenotype is thus 80% or 0.8.

Their relative disadvantage can also be expressed as a *selection coefficient*, (s), where

[s = 1 − w]

In this case,

[s = 1 − 0.8 = 0.2.]

The change in frequency of the dominant allele ((Δp)) after one generation is expressed by the equation

[Δp = dfrac{s p_0 q_0^2}{1 - s q_0^2}]

where (p_0) and (q_0) are the initial frequencies of the dominant and recessive alleles respectively. Substituting, we get

[egin{align} Δp & = dfrac{(0.2)(0.6)(0.4)^2}{1 − (0.2)(0.4)^2} [5pt] &=dfrac{0.019}{0.968} [5pt] &=0.02 end{align}]

So, in one generation, the frequency of allele **A** rises from its initial value of 0.6 to 0.62 and that of allele **a** declines from 0.4 to 0.38 ((q = 1 − p)).

The new equilibrium produces a population of

- 38.4% homozygous dominants (an increase of 2.4%) (
**p**= 0.384)^{2} - 47.1% heterozygotes (a decline of 0.9%)(
**2pq**= 0.471) and - 14.4% homozygous recessives (a decline of 1.6%)(
**q**= 0.144)^{2}

If the fitness of the homozygous recessives continues unchanged, the calculations can be reiterated for any number of generations. If you do so, you will find that although the frequency of the recessive genotype declines, the **rate** at which **a** is removed from the gene pool declines; that is, the process becomes less efficient at purging allele **a**. This is because when present in the heterozygote, **a** is protected from the effects of selection.

## Essay @ Hardy-Weinberg Law | Genetics

In this essay we will discuss about the Hardy-Weinberg law of population genetics.

The formula (p + q) 2 = p 2 + 2pq + q 2 is expressing the genotypic expectations of progeny in terms of gametic or allelic frequencies of the parental gene pool and is originally formulated by a British mathematician Hardy and a German physician Weinberg (1908) independently.

Both forwarded the idea, called Hardy-Weinberg law or equilibrium after their names. That both gene frequencies and genotype frequencies will remain constant from generation to generation in an infinitely large interbreeding population in which mating is at random and no selection, migration or mutation occur.

Should a population initially be in disequilibrium, one generation or random mating is sufficient to bring it into genetic equilibrium and thereafter the population will remain in equilibrium (unchanged in gametic and zygotic frequencies) as long as Hardy-Weinberg condition persists.

### Assumptions of Hardy-Weinberg Equilibrium:

We will consider a population of diploid, sexually reproducing organisms with a single autosomal locus segregating two alleles (i.e., every individual is one of three genotypes – MM, MN and AW).

**The following major assumptions are necessary for the Hardy-Weinberg equilibrium to hold:**

3. No Mutation or Migration, and

#### 1. Random Mating:

The first assumption of Hardy-Weinberg equilibrium is random mating which means that the probability that two genotypes will mate is the product of the frequencies (or probabilities) of the genotypes in the population.

If an MM genotypes makes up 90% of a population, then any individual has a 90% chance (probability = 0.9) of mating with a person with an MM genotype. The probability of an MM by MM mating is (0.9) (0.9), or 0.81.

Any deviation from random mating comes about for two reasons: choice or circumstance. If members of a population choose individuals of a particular phenotype as mates more or less often than at random, the population is engaged in assortative mating.

If individuals with similar phenotypes are mating more often than at random, positive assortative mating is in force if matings occur between individuals with dissimilar phenotypes more often than at random, negative assortative mating or disassortative mating is at work.

Further, deviation from random mating also arise when mating individuals are either more closely related genetically or more distantly related than individuals chosen at random from the population.

Inbreeding is the mating of related individuals, and outbreeding is the mating of genetically unrelated individuals. Inbreeding is a consequence of pedigree relatedness (e.g., cousins) and small population size.

One of the first distinct observations of population genetics is that deviation from random mating alter genotypic frequencies but not allelic frequencies. Imagine a population in which every individual is the parent of two children on the average, each individual will pass on one copy of each of his or her alleles.

Assortative mating and inbreeding will change the zygotic (genotypic) combinations from one generation to the next, but will not change which alleles are passed into the next generation. Thus, genotypic, but not allelic frequencies change under non-random mating.

#### 2. Large Population Size:

Although an extremely large number of gametes are produced in each generation, each successive generation is the result of a sampling of a relatively small portion of the gametes of the previous generation. A sample may not be an accurate representation of a population, especially if the sample is small.

Thus, the second assumption of the Hardy-Weinberg equilibrium is that the population is infinitely large. A large population produces a large sample of successful gametes. The larger the sample of successful gametes, the greater the probability that the allelic frequencies of the offspring will accurately represent the allelic frequencies in the parental population.

When populations are small or when alleles are rare, changes in allelic frequencies take place due to chance alone. These changes are referred to as random genetic drift or just genetic drift.

#### 3. No Mutation or Migration:

Allelic and genotypic frequencies may change through the loss or addition of alleles through mutation or migration (immigration or emigration) of individuals from or into a population, ne third and fourth assumptions of the Hardy-Weinberg equilibrium are that neither mutation nor migration causes such allelic loss or addition in the population.

#### 4. No Natural Selection:

The final assumption necessary to the Hardy-Weinberg equilibrium is that no individual will have a reproductive advantage over another individual because of its genotype. In other words, no natural selection in occurring. (Note. Artificial selection, as practised by animal and plant breeders, will also perturb the Hardy-Weinberg equilibrium of captive population).

The significance of Hardy-Weinberg equilibrium was not immediately appreciated. A rebirth of biometrical genetics was later brought about with the classical papers of R.A. Fisher, beginning in 1918 and those of Sewall Wright, beginning in 1920.

Under the leadership of these mathematicians, emphasis was placed on the population rather than on the individual or family group, which had previously occupied the attention of most Mendelian geneticists. In about 1935, T. Dobzhansky and others started to interpret and to popularize the mathematical approach for studies of genetics and evolution.

### Genetic Equilibrium:

As shown by Hardy and Weinberg, alleles segregating in a population tend to establish an equilibrium with reference to each other. Thus, if two alleles should occur in equal proportion in a large, isolated breeding population and neither had a selective or mutational advantage over the other, they would be expected to remain in equal proportion generation after generation. This would be a special case because alleles in natural populations seldom if ever, occur in equal frequency.

They may, however, be expected to maintain their relative frequency, whatever it is, subject only to such factors as chance, natural selection, differential mutation rates or mutation pressure, meiotic drive and migration pressure, all of which alter the level of the allele frequencies. A genetic equilibrium is maintained through random mating.

### Youreka Science

Youreka Science was created by Florie Mar, PhD, while she was a cancer researcher at UCSF. While teaching 5th graders about the structure of a cell, Mar realized the importance of incorporating scientific findings into classroom in an easy-to-understand way. From that she started creating whiteboard drawings that explained recent papers in the scientific literature… Continue Reading

## Hardy-Weinberg Law With Its Applications | Genetics

In this article we will discuss about the Hardy-Weinberg law with its applications.

In 1908, the mathematician G. H. Hardy in England and the physician W. Weinberg in Germany independently developed a quantitative theory for defining the genetic structure of populations. The Hardy-Weinberg Law provides a basic algebraic formula for describing the expected frequencies of various genotypes in a population.

The similarity of their work however, remained unnoticed until Stern (1943) drew attention to both papers and recommended that names of both discoverers be attached to the population formula. The Law states that gene frequencies in a population remain constant from generation to generation if no evolutionary processes like migration, mutation, selection and drift are operating.

Thus if matings are random, and no other factors disturb the reproductive abilities of any genotype, the equilibrium genotypic frequencies are given by the square of the allelic frequencies.

**If there are only two alleles A and a with frequencies p and q respectively, the frequencies of the three possible genotypes are:**

If there are 3 alleles say A1, A2 and A3 with frequencies p, q and r, the genotypic frequencies would be

(p + q + r) 2 = p 2 + q 2 + r 2 + 2pq + 2pr + 2qr

This square expansion can be used to obtain the equilibrium genotypic frequencies for any number of alleles.

It must also be noted that the sum of all the allelic frequencies, and of all the genotypic frequencies must always be 1. If there are only two alleles p and q, then p + q = 1, and therefore p 2 + 2pq + q 2 = (p + q) 2 = 1. If there are 3 alleles with frequencies p, q, and r, then p + q + r = 1, as well as (p + q + r) 2 = 1.

The time required for attaining equilibrium frequencies has been determined. If a certain population of individuals with one set of allele frequencies mixes with another set and complete panmixis occurs (that is, random mating), then the genotypes of the next generation will be found in the proportion p 2 + 2pq + q 2 where p and q are allele frequencies in the new mixed populations.

Thus it takes only one generation to reach Hardy-Weinberg equilibrium provided the allelic frequencies are the same in males and females. If the allelic frequencies are different in the two sexes, then they will become the same in one generation in the case of alleles on autosomes, and genotypic frequencies will reach equilibrium in two generations.

In general equilibrium is arrived at within one or at the most a few generations. Once equilibrium is attained it will be repeated in each subsequent generation with the same frequencies of alleles and of genotypes.

The Hardy-Weinberg law is applicable when there is random mating. Random mating occurs in a population when the probability of mating between individuals is independent of their genetic constitution. Such a population is said to be panmictic or to undergo panmixis. The matings between the genotypes occur according to the proportions in which the genotypes are present.

The probability of a given type of mating can be found out by multiplying the frequencies of the two genotypes that are involved in the mating. Matings are not random for instance when a population consists of different races such as blacks and whites in the U.S., or different communities as in India as there are preferred matings between members of the same racial or communal group.

#### Applications of the Hardy-Weinberg Law:

**(a)** **Complete Dominance:**

When Hardy-Weinberg equilibrium exists, allele frequencies can even be found out in presence of complete dominance where two genotypes cannot be distinguished. If two genotypes AA and Aa have the same phenotype due to complete dominance of A over a the allele frequencies can be determined from the frequencies of individuals showing the recessive phenotype aa.

The frequency of aa individuals must be equal to the square of the frequency of the recessive allele q. Let us suppose q = 0.5, then q 2 – (0.5) 2 = 0.25. In other words when aa phenotype is 0.25 in the population, then it follows that the frequency of the recessive allele a is √0.25 – 0.5. The frequency of the dominant allele A would be 1 – q or 1 – 0.25 = 0.75.

**(b)** **Frequencies of Harmful Recessive Alleles:**

The Hardy-Weinberg Law can also be used to calculate the frequency of heterozygous carriers of harmful recessive genes. If there are two alleles A and a at an autosomal locus with frequencies p and q in the population and p + q = 1, then the frequency of AA, Aa, and aa genotypes would be p 2 + 2pq + p 2 .

If the aa genotype expresses a harmful phenotype such as cystic fibrosis, then the proportion of affected individuals in the population would be q 2 , and the frequency of the heterozygous carriers of the recessive allele would be 2pq.

To illustrate with figures, suppose one out of 1,000 children is affected with cystic fibrosis, then the frequency q 2 = 0.001, so that q = √0.001 which is about 0.032, then 2pq = 2 x 0.032 x 0.968 = 0.062. This means that about 62 individuals out of 1000 or one out of 16 is a carrier of the allele for cystic fibrosis.

As already mentioned the number of individuals (aa) who are actually affected is one out of 1000. This implies that the frequency of heterozygous carriers is much higher than that of affected homozygotes.

Similar calculation shows that when an allele is very rare in the population the proportion of carriers is still much higher and of affected homozygotes much lower. Thus, lower the frequency of an allele, greater the proportion of that allele that exists in the heterozygotes.

**(c)** **Multiple Alleles:**

The Hardy-Weinberg Law permits calculation of genotypic frequencies at loci with more than two alleles, such as the ABO blood groups. There are 3 alleles I A , I B and I° with frequencies p, q and r. Here p + q + r = 1. The genotypes of a population with random mating would be (p + q + r) 2 .

**(d)** **Sex-Linked Loci:**

It is possible to apply Hardy-Weinberg Law for calculating gene frequencies in case of sex-linked loci in males and females. Red green color blindness is a sex- linked recessive trait. Let r denote the recessive allele which produces affected individuals, and R the normal allele. The frequency of R is p and of r is q where p + q = 1. The frequencies of females having RR, Rr, rr genotypes would be p 2 , 2pq, q 2 respectively.

Males are different as they are hemizygous, have only one X chromosome derived from the mother with a single allele either R or r. The frequency of affected r males would be the same as the frequency of the r allele among the eggs that is q. The frequency of normal R males would be p. Suppose the frequency of r alleles is 0.08, then the incidence of affected males would be 0.08 or about 8%.

The frequency of affected rr females would be (0.08) 2 = 0.0064 or 0.64%. Thus the Hardy-Weinberg Law explains that males would be affected a hundred times more frequently than females. This is actually what is observed. Males are more affected by sex-linked recessive traits than females.

The difference between the sexes is even more pronounced if the recessive allele is still more rare. The incidence of a common form of haemophilia is one in a thousand males thus q = 0.001. However, only one in 1000,000 females will be affected. Thus males could have haemophilia one thousand times more often than females.

**(e)** **Linkage Disequilibrium:**

Consider two or more alleles at one locus and another locus on the same chromosome with two or more alleles. Due to genetic exchange by recombination occurring regularly over a period of time, the frequencies of the allelic combinations at the two syntenic loci will reach equilibrium.

If equilibrium is not reached, the alleles are said to be in linkage disequilibrium. The effect is due to tendency of two or more linked alleles to be inherited together more often than expected. Such groups of genes have also been referred to as supergenes.

## Departure from Hardy Weinberg Equilibrium and Genotyping Error

**Objective:** Departure from Hardy Weinberg Equilibrium (HWE) may occur due to a variety of causes, including purifying selection, inbreeding, population substructure, copy number variation or genotyping error. We searched for specific characteristics of HWE-departure due to genotyping error. **Methods:** Genotypes of a random set of genetic variants were obtained from the Exome Aggregation Consortium (ExAC) database. Variants with <80% successful genotypes or with minor allele frequency (MAF) <1% were excluded. HWE-departure (d-HWE) was considered significant at *p* < 10E-05 and classified as d-HWE with loss of heterozygosity (LoH d-HWE) or d-HWE with excess heterozygosity (gain of heterozygosity: GoH d-HWE). Missing genotypes, variant type (single nucleotide polymorphism (SNP) vs. insertion/deletion) MAF, standard deviation (SD) of MAF across populations (MAF-SD) and copy number variation were evaluated for association with HWE-departure. **Results:** The study sample comprised 3,204 genotype distributions. HWE-departure was observed in 134 variants: LoH d-HWE in 41 (1.3%), GoH d-HWE in 93 (2.9%) variants. LoH d-HWE was more likely in variants located within deletion polymorphisms (*p* < 0.001) and in variants with higher MAF-SD (*p* = 0.0077). GoH d-HWE was associated with low genotyping rate, with variants of insertion/deletion type and with high MAF (all at *p* < 0.001). In a sub-sample of 2,196 variants with genotyping rate >98%, LoH d-HWE was found in 29 (1.3%) variants, but no GoH d-HWE was detected. The findings of the non-random distribution of HWE-violating SNPs along the chromosome, the association with common deletion polymorphisms and indel-variant type, and the finding of excess heterozygotes in genomic regions that are prone to cross-hybridization were confirmed in a large sample of short variants from the 1,000 Genomes Project. **Conclusions:** We differentiated between two types of HWE-departure. GoH d-HWE was suggestive for genotyping error. LoH d-HWE, on the contrary, pointed to natural variabilities such as population substructure or common deletion polymorphisms.

**Keywords:** Hardy Weinberg Equilibrium (HWE) SNP quality control association studies in genetics heterozygosity.

### Figures

Analysis of genotyped short variations…

Analysis of genotyped short variations in genomic region 17:56,000,000–64,000,000 **(A,B)** and genomic region…

## Hardy &ndash Weinberg’s Principle (With 5 Assumptions) | Genetics

In this article we will discuss about the principle of Hardy and Weinberg which requires five assumptions for explaining the equilibrium state of gene and genotype frequency.

It was the year 1908, when an English mathematician — G. H. Hardy — and a German physician, W. Weinberg independently discovered the principle concerned with the frequency of alleles in a population, which is now known as Hardy-Weinberg equilibrium principle.

This principle states that genotypes in a Mendelian population tend to establish an equilibrium with reference to each other and, at equilibrium, both allele and genotype frequencies remain constant from generation to generation. This equilibrium state occurs among diploid, sexually reproducing organisms with non-overlapping generation and in large, random, panmictic populations where no selection or other factors are present.

**Therefore, the principle of Hardy and Weinberg requires 5 assumptions for explaining the equilibrium state of gene and genotype frequency, which are:**

(a) Individuals of each genotype must be as reproductively fit as those of any other genotype in the population

(b) The population must consist of an infinitely large number of individuals

(c) Random mating must occur throughout the population

(d) Individuals must not migrate into or out of the population

(e) There must be mutation equilibrium.

Before going into the details of Hardy- Weinberg principle, first we consider the Mendelian principle of heredity in mathematical terms. The principle of segregation can be represented by the binomial expansion of (a+b) n : where “a” is the probability that an event will occur and “b” is the probability that it will not occur.

The segregation of a single pair of alleles (A_{a}) in a monohybrid cross may also be represented by the simple expansion of (a+b) n = (A + a) 2 = 1AA + 2Aa + 1aa. Now, if we consider the frequency of these two alleles (A and a) in a population is p and q, respectively, then at equilibrium the frequencies of each genotype class is p 2 (AA), 2pq(A_{a}), and q 2 (aa). Frequency means the ratio of the actual number of individuals falling in single class to the total number of individuals. The genetic proportions of an equilibrium population are entirely determined by its allele frequencies.

**Following is the algebraic proof of genetic equilibrium for any two alleles in a population:**

So, when two alleles are involved, the p + q = 1, and, if we deduce it mathematically:

Now, if 1 – q is substituted by ‘p’ then all the relationship of the formula can be represented in terms of q which is:

Therefore, if an allele ‘A’ has a frequency of 1-q and another allele ‘a’ has a frequency of q then the expected distribution of these alleles under panmictic conditions in succeeding generations may be calculated.

In this regard it should be remembered that dominance and recessiveness of the alleles do not directly influence allele frequency and dominance alone does not make an allele occur more frequently in the population. If some phenotype has a selective advantage over another, dominance could indirectly influence allele frequency.

The above-mentioned equation p + q = 1 applies when only two autosomal alleles in a population occur at a given locus, but if the system includes more alleles, more symbols must be added to the equation.

**For example, in case of AB blood group which is controlled by three alleles — namely I A , I B and i — the equation would be:**

or, p 2 + q 2 + r 2 + 2pq + 2pr + 2qr (sum of the genotype frequency).

Now, if we consider the alleles which are present in the sex chromosomes then it shows some different frequency from that discussed above. This difference in frequency of alleles in sex chromosome is due to the arrangements of sex chromosomes in the two sexes.

**For example, if we consider two alleles (Aa) are present in X-chromosome then the genotypic values at equilibrium will be:**

(a) For females: p 2 + 2pq + q 2 i.e. AA + 2Aa + aa

(Due to the double dose of X-chromosome)

(Due to the single dose of X-chromo- some)

Sex-linked (X-linked) genes (alleles) shows criss-cross pattern because in human the X- chromosome is transmitted from a father through his daughters) to half of her sons (her father’s grandsons). This criss-cross mechanism of inheritance indicates that the allele frequency in males in any generation equals the frequency in females in the previous generation i.e. in mathematical term:

If m_{n} and f_{n} represent the frequency of an allele in males and females, respectively, in generation n, then

m_{n} = f_{n-1}(because a male receives his X-chromosome from “his mother)

The frequency in females in any generation equals the average allele frequency to the previous generation, because females receive one X-chromosome from each parent, so:

It is interesting to note that naturally- occurring human populations are not genotypically in equilibrium state i.e. human population does not always follow the Hardy- Weinberg principle.

If we consider the protein and enzyme polymorphism in human population then it would be clear to us that the degree of allelic variation may occur at different gene loci and although multiple allelism may be demonstrable there is one allele that can be regarded as the standard or normal form which is almost universally present, while others are extremely rare. This type of change in the allele frequency may ultimately affect the population causing population change or population dynamics.