Contents

- 1 Hardy-Weinberg Equilibrium in the Large-Scale Genomic Sequencing Era
- 1.1 Salient Features of Hardy Weinberg’s Principle
- 1.2 Genotype Equilibrium
- 1.3 Application of Hardy Weinberg Principle in Natural Population
- 1.4 Assessment of Allelic Frequencies in Population
- 1.5 Significance of Hardy Weinberg’s Principle
- 1.6 Importance of Random Mating
- 1.7 Hardy Weinberg Principle and Evolution

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## Hardy-Weinberg Equilibrium in the Large-Scale Genomic Sequencing Era

After, the rediscovery of Mendel in 1900 by Correns, Tschermack, and de Vries investigators tried to observe Mendelian inheritance in man as well as other organisms. According to Mendelian principles of inheritance, the dominant and recessive allele in the population should exist in 3 : 1 proportion. With the help of a mathematical model Hardy and Weinberg, provided a clue for estimating allele frequency in the population.

The fundamental idea of population genetics was offered by G. H. Hardy, a British mathematician, and W. Weinberg, a German Physician-independently in 1908. It is known as Hardy-Weinberg’s law. The Hardy-Weinberg’s law is divided into two parts:

- In an infinitely large randomly mating population, free from mutation, migration, and natural selection the allelic frequency will not change from one generation to the next.
- As long as mating is random, the genotype frequency will not change but remain in the proportion p
^{2}(frequency of ‘AA’), 2pq (frequency of ‘Aa’), and q^{2}(frequency of ‘aa’) where p is the allele frequency of ‘A’ and q is the allelic frequency of ‘a’.

### Salient Features of Hardy Weinberg’s Principle

According to Hardy Weinberg’s Principle, the gene and genotype frequency of each allele in a population remains at an equilibrium generation after generation if that population exhibits the following attributes:

**1. Random Mating:**

Each individual in the population has an equal opportunity of mating with any other individual in that population.

**2. Large Population Size:**

The population is sufficiently large so that the frequencies of alleles do not change from generation to generation. In a small population, there will be significant random fluctuation in allele frequencies due to sampling error.

**3. Absence of Evolutionary Force:**

- No mutation: A mutation in a gene or chromosome may create a new Slide or convert one allele to another, thereby modifying the gene pool.
- No migration: Allelic and genotype frequencies may change through the loss or addition of alleles through migration, (immigration and emigration).
- No Natural Selection: That is individuals of all genotypes have an equal rate of survival and equal reproductive success.

With the help of a mathematical model, they could be able to establish the validity of their proposition. Let there be two alleles of a gene in the population namely, A and a determining certain character. Among these two alleles, A is dominant over a and their frequencies say are p and q respectively. Suppose, in the initial population we get the idea of allelic frequency distribution without knowing the size of the population and the frequency of different genotypes. Then, the individuals of the population both male and female may produce two types of gametes having A and A respectively with the frequency p and q only. The union of male and female gametes in the case of random mating in this population may produce three genotypic populations with AA, Aa, and aa genotypes. In the next generation population structure with the corresponding frequencies may be indicated by the following checkerboard.

From this, we get the different genotypes with the frequencies as AA : p^{2}, Aa : 2pq and aa : q^{2}

From the above distribution of the genotypes, the allelic frequencies may be calculated in the following manner:

A = p^{2} + \(\frac{1}{2}\) × 2pq

= p^{2} + pq

= p(p + q)

= p (∵ p + q = 1)

and a = q^{2} + \(\frac{1}{2}\) × 2pq

= q^{2} + pq

= q(p + q)

= q (∵ p + q = 1)

It appears, from the above that the frequencies of the two alleles A and a remain unchanged as p and q respectively as before. Further, random mating in the new population may also be shown with the help of a checkerboard as under.

The frequency distribution of different genotypes available from the matings as shown in the checkerboard above may be represented as under.

Therefore, the total frequency of the genotype AA = p^{4} + 2p^{3}q + p^{2}q^{2}

= p^{2} (p^{2} + 2pq + q^{2})

= p^{2} (p + q)^{2}

= p^{2} (∵ p + q = 1)

and, the total frequency of the genotype Aa = 2p^{3}q + 4p^{2}q^{2} + 2pq^{3}

= 2pq (p^{2} + 2pq + q^{2})

= 2pq (p +q)^{2}

= 2pq (∵ p + q = 1)

and, total frequency of the genotype aa = p^{2}q^{2} + 2pq^{3} + q^{4}

= q^{2} (p^{2} + 2pq + q^{2})

= q^{2} (p + q)^{2}

= q^{2} (∵ p + q = 1)

From these, again the determination of the allelic frequencies may be carried out in the following manner.

The frequency of A gene in the second generation = p^{2} + \(\frac{1}{2}\) × 2pq

= p^{2} + pq

= p(p + q)

= p

and, the frequency of a gene in the second generation = q^{2} + \(\frac{1}{2}\) × 2pq

= q^{2} + pq

= q(p + q)

= q

The above derivation supports the validity of Hardy Weinberg’s principle exhibiting no change in allelic frequencies in successive generations in the case of random mating.

### Genotype Equilibrium

The genetic equilibrium may be defined as, “The relative frequencies of various kinds of genes in a large and randomly mating sexual population tend to remain constant from generation to generation in the absence of mutation, selection, and gene flow”. This is called Hardy Weinberg’s law of equilibrium.

The population in which the frequency of the alleles of a certain gene remains constant from generation to generation shows the state of genetic equilibrium. This equilibrium condition is expressed through genotypic frequency distribution. From the above derivation, the state of equilibrium for two alleles may be understood. The alleles A and a when remain with the frequency p and q, three possible genotypes from them show frequency distribution as p^{2} for AA, 2pq for Aa, and q^{2} for aa and this distribution pattern comes from the binomial expansion of (p + q)^{2}. When the frequency of the three possible genotypes for two alleles exhibit a frequency distribution in the pattern p^{2}, 2pq, and q^{2}, it is called the equilibrium condition for the three genotypes in the population, and such a population is said to be at Hardy Weinberg equilibrium.

### Application of Hardy Weinberg Principle in Natural Population

**1. Recessive Allele:**

Provided all the conditions are fulfilled as per the suggestion of Hardy and Weinberg, Hardy Weinberg’s principle will be applicable to the natural population. Some examples from a human population can be cited to evaluate the validity of Hardy Weinberg’s principle in the natural population. Albinism appears in the human population as an effect of the mutation of a normal gene. The albino individuals are homozygous for a recessive gene that appears as a mutation of its normal allele. If the gene for albinism is considered as a, then its wildtype allele may be denoted as A which is dominant over a. Therefore, individuals having the genotype AA and Aa are normal by phenotype and capable of producing melanin in the skin. The albino individuals lack the power to produce melanin and therefore, their skin is pigmentless. Such individuals face many inconveniences and bright sunlight is detrimental to their bodies.

In the human population albinism appears as 1 among 20000 individuals. From this, we may easily calculate the frequency of this deleterious gene present in the population. Say, the frequency of the gene is q then the frequency of the albino in the population may be represented by q^{2}. Hence, q^{2} = \(\frac{1}{2000}\) or, q^{2} = 0.00005. Then, q = \(\sqrt{0.00005}\) = 0.007 approximately and the frequency of A gene (p) comes to = 1 – 0.007 = 0.993. Because of the deleterious nature, the recessive allele a may be eliminated from the population by natural way of negative selection. In spite of this, the population will not be able to eliminate this detrimental gene fully because in the heterozygote (Aa) the gene remains in suppressed condition. The frequency of the heterozygote in such a population (in case of genetic equilibrium) will be equal to 2pq or 2 × 0.993 × 0.007 = 0.014 (approx.). In other words, the carrier of this deleterious gene in the population comes to about 0.014/0.00005 = 280 times more than the affected individual in the population.

Cystic fibrosis is another serious genetic disorder in man and this occurs due to a recessive gene. The affected individuals among the newborn babies appear to be about 1 among 1700. A calculation of the frequency of the genes and genotype (as shown in the case of albinism) may indicate that 1 among 21 individuals appears to be a carrier for the gene for cystic fibrosis. In the human population, the determination of the gene frequencies as well as genotypic frequencies may help us to assess the future of our life in the population.

**2. Multiple Allele:**

Hardy Weinberg formula will also be applicable in the case of multiple alleles when there are more than two allelic forms for a gene. In the case of the ABO blood group in man there are three alleles namely I^{A}, I^{B} and i controlling the expression of four ABO phenotypes as A, B, AB, and O. Considering three alleles for gene and their frequency as p of I^{A}, q for I^{B} and r for i, the genotypic expression in the population may be expressed through expansion of (p + q + r)^{2} as p^{2} + q^{2} + r^{2} + 2pq + 2pr + 2qr. p^{2}, q^{2}, and r^{2} represent the frequency of the homozygotes (I^{A}I^{A}, I^{B}I^{B}, and ii) and 2pq, 2pr, and 2qr the frequencies of the heterozygotes (I^{A}I^{B}, I^{A}i, and I^{B}i) in the population. The additive frequency of I^{A}, I^{B}, and i (i.e., p + q + r) is always 1, and hence p + q + r = 1. As usual (p + q + r)^{2} or p^{2} + q^{2} + r^{2} + 2pq + 2pr + 2qr = 1. To calculate the frequency of individual genes in the population, the following formulae may be applied.

Frequency of I^{A} = p^{2} + \(\frac{1}{2}\) × 2pq + \(\frac{1}{2}\) × 2pr

Frequency of I^{B} = q^{2} + \(\frac{1}{2}\) × 2pq + \(\frac{1}{2}\) × 2qr

and Frequency of i = r^{2} + \(\frac{1}{2}\) × 2pr + \(\frac{1}{2}\) × 2qr

The frequency of the A blood group phenotype of a population may be represented by p2^{2} + 2pr, that for the B phenotype by q^{2} + 2qr, and the O phenotype by r^{2}. If the number of O group individuals in the population is known, then from that number r^{2} value may easily be determined and so r will be the square root of that value. When r is obtained, from that other frequencies of the genes (I^{A} and I^{B}) may be derived easily.

**3. Sex Linked Alleles:**

In the human population, sex-linked alleles remain distributed in a unique fashion. Because males in the population contain only one X chromosome, while females contain two X-chromosomes. Therefore, for two alleles of a gene (sex-linked), when females show three genotypes (2 for homozygotes and 1 for heterozygotes), the males contain only 2 genotypes lacking a heterozygous condition. In this case for two sex-linked alleles A and a with frequencies p and q respectively, the genotypic frequencies in the females of the population become p^{2} for AA, q^{2} for aa, and 2pq for Aa. While the genotypes in the male population become A and a with frequencies p and q respectively. To determine the frequency of individual genes from the total population the following formulae may be applied.

Frequency of A = \(\frac{2 \mathrm{pF}+\mathrm{pM}}{3}\) (pF = Frequency of A in females and pM = Frequency of A in males) and Frequency of a = \(\frac{2 \mathrm{qF}+\mathrm{qM}}{3}\) (qF = Frequency 0f a in females and qM = Frequency of a in males)

When the frequency of sex-linked alleles remains the same both in the male and female population, it denotes the equilibrium state of the population.

### Assessment of Allelic Frequencies in Population

Based on Hardy Weinberg’s principle, the allelic frequency of the genes in a population may easily be calculated. The frequency of alleles may help to determine the genotypic frequency at equilibrium. If the frequencies do not alter in a population as per the concepts in the Hardy-Weinberg principle, a population may be called stable and such stability does not allow evolution to occur but in actual situation population is dynamic and it faces a huge amount of pressure that leads to change in gene frequency. Various sorts of phenomena are involved in changing the allelic frequencies in a population and these are mutation, selection, genetic drift, migration, etc. Therefore, the stability of a population in consideration of its genetic makeup is an abstract idea.

However, the allelic frequency indicates the momentary state of a population and that may assess a population to understand the magnitude of the expression of a particular gene as a temporary assumption. As evolution is a long-standing event whose impact may only be realized after thousands or billions of years, so assessment of the genotypic and phenotypic frequencies for the alleles in a population appear to be of much importance for the evaluation of the detrimental condition.

On the basis of the allelic frequencies, the heterozygote frequency in a population may help us to realize the presence of a deleterious recessive gene within the carrier. The carriers on the other hand promote the transmission of a deleterious gene to the next generation. The frequency of the heterozygotes appears to be more when the allelic frequencies remain between 0.33 and 0.66 and that comes to a maximum when each of the allelic frequencies (in the case of a pair of alleles) becomes 0.5. When the population is at Hardy Weinberg equilibrium, the heterozygote frequency cannot exceed 0.5.

When the frequency of one allele is low enough, the homozygote frequency for that allele will be very low and the frequency of the homozygotes for that allele in the population will be rare. However, the same allele may remain in the heterozygote in the population in suppressed condition. Considering the frequency of a deleterious allele in a population as 0.2 we may have 0.04 or 4% individuals as affected by that deleterious gene in a population. Suppose the population is at equilibrium, in that case, the frequency of the normal allele will be 0.8 giving a heterozygote population 32% (2pq = 2 × 0.8 × 0.2 or 0.32) and a normal individual homozygote will be represented by 64%. In comparison to the affected individuals in the population, the heterozygotes will be 0.32/0.04 or 0.8 or 80% more.

### Significance of Hardy Weinberg’s Principle

- Hardy Weinberg’s principle provides a theoretical baseline for measuring evolutionary changes.
- It helps to determine whether a population is in equilibrium or not. However, no natural population is in complete equilibrium.
- It shows that dominant traits do not necessarily increase from one generation to the next.
- Equilibrium maintains heterozygosity in the population.
- Equilibrium prevents evolutionary progress.

### Importance of Random Mating

The validity of Hardy Weinberg’s principle is greatly dependent on random mating which permits logical and even distribution of the alleles in a population. According to the principle of panmixis, there will be no choice of selection of partner by an individual in a population for mating. This sort of practice by the male and female individuals in the population promotes the establishment of genotypic equilibrium. In the case of autosomal genes a population remaining at disequilibrium, may attain equilibrium through random mating and only one generation of random mating may help in attaining equilibrium. With an example, the validity of random mating may be presented.

In the human population tasting of phenylthiocarbamide (PTC) is determined by a single gene, say T. A mutation of this gene which is recessive in nature confers the non-tasting property of this chemical to an individual. Therefore, in the human population two groups of individuals may be available, one group is capable of tasting PTC which tastes bitter to the individuals and the other group appears non-tasters to whom the chemical is tasteless. Therefore, the non-tasters are homozygous recessive in nature (Because T is dominant over t). For these pair of alleles, T and t in the population there may be three possible genotypes TT, Tt, and tt. Out of these three genotypes, TT and Tt are tasters and those with tt are non-tasters. Suppose in a population the frequencies of the three genotypes appear as 0.30 TT, 0.60 Tt, and 0.10 tt. Then, the frequency of the two alleles in the population will be as under.

- Frequency of T(p) = 0.30 + \(\frac{1}{2}\) × 0.60 = 0.30 + 0.30 = 0.60
- Frequency of t(q) = 0.10 + \(\frac{1}{2}\) × 0.60 = 0.10 + 0.30 = 0.40

With this frequency distribution of the alleles frequencies of the three genotypes at equilibrium should be p^{2} for TT = (0.6)^{2} = 0.36, and 2pq for Tt = 2 × 0.6 × 0.4 = 0.48 and q^{2} for tt = (0.4)^{2} = 0.16. But the given population with the frequency distribution for the three genotypes 0.30 (TT), 0.60 (Tt), and 0.10 (tt) respectively does not fit with the frequencies at equilibrium. Hence, the given population may be stated to be in disequilibrium. Random mating in this population may be shown in the following checkerboard in which a clear alteration of the genotypic frequencies may be noted. Distribution of the genotypic frequencies as obtained from different mating types may be shown as under.

From the above analysis, it appears that following random mating for a generation the frequencies of three genotypes TT, Tt, and tt appear as 0.36, 0.48, and 0.16 respectively which is equal to the genotypic frequencies at equilibrium as calculated before. This means that a population at disequilibrium may attain equilibrium after only one generation of random mating. This also ensures that the allelic frequencies remain unaltered in the successive generations though variation in the genotypic frequencies may be noted in two succeeding generations. Hence equilibrium in the genotypic frequency distribution does not depend on the genotypic frequencies of the previous generation, rather it depends on the allelic frequencies of the previous generation.

Though random mating permits a population at disequilibrium to achieve equilibrium by one generation of random mating this is not true for the sex-linked alleles. This is due to differential patterns of genotypic distribution in the male and female populations. Say, one X-linked gene is X^{A} and its recessive allele is X^{a} with the frequencies as p for X^{A} and q for X^{a} in both populations. From these we get the female population having genotypic frequencies as p^{2} : X^{A}X^{A}, 2pq : X^{A}X^{a}, and q^{2} : X^{a}X^{a} and that in the male population, the same remains as p : X^{A} and q : X^{a}. From this proportion, we may calculate the individual allelic frequency for the whole population as, p[f(XA)] = \(\frac{2 \mathrm{pF}+\mathrm{pM}}{3}\) and q[f(Xa)] = \(\frac{2 \mathrm{qF}+\mathrm{qM}}{3}\)

Suppose, in a population, the frequencies of a sex-linked gene X^{a} in males and females appear as 0.2 and 0.8 respectively. Then, the frequency of this gene in the whole population at equilibrium should be as under.

Frequency of X^{a}(q) = \(\frac{2 \mathrm{qF}+\mathrm{qM}}{3}\)

= \(\frac{2 \times 0.2+0.8}{3}\)

= \(\frac{0.4+0.8}{3}\)

= 0.4

Hence, the frequency of other allele X^{A} will be 1 – 0.4 = 0.6. Under this condition, the given population does not show an equilibrium condition for this sex-linked allelic distribution. However, random mating of this population may only permit the attainment of equilibrium. But only one generation of random mating will not be sufficient for promoting equilibrium, rather it needs random mating for several successive generations in order to achieve equilibrium. This may be shown graphically in the following diagram.

The females of the next generation inherit one X chromosome from the male parent and the other X chromosome from the female parent. Therefore the frequency of the gene in the female progeny population will be the average of the frequencies in male and female populations considered separately. Taking the example of a given population in which the frequency of the gene X^{a} appears 0.8 in females and 0.2 in males. It becomes \(\frac{0.8+0.2}{2}\) = 0.5 in the females of the next generation (as shown in the diagram). On the other hand, the male progeny inherits the X chromosome only from the parental female or mother. The frequency of the gene in females of the initial generation becomes the frequency of the gene in the males of the next generation.

Graphical representation of the frequency distribution of a sex-linked allele having differential value at the initial stage. Following random mating in successive generations, the difference between the values in the male and female population gradually diminishes, ultimately achieving a stable condition, when the frequency of the gene both in the male and female population becomes the same. The frequency of the gene in the male population is shown by a dotted line and the frequency of the same gene in the female population is shown by a continuous line.

### Hardy Weinberg Principle and Evolution

Hardy Weinberg’s principle gives a model of a population in consideration to its genetic structure and it reflects an equilibrium condition of the population when a species of population does not face evolution. But such a situation is quite abstract and appears to be a theoretical model. In nature, there are various forces that cause evolution to be propagated in the population. As per the modern concept, population is the unit of evolution, and evolution in a population may be measured by its gene frequencies. As long as gene frequencies remain constant in nature, a genotypic equilibrium will be maintained in the population when no evolution is possible.

However, evolutionary forces like mutation, selection, genetic drift, non-random mating, migration, recombination, etc. are in force on the population causing changes in gene frequencies in the population and as a result, evolution has become a continuous process. Variation is the ultimate source of evolution and the evolutionary forces as mentioned above cause variations to appear in the members of a population. When genetic equilibrium in a population will be in jeopardy, the population is to face evolution inevitably. In this consideration, it may be surmised that various ways that may promote change of gene frequencies in a population are the mechanisms by which evolution may occur. Therefore, the mechanisms involved in the causation of evolution may be represented in the following table.

**Mechanisms Involved in Evolution:**

Mechanism |
Occurrence |
Results |
Inference |

1. Mutation | Formation of a new allele of a gene through alteration | Origin of new character | It develops variation among individuals in a population |

2. Recombination | Reorganization or rearrangement of genes through crossing over | Formation of a new combination of characters | It develops variations among related members in a population |

3. Natural Selection | Detection of adaptability of an allele in the environment | Stability of an allele or its removal | It promotes change in gene frequency |

4. Genetic Drift | Sampling Error | Chance-dependent removal of an allele | It also promotes a change in gene frequency |

5. Gene Flow | Transfer of some genes from the main population to a different population | Depletion of a gene frequency in one population and increase of the same in other | It causes disequilibrium of H-W. genotypic equilibrium |

6. Non-random mating | Selective transfer | Increase in homozygosity | The population faces detrimental effects of genetic load |