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# Hardy-Weinberg Principle

In our articles on Genetics and Inheritance, we looked at the basic principles of how genes are passed on from generation to generation. Here, we’ll discuss what a gene pool is and explain how to calculate the frequency of an allele based on the Hardy Weinberg equation.

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Gene pool: All the alleles of the genes of all individuals in a population at a given time.

Allelic frequency: The number of times an allele appears in a population.

## What is the Hardy-Weinberg principle?

The Hardy-Weinberg principle, named after English mathematician G. H. Hardy and German doctor Wilhelm Weinburg, can calculate the frequency of an allele in a population at equilibrium. It is a null model in genetics. According to the Hardy-Weinberg principle, at equilibrium, the allele frequencies of a gene within a population will not differ from one generation to the next. Unless an allele leads to a phenotype with a significant advantage or disadvantage over other alleles, its frequency in a population is unlikely to change. A population in Hardy-Weinburg equilibrium is not evolving means it is described as ‘stable’.

The industrial revolution (transition to new manufacturing processes in Great Britain) created many changes to the environment, one of these was in moths. In the early years of the industrial revolution, the first black peppered moth appeared. Usually, peppered moths would be lightly coloured. As the industrial revolution progressed, dark moths became more and more prevalent. This phenomenon was due to directional selection. Dust, soot, and pollution filled the air in large cities, and hardly any lightly coloured moths were found. Lightly coloured moths on sooty, dirty buildings would be easily spotted and more susceptible to predation. At the level of each city, dark varieties of moths are selected - this is directional selection.

As we will see below, the Hardy-Weinberg principle provides us with a mathematical equation to calculate the expected frequency of an allele in a population.

$\phantom{\rule{0ex}{0ex}}{\mathrm{P}}^{2}+2\mathrm{PQ}+{\mathrm{Q}}^{2}=1\phantom{\rule{0ex}{0ex}}$

P: dominant homozygous frequency (AA)

2 PQ: heterozygous frequency (Aa)

: recessive homozygous frequency (aa)

1: 100% of the population

### What are the assumptions of the Hardy-Weinberg principle?

Imagine a population of diploid organisms that reproduce sexually. Let’s assume that there is no overlap between generations and that the frequencies of all alleles are equal in males and females.

There are five conditions for Hardy-Weinberg equilibrium. These are:

1. There is no selection: all alleles are equally likely to be passed on to the next generation.
2. No mutations arise.
3. There is no migration: the population is isolated.
4. The population size is infinitely large.
5. Mating is random.

Unless the population exists in a lab, finding a population that fulfils all of these criteria is improbable.

These exact conditions are unlikely to be met in natural populations; however, the Hardy-Weinberg principle provides an essential and valuable null model which we can use to study gene frequencies. In biology, null models attempt to describe what would happen in a system not influenced by any biological processes of interest (such as the conditions above!).

If the predicted frequencies do not match the observed frequencies, we can conclude that at least one of the Hardy-Weinberg equilibrium conditions has not been met. For instance, that population might be undergoing directional selection, wherein an extreme phenotype is favoured over the mean or another phenotype.

The inheritance pattern for the gene in question might also be non-Mendelian, which affects its chances of being passed on to the next generation. (For more information, check out our articles on Selection and Inheritance!)

## How can we calculate the allelic frequency?

Let’s take, for example, a population of 10,000 humans.

• The hypothetical gene B controls hair colour; the dominant allele B results in brown hair, while the recessive allele b results in blonde hair.
• Recall from the material on Genetics that every individual has two copies of this gene, and therefore, can have two of either allele (a homozygous genotype) or one of each allele (a heterozygous genotype).
• Therefore (because each individual has two copies of the gene), our population has (10,000 x 2 =) 20,000 alleles of this gene.
• If all individuals in this population had blonde hair (homozygous recessive), the probability of anyone being bb would be 1.0 and BB, 0.0. The allele b appears at a frequency of 100%, while B appears at a frequency of 0%.
 b b b bb bb b bb bb
• It gets trickier if you throw in heterozygous individuals. Take, for example, a cross of two heterozygous individuals (shown in the table below). To figure out the frequency of “b”, take the number of “b” alleles, and divide by the number of alleles for this gene. Since there are 10,000 individuals, there are a total of 20,000 alleles. Two individuals produce 4 B alleles and 4 b alleles. If there are 20,000 alleles, 10,000 would be B and 10,000 would be b, giving an allele frequency of 0.5 or 50%.
 B b B BB Bb b Bb bb

### What is the Hardy-Weinberg equation?

$\phantom{\rule{0ex}{0ex}}{\mathrm{P}}^{2}+2\mathrm{PQ}+{\mathrm{Q}}^{2}=1.0\phantom{\rule{0ex}{0ex}}$

Where p² = BB, 2pq = Bb, and b² = aa.

• Consider gene B. It has the dominant allele B and recessive allele b.
• Let the probability of allele B = p and allele b = q.
• There are only two alleles, so the probability of one plus the other must be 1.0/ Thus, p+q = 1.0

If you are told that the frequency of a dominant allele in a population is 70%, you have been given p directly, p = 0.7. Therefore q = 0.3, as p + q = 1.0

In the article on Inheritance, we learned that there are only four possible arrangements of these two alleles. It follows that all together, the probability of all four will equal 1.0.

#### How can we calculate the probability of a genotype in a population?

Now that we have the Hardy-Weinberg equation, we can calculate the frequency of an allele in a population.

Suppose that blondeness is the result of recessive allele b, and only one in 10 people display this phenotype. What is the probability that an individual in this population is a heterozygote?

1. Since this trait is recessive, it will only appear in individuals with the genotype bb.

2. Only 1 in 10 individuals have this phenotype; the probability of bb is ${q}^{2}$ = 0.10.

3. q is the square root of 0.10. q = 0.3162.

4. p + q = 1.0. Therefore, p = 1.0 - q = 1.0 - 0.3162 = 0.6838.

5. The probability of heterozygotes is 2pq. 2pq = 0. 4324

## Hardy-Weinberg Principle - Key takeaways

• The gene pool consists of all the alleles of all the genes of all the individuals in a population at a given time.

• The number of times that an allele appears in a population is known as its allelic frequency.

• The dominance of an allele has nothing to do with whether it is deleterious (harmful) or beneficial.

• According to the Hardy-Weinberg principle, the allele frequencies of a gene within a population will not change from one generation to the next.

• The Hardy-Weinberg equation is expressed as p2 + 2pq + q2 = 1.0. In biology, null models attempt to describe what would happen in a system that is not influenced by any biological processes of interest.

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What does the Hardy-Weinberg principle predict?

The Hardy-Weinberg principle predicts that at equilibrium, the allele frequencies of a gene within a population will not change from one generation to the next.

What is the Hardy-Weinberg principle used for?

It can be used to calculate the frequency of an allele in a population at equilibrium. The Hardy-Weinberg principle is also used as a null model in genetics. If the predicted allelic frequencies do not match the observed frequencies, we can conclude that at least one of the Hardy-Weinberg equilibrium conditions has not been met.

What are the conditions of the Hardy-Weinberg principle?

There are 5 conditions for Hardy-Weinberg equilibrium. These are:

-        Mating is random.

-        The population size is infinitely large.

-        There is no migration: the population is isolated.

-        There is no selection: all alleles are equally likely to be passed on to the next generation.

-        No mutations arise.

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