Hardy-Weinberg Law - Notes
The Hardy-Weinberg Law (also Hardy Weinberg Equilibrium or Principle) states that the allele frequency (and genotype frequency) of a population remains constant over generations, unless a specific factor or combination of factors disrupts this equilibrium. Such factors might include non-random mating, mutation, natural selection, genetic bottlenecks leading to increased genetic drift, the immigration or emigration of individuals (gene flow) or meiotic drive. The Hardy-Weinberg Equilibrium does not actually exist in nature because one or more of these factors is always in play. The concept of an equilibrium exists instead as a baseline against which to measure genetic change between generations.
According to the Hardy-Weinberg principle, then, changes in allele frequency (and thus evolution) would be theoretically impossible if the following conditions were met:
1. There was no mutation
2. There were no selective pressures
3. The population size was infinite (this would bring the rate of genetic drift infinitely close to zero)
4. All members of the population were breeding
5. All breeding was random
6. All individuals produced the same amount of offspring
7. There was no migration of individuals into (immigration), or out of (emigration), the population (i.e. the rate of gene flow was zero)
Representing the equilibrium mathematically:
When considering a locus that has two alleles, A and a:
The frequency of A (the dominant allele) is denoted p
The frequency of a (the recessive allele) is denoted q
And p + q = 1, since every locus in the population must carry either allele.
p = AA + ½ Aa
q = aa + ½ Aa
If the population is in Hardy-Weinberg equilibrium, then the alleles are distributed evenly among heterozygotes and homozygotes:
Dominant homozygotes (genotype AA) are denoted p^2 because the probability of inheriting two dominant alleles is p*p
Recessive homozygotes (genotype aa) are denoted q^2 because the probability of inheriting two recessive alleles is q*q
Heterozygotes (genotype Aa) are denoted 2pq because the probability of inheriting both alleles is (p*q) + (q*p)
And p^2 + 2pq + q^2 = 1, since every individual in the population must be one of these genotypes
p² + 2pq + q² = 1
Step-by-Step Example:
Albinism is a rare genetically inherited trait that is only expressed in the phenotype of homozygous recessive individuals (aa). The most characteristic symptom is a marked deficiency in the skin and hair pigment melanin. This condition can occur among any human group as well as among other animal species. The average human frequency of albinism in North America is only about 1 in 20,000.
Referring back to the Hardy-Weinberg equation (p² + 2pq + q² = 1), the frequency of homozygous recessive individuals (aa) in a population is q². Therefore, in North America the following must be true for albinism:
q² = 1/20,000 = .00005
By taking the square root of both sides of this equation, we get: (Note: the numbers in this example are rounded off for simplification.)
q = .007
In other words, the frequency of the recessive albinism allele (a) is .00707 or about 1 in 140. Knowing one of the two variables (q) in the Hardy-Weinberg equation, it is easy to solve for the other (p).
p = 1 - qp = 1 - .007
p = .993
The frequency of the dominant, normal allele (A) is, therefore, .99293 or about 99 in 100.
The next step is to plug the frequencies of p and q into the Hardy-Weinberg equation:
p² + 2pq + q² = 1(.993)² + 2 (.993)(.007) + (.007)² = 1
.986 + .014 + .00005 = 1
This gives us the frequencies for each of the three genotypes for this trait in the population:
p² = / predicted frequencyof homozygous
dominant individuals / = .986 = 98.6%
2pq = / predicted frequency
of heterozygous
individuals / = .014 = 1.4%
q² = / predicted frequency
of homozygous
recessive individuals
(the albinos) / = .00005 = .005%
With a frequency of .005% (about 1 in 20,000), albinos are extremely rare. However, heterozygous carriers for this trait, with a predicted frequency of 1.4% (about 1 in 72), are far more common than most people imagine. There are roughly 278 times more carriers than albinos. Clearly, though, the vast majority of humans (98.6%) probably are homozygous dominant and do not have the albinism allele.
Reminders:
p + q = 1
p2 + 2pq + q2 = 1
p = frequency of the dominant allele in the population
q = frequency of the recessive allele in the population
p2 = percentage of homozygous dominant individuals
q2 = percentage of homozygous recessive individuals
2pq = percentage of heterozygous individuals
Other Examples:
This equation can be used, for example, to predict the frequency of carriers of a disease in a population. Consider the autosomal recessive disease, phenylketonuria. If the frequency of the recessive (in this case, harmful) allele is 1% (q = 0.01), then the number of people who suffer (i.e. who are homozygous recessive) is q^2 = 0.0001 or 0.01% of the population. The number of carriers - or heterozygotes - is 2pq = 2 x 0.99 x 0.01 = 0.198, or 1.98% of the population. That means that carriers of the disease exist in the population at a frequency of almost 200 times more than actual sufferers. This can help us to identify the likelihood of one carrier mating with another, and potentially producing an offspring who suffers from the condition.
In a population of pea plants, the gene for flower color has two alleles, F and f. Individuals with two F alleles have white flowers, those with two f alleles have purple flowers, and heterozygotes (having one F and one f allele) have red flowers. A scientist calculates the allele frequencies in one generation of the population. The frequency of F is p = 0.9 and the frequency of f is q = 0.1. Based on these findings, we know that if the system is at Hardy-Weinberg equilibrium the frequency of heterozygotes in the next generation should be 2pq or 2(0.9)(0.1) or 0.18. This means that 18% of the population should have red flowers. If we then allow a selection pressure, such as pollinator preference for flower color, to enter the system, we will see that the percentage of the population that has red flowers is not 18%. It has changed due to selection pressure. The effect of this selection pressure can be quantified by comparing the new frequency of red flowers with the expected 18%.
In a given plant population, the gene that determines height has two alleles, H and h. In one generation, H has a frequency of p=0.8 and h has a frequency of q=0.2. in the next generation, p=0.7 and q=0.3. Is this population in Hardy- Weinberg equilibrium? Why or why not? This population is not in Hardy-Weinberg equilibrium because the allelic frequencies of H and h change from one generation to the next.
In a population of giraffes, the gene that determines neck length has two alleles, N for longer necks and n for shorter necks. Only giraffes with two N alleles will have long necks. In one generation, the frequency of N is p=0.4 and the frequency of n is q=0.6. This generation experiences a large forest fire that kills all low-growing plants, leaving tall trees with high branches as the only food source. In the next generation, 49% of giraffes have long necks. How much has selection contributed to the allele frequencies in the second generation? The Hardy-Weinberg equation for a two-allele system is (p + q)^2 = 1 or p^2 +2pq +q^2 = 1. The frequency of giraffes with two N alleles is determined by p^2. In the first generation, this is (0.4)^2 = .16 or 16%. Since 49% of giraffes in the second generation have two N alleles, selection has increased the frequency of N in the population. Working backwards we find that 49% = 0.49 = p^2, so p = 0.7 in the second generation. Selection has caused the frequency of N to change from 0.4 to 0.7 in one generation.
In a population of giraffes, the gene that determines spot size has two alleles, S for larger spots and s for smaller spots. Giraffes with one of each allele have medium-sized spots. In one generation, the frequency of S is p=0.4 and the frequency of s is q=0.6. Since spots help the giraffes blend in with their surroundings, they are acted upon by natural selection. In the next generation, 64% of giraffes have small spots. What percentage of giraffes in the second generation will have medium spots? Large spots? The Hardy-Weinberg equation for a two-allele system is (p + q)^2 = 1 or p^2 +2pq +q^2 = 1. The frequency of giraffes with twos alleles is determined by q^2. From the problem, we see that q^2 is 64% or 0.64. From this we find that the frequency of s in the second generation is q=0.8. Using the original equation, we plug in this value of q and solve for p, which we find is 0.2. The frequency of giraffes with large spots is p^2 = 0.04 or 4%. The frequency of giraffes with medium spots is 2pq = 0.32 or 32%.