How To Tell If A Population Is In Hardy-weinberg Equilibrium

In population genetics, understanding whether a population is in Hardy-Weinberg equilibrium is fundamental to assessing its evolutionary status. This principle, named after Godfrey Harold Hardy and Wilhelm Weinberg, provides a baseline against which to measure changes in allele and genotype frequencies, indicating whether evolutionary forces are at play.

The Hardy-Weinberg Principle: A Foundation of Population Genetics

The Hardy-Weinberg principle states that in a large, randomly mating population, the allele and genotype frequencies will remain constant from generation to generation in the absence of other evolutionary influences. These influences include:

• Mutation: Changes in the DNA sequence.
• Non-random mating: Mating patterns where individuals choose mates based on specific traits.
• Gene flow: The movement of genes into or out of a population.
• Genetic drift: Random changes in allele frequencies, particularly significant in small populations.
• Natural selection: Differential survival and reproduction based on heritable traits.

If a population is in Hardy-Weinberg equilibrium, it suggests that these evolutionary forces are not significantly affecting the gene pool, allowing us to predict genotype frequencies based on allele frequencies.

The Hardy-Weinberg Equations

The Hardy-Weinberg principle is expressed through two primary equations:

1. Allele Frequency Equation: p + q = 1
   • p represents the frequency of the dominant allele.
   • q represents the frequency of the recessive allele.
   • This equation states that the sum of the frequencies of all alleles for a particular trait in a population must equal 1 (or 100%).
2. Genotype Frequency Equation: p² + 2pq + q² = 1
   • p² represents the frequency of the homozygous dominant genotype.
   • 2pq represents the frequency of the heterozygous genotype.
   • q² represents the frequency of the homozygous recessive genotype.
   • This equation states that the sum of the frequencies of all possible genotypes for a particular trait in a population must equal 1.

Steps to Determine if a Population is in Hardy-Weinberg Equilibrium

To determine if a population is in Hardy-Weinberg equilibrium, you need to follow a series of steps that involve calculating allele and genotype frequencies and comparing observed frequencies with expected frequencies. Here's a detailed guide:

1. Gather Data: Observed Genotype Frequencies

The first step is to collect data on the observed number of individuals with each genotype for the trait you are studying. This data is crucial for calculating the observed genotype frequencies. For example, let’s consider a population of butterflies with two alleles for wing color: R (dominant, red wings) and r (recessive, white wings). You sample 500 butterflies and observe the following:

• RR (homozygous dominant, red wings): 320
• Rr (heterozygous, red wings): 160
• rr (homozygous recessive, white wings): 20

2. Calculate Observed Genotype Frequencies

To calculate the observed genotype frequencies, divide the number of individuals with each genotype by the total number of individuals in the sample.

• Frequency of RR: 320 / 500 = 0.64
• Frequency of Rr: 160 / 500 = 0.32
• Frequency of rr: 20 / 500 = 0.04

Make sure that the sum of these frequencies equals 1. 0.64 + 0.32 + 0.04 = 1

3. Calculate Observed Allele Frequencies

Using the observed genotype frequencies, calculate the observed allele frequencies for both the dominant (R) and recessive (r) alleles.

• Calculate the frequency of the recessive allele (q): Since the rr genotype represents individuals with two copies of the r allele, the frequency of q is the square root of the frequency of rr.
  • q = √frequency of rr = √0.04 = 0.2
• Calculate the frequency of the dominant allele (p): Use the equation p + q = 1 to find the frequency of p.
  • p = 1 - q = 1 - 0.2 = 0.8

4. Calculate Expected Genotype Frequencies

Now that you have the observed allele frequencies, use the Hardy-Weinberg equation p² + 2pq + q² = 1 to calculate the expected genotype frequencies, assuming the population is in equilibrium.

• Expected frequency of RR: p² = (0.8)² = 0.64
• Expected frequency of Rr: 2pq = 2 * 0.8 * 0.2 = 0.32
• Expected frequency of rr: q² = (0.2)² = 0.04

5. Calculate Expected Number of Individuals for Each Genotype

Multiply the expected genotype frequencies by the total number of individuals in the sample to find the expected number of individuals for each genotype.

• Expected number of RR: 0.64 * 500 = 320
• Expected number of Rr: 0.32 * 500 = 160
• Expected number of rr: 0.04 * 500 = 20

6. Perform a Chi-Square (χ²) Test

A chi-square test is used to determine if there is a statistically significant difference between the observed and expected values. The chi-square formula is:

χ² = Σ [(Observed - Expected)² / Expected]

Where:

• Σ means "sum of"
• Observed is the observed number of individuals for each genotype
• Expected is the expected number of individuals for each genotype

Calculate the Chi-Square Value:

1. For RR: (320 - 320)² / 320 = 0
2. For Rr: (160 - 160)² / 160 = 0
3. For rr: (20 - 20)² / 20 = 0

χ² = 0 + 0 + 0 = 0

Determine the Degrees of Freedom (df):

The degrees of freedom for a Hardy-Weinberg equilibrium test are calculated as the number of genotype classes minus the number of alleles. In this case, there are three genotype classes (RR, Rr, rr) and two alleles (R, r). However, because we are estimating allele frequencies from the data, we lose one degree of freedom. Therefore, the degrees of freedom are:

df = (number of genotype classes) - (number of alleles - 1) = 3 - (2 - 1) - 1 = 1

Find the Critical Value:

Using a chi-square distribution table, find the critical value for α = 0.05 (the standard significance level) and df = 1. The critical value is approximately 3.841.

Compare the Chi-Square Value to the Critical Value:

• If the calculated χ² value is less than the critical value, you fail to reject the null hypothesis. This suggests that the population is in Hardy-Weinberg equilibrium.
• If the calculated χ² value is greater than the critical value, you reject the null hypothesis. This suggests that the population is not in Hardy-Weinberg equilibrium, and evolutionary forces may be at play.

In our example, the calculated χ² value is 0, which is much less than the critical value of 3.841. Therefore, we fail to reject the null hypothesis, and we can conclude that the butterfly population is likely in Hardy-Weinberg equilibrium.

Example 2: A Population Not in Equilibrium

Let's consider another example where the population is not in Hardy-Weinberg equilibrium. Suppose you sample 500 beetles and observe the following genotypes for a trait determined by alleles A (dominant, black) and a (recessive, brown):

• AA: 200
• Aa: 250
• aa: 50

1. Calculate Observed Genotype Frequencies

• Frequency of AA: 200 / 500 = 0.40
• Frequency of Aa: 250 / 500 = 0.50
• Frequency of aa: 50 / 500 = 0.10

2. Calculate Observed Allele Frequencies

• q = √frequency of aa = √0.10 ≈ 0.316
• p = 1 - q = 1 - 0.316 ≈ 0.684

3. Calculate Expected Genotype Frequencies

• Expected frequency of AA: p² = (0.684)² ≈ 0.468
• Expected frequency of Aa: 2pq = 2 * 0.684 * 0.316 ≈ 0.432
• Expected frequency of aa: q² = (0.316)² ≈ 0.10

4. Calculate Expected Number of Individuals for Each Genotype

• Expected number of AA: 0.468 * 500 ≈ 234
• Expected number of Aa: 0.432 * 500 ≈ 216
• Expected number of aa: 0.10 * 500 = 50

5. Perform a Chi-Square (χ²) Test

1. For AA: (200 - 234)² / 234 ≈ 6.12
2. For Aa: (250 - 216)² / 216 ≈ 5.56
3. For aa: (50 - 50)² / 50 = 0

χ² ≈ 6.12 + 5.56 + 0 ≈ 11.68

Determine the Degrees of Freedom (df):

df = 1 (as explained in the previous example)

Find the Critical Value:

Using a chi-square distribution table, the critical value for α = 0.05 and df = 1 is approximately 3.841.

Compare the Chi-Square Value to the Critical Value:

The calculated χ² value of 11.68 is greater than the critical value of 3.841. Therefore, we reject the null hypothesis, and we can conclude that the beetle population is not in Hardy-Weinberg equilibrium. This suggests that evolutionary forces are likely influencing the allele and genotype frequencies in this population.

Potential Reasons for Deviations from Hardy-Weinberg Equilibrium

If a population is not in Hardy-Weinberg equilibrium, it indicates that one or more of the assumptions of the principle are being violated. Here are some common reasons:

• Natural Selection: If certain genotypes have higher survival or reproductive rates than others, allele and genotype frequencies will change over time.
• Non-Random Mating: Assortative mating (where individuals with similar traits mate more frequently) and inbreeding can alter genotype frequencies.
• Mutation: While mutation rates are typically low, they can introduce new alleles or change existing allele frequencies, especially over long periods.
• Gene Flow: Migration of individuals into or out of a population can introduce or remove alleles, altering allele and genotype frequencies.
• Genetic Drift: Random fluctuations in allele frequencies, particularly in small populations, can lead to deviations from Hardy-Weinberg equilibrium. The founder effect and bottleneck effect are specific examples of genetic drift that can have significant impacts.

Importance of Hardy-Weinberg Equilibrium

Understanding Hardy-Weinberg equilibrium is crucial for several reasons:

• Baseline for Evolutionary Studies: It provides a null hypothesis against which to test whether a population is evolving. Significant deviations from equilibrium suggest that evolutionary forces are at play.
• Predicting Genotype Frequencies: If a population is in equilibrium, we can predict genotype frequencies from allele frequencies, which is useful for understanding the genetic structure of populations.
• Identifying Genetic Disorders: In human genetics, the Hardy-Weinberg principle can be used to estimate the frequency of carriers for genetic disorders in a population.
• Conservation Biology: It helps in managing and conserving endangered species by assessing the genetic diversity and evolutionary potential of populations.

Common Pitfalls and Considerations

• Sample Size: Ensure that your sample size is large enough to provide accurate estimates of allele and genotype frequencies. Small sample sizes can lead to biased results.
• Random Sampling: The sample should be randomly selected to avoid bias. Non-random sampling can lead to inaccurate estimates of population frequencies.
• Accurate Genotyping: Ensure that genotyping is accurate and reliable. Errors in genotyping can lead to incorrect frequency calculations.
• Assumptions: Be aware of the assumptions of the Hardy-Weinberg principle and consider whether they are likely to be violated in the population you are studying. If assumptions are violated, deviations from equilibrium may not necessarily indicate natural selection.
• Other Factors: Consider other factors that may influence allele and genotype frequencies, such as population structure, migration patterns, and environmental conditions.

Conclusion

Determining whether a population is in Hardy-Weinberg equilibrium is a fundamental step in population genetics. By following the steps outlined above—collecting data, calculating observed and expected frequencies, and performing a chi-square test—you can assess whether a population is evolving or remaining stable. Understanding the reasons for deviations from equilibrium can provide valuable insights into the evolutionary forces shaping the genetic structure of populations. This knowledge is essential for various applications, including conservation biology, human genetics, and evolutionary research.