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

The Hardy-Weinberg principle, a cornerstone of population genetics, describes the conditions under which allele and genotype frequencies in a population will remain constant from generation to generation. This principle serves as a null hypothesis to test whether evolution is occurring in a population. In simpler terms, it helps us understand if a population is evolving or staying genetically the same over time. Determining whether a population is in Hardy-Weinberg equilibrium (HWE) involves a series of calculations and comparisons between observed and expected genotype frequencies. This article provides a comprehensive guide on how to assess if a population adheres to the Hardy-Weinberg equilibrium, covering the underlying principles, necessary calculations, potential deviations, and their implications.

Understanding the Hardy-Weinberg Principle

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 that is not random, such as assortative mating (individuals with similar phenotypes mate more frequently).
• Gene flow: The movement of alleles into or out of a population.
• Genetic drift: Random changes in allele frequencies due to chance events.
• Natural selection: Differential survival and reproduction of individuals with different genotypes.

The Hardy-Weinberg equilibrium is described by two equations:

1. Allele frequency equation: p + q = 1

   • Where p is the frequency of one allele (e.g., A), and q is the frequency of the other allele (e.g., a) for a particular gene.

2. Genotype frequency equation: p² + 2pq + q² = 1

   • Where p² is the frequency of the homozygous genotype AA, 2pq is the frequency of the heterozygous genotype Aa, and q² is the frequency of the homozygous genotype aa.

These equations allow us to predict the expected genotype frequencies in a population if it is in equilibrium, and then compare these expected frequencies with the observed frequencies to determine if the population is indeed in HWE.

Steps to Determine Hardy-Weinberg Equilibrium

To determine if a population is in Hardy-Weinberg equilibrium, you need to follow these steps:

1. Collect Data on Genotype Frequencies

The first step is to collect data on the number of individuals with each genotype in the population. This typically involves sampling a representative portion of the population and genotyping them for the gene of interest.

• Example: Suppose you are studying a population of butterflies and are interested in a gene that controls wing color. You collect a sample of 500 butterflies and find the following genotype counts:

  • AA (homozygous dominant): 245
  • Aa (heterozygous): 210
  • aa (homozygous recessive): 45

2. Calculate Observed Genotype Frequencies

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

• Example: Using the butterfly data:

  • Frequency of AA (P): 245 / 500 = 0.49
  • Frequency of Aa (H): 210 / 500 = 0.42
  • Frequency of aa (Q): 45 / 500 = 0.09

3. Calculate Allele Frequencies

Calculate the allele frequencies (p and q) from the observed genotype frequencies. The frequency of the A allele (p) can be calculated as the frequency of the AA genotype plus half the frequency of the Aa genotype. Similarly, the frequency of the a allele (q) can be calculated as the frequency of the aa genotype plus half the frequency of the Aa genotype.

• Equations:

  • p = P + (1/2)H
  • q = Q + (1/2)H

• Example: Using the butterfly data:

  • p = 0.49 + (1/2)*0.42 = 0.49 + 0.21 = 0.70
  • q = 0.09 + (1/2)*0.42 = 0.09 + 0.21 = 0.30

  Note that p + q should equal 1. In this case, 0.70 + 0.30 = 1.00, which confirms the calculation.

4. Calculate Expected Genotype Frequencies

Using the calculated allele frequencies, calculate the expected genotype frequencies under Hardy-Weinberg equilibrium using the equation p² + 2pq + q² = 1.

• Equations:

  • Expected frequency of AA: p²
  • Expected frequency of Aa: 2pq
  • Expected frequency of aa: q²

• Example: Using the butterfly data:

  • Expected frequency of AA: (0.70)² = 0.49
  • Expected frequency of Aa: 2 * 0.70 * 0.30 = 0.42
  • Expected frequency of aa: (0.30)² = 0.09

5. Calculate Expected Genotype Counts

Multiply the expected genotype frequencies by the total number of individuals in the sample to obtain the expected genotype counts.

• Example: Using the butterfly data (sample size = 500):

  • Expected count of AA: 0.49 * 500 = 245
  • Expected count of Aa: 0.42 * 500 = 210
  • Expected count of aa: 0.09 * 500 = 45

6. Perform a Chi-Square Test

To determine if the observed genotype counts differ significantly from the expected genotype counts, perform a chi-square (χ²) test. The chi-square test is a statistical test that determines if there is a significant difference between the expected and observed frequencies.

• Equation:

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

• Steps:

  1. Calculate the chi-square value for each genotype:

     • For AA: [(245 - 245)² / 245] = 0
     • For Aa: [(210 - 210)² / 210] = 0
     • For aa: [(45 - 45)² / 45] = 0

  2. Sum the chi-square values for all genotypes:

     • χ² = 0 + 0 + 0 = 0

  3. Determine the degrees of freedom (df):

     • For Hardy-Weinberg equilibrium, the degrees of freedom is typically the number of genotypes minus the number of alleles. With 2 alleles and 3 genotypes, df = (number of genotypes) - (number of alleles) = 3-2 = 1. However, because the allele frequencies are estimated from the data, one degree of freedom is lost, so df = 1 - 1 = 0. However, if we are testing for HWE using genotype counts with known allele frequencies, the degrees of freedom is 1.

  4. Find the critical value from the chi-square distribution table:

     • Look up the critical value in a chi-square distribution table using the calculated degrees of freedom and a significance level (α). A common significance level is 0.05. If df=0, there is no p-value.
     • A chi-square table is required to obtain the critical value. For df=1 and α=0.05, the critical value is 3.841.

  5. Compare the calculated chi-square value to the critical value:

     • If the calculated chi-square value is less than the critical value, the null hypothesis (that the population is in Hardy-Weinberg equilibrium) is not rejected.
     • If the calculated chi-square value is greater than the critical value, the null hypothesis is rejected, suggesting that the population is not in Hardy-Weinberg equilibrium.

• Example: Using the butterfly data:

  • Calculated χ² = 0
  • Degrees of freedom = 1
  • Critical value (α = 0.05) = 3.841
  • Since 0 < 3.841, we do not reject the null hypothesis. The population is likely in Hardy-Weinberg equilibrium.

Detailed Example with Deviation

Let's consider another example where the population may not be in Hardy-Weinberg equilibrium. Suppose we sample 500 individuals from a population of wildflowers and observe the following genotype counts for a gene controlling flower color:

• RR (red): 400
• RW (pink): 80
• WW (white): 20

1. Calculate Observed Genotype Frequencies

• Frequency of RR (P): 400 / 500 = 0.80
• Frequency of RW (H): 80 / 500 = 0.16
• Frequency of WW (Q): 20 / 500 = 0.04

2. Calculate Allele Frequencies

• p (frequency of R) = 0.80 + (1/2)*0.16 = 0.80 + 0.08 = 0.88

• q (frequency of W) = 0.04 + (1/2)*0.16 = 0.04 + 0.08 = 0.12

  Check: 0.88 + 0.12 = 1.00 (confirms calculation)

3. Calculate Expected Genotype Frequencies

• Expected frequency of RR: (0.88)² = 0.7744
• Expected frequency of RW: 2 * 0.88 * 0.12 = 0.2112
• Expected frequency of WW: (0.12)² = 0.0144

4. Calculate Expected Genotype Counts

• Expected count of RR: 0.7744 * 500 = 387.2
• Expected count of RW: 0.2112 * 500 = 105.6
• Expected count of WW: 0.0144 * 500 = 7.2

5. Perform a Chi-Square Test

1. Calculate the chi-square value for each genotype:

   • For RR: [(400 - 387.2)² / 387.2] = (12.8)² / 387.2 = 163.84 / 387.2 ≈ 0.423
   • For RW: [(80 - 105.6)² / 105.6] = (-25.6)² / 105.6 = 655.36 / 105.6 ≈ 6.206
   • For WW: [(20 - 7.2)² / 7.2] = (12.8)² / 7.2 = 163.84 / 7.2 ≈ 22.756

2. Sum the chi-square values for all genotypes:

   • χ² = 0.423 + 6.206 + 22.756 = 29.385

3. Determine the degrees of freedom (df):

   • Degrees of freedom = 1

4. Find the critical value from the chi-square distribution table:

   • For df = 1 and α = 0.05, the critical value is 3.841.

5. Compare the calculated chi-square value to the critical value:

   • Since 29.385 > 3.841, we reject the null hypothesis. The population is not in Hardy-Weinberg equilibrium.

In this case, the significant chi-square value indicates that the observed genotype frequencies deviate significantly from the expected frequencies under Hardy-Weinberg equilibrium.

Interpreting Deviations from Hardy-Weinberg Equilibrium

When a population is found to deviate from Hardy-Weinberg equilibrium, it suggests that one or more of the assumptions of the principle are being violated. This could be due to:

• Non-random mating:

  • Inbreeding: Increases the frequency of homozygous genotypes and decreases the frequency of heterozygous genotypes.
  • Assortative mating: Individuals with similar phenotypes mate more frequently, which can lead to an increase in homozygous genotypes for the genes controlling those phenotypes.

• Natural selection:

  • If certain genotypes have higher survival or reproductive rates, their frequencies will increase over time, leading to deviations from HWE.

• Mutation:

  • Although mutation rates are generally low, they can introduce new alleles into the population, altering allele frequencies over time.

• Gene flow:

  • The movement of individuals (and their genes) between populations can alter allele frequencies, especially if the populations have different allele frequencies to begin with.

• Genetic drift:

  • In small populations, random chance events can cause allele frequencies to fluctuate significantly from generation to generation.

Common Pitfalls and Considerations

When assessing Hardy-Weinberg equilibrium, there are several common pitfalls to avoid:

• Small Sample Sizes: Small sample sizes can lead to inaccurate estimates of allele and genotype frequencies. Use larger samples to improve the reliability of your results.
• Non-Random Sampling: Ensure that your sample is representative of the entire population. Biased sampling can lead to incorrect conclusions about HWE.
• Incorrect Genotyping: Errors in genotyping can lead to inaccurate genotype counts and frequencies. Use reliable genotyping methods and quality control measures.
• Ignoring Population Structure: If the population is subdivided into smaller subpopulations with different allele frequencies, the overall population may appear to deviate from HWE even if each subpopulation is in equilibrium.
• Assuming Diploidy: Hardy-Weinberg equilibrium applies to diploid organisms. If you are studying organisms with different ploidy levels (e.g., haploid or polyploid), the calculations and interpretations will differ.

Applications of Hardy-Weinberg Equilibrium

The Hardy-Weinberg principle has numerous applications in various fields, including:

• Population Genetics: Understanding the genetic structure of populations and identifying factors that cause evolutionary change.
• Conservation Biology: Assessing the genetic health of endangered species and managing populations to maintain genetic diversity.
• Medical Genetics: Calculating the risk of inheriting genetic disorders and predicting the frequency of disease alleles in populations.
• Forensic Science: Estimating the frequency of DNA profiles in different populations for use in forensic investigations.
• Agriculture: Improving breeding programs by understanding the genetic basis of traits and predicting the response to selection.

Advanced Considerations

Exact Tests for Hardy-Weinberg Equilibrium

While the chi-square test is commonly used, it is an approximation and may not be accurate for small sample sizes or when expected genotype counts are low. In such cases, exact tests, such as Fisher's exact test, may be more appropriate. Exact tests calculate the exact probability of observing the data (or more extreme data) under the null hypothesis of Hardy-Weinberg equilibrium.

Multiple Alleles

The Hardy-Weinberg principle can be extended to genes with more than two alleles. For example, if a gene has three alleles (A, B, and C) with frequencies p, q, and r, respectively, the expected genotype frequencies under HWE are:

• AA: p²
• BB: q²
• CC: r²
• AB: 2pq
• AC: 2pr
• BC: 2qr

The sum of the allele frequencies should still equal 1 (p + q + r = 1), and the sum of the genotype frequencies should also equal 1.

X-linked Genes

For X-linked genes in species with sex chromosomes (e.g., humans), the calculations for Hardy-Weinberg equilibrium differ slightly because males have only one X chromosome. In females (XX), the genotype frequencies are calculated as usual. In males (XY), the allele frequency is equal to the genotype frequency. For example, if p is the frequency of allele A and q is the frequency of allele a, then:

• Females: p² (AA), 2pq (Aa), q² (aa)
• Males: p (A), q (a)

Accounting for Null Alleles

In some cases, one or more alleles may be "null alleles" that do not produce a functional product. These alleles can be difficult to detect using standard genotyping methods. If a null allele is present, it can affect the observed genotype frequencies and lead to deviations from HWE. Special statistical methods are available to account for null alleles when testing for HWE.

Conclusion

Determining whether a population is in Hardy-Weinberg equilibrium is a fundamental step in understanding the genetic dynamics of that population. By following the steps outlined in this article—collecting data, calculating observed and expected genotype frequencies, and performing a chi-square test—you can assess whether the population deviates significantly from HWE. Understanding the potential reasons for such deviations, such as non-random mating, natural selection, mutation, gene flow, and genetic drift, provides valuable insights into the evolutionary processes shaping the population. The Hardy-Weinberg principle serves as a crucial tool for researchers and practitioners in various fields, from population genetics to conservation biology, medical genetics, and beyond. By applying this principle carefully and considering its limitations, we can gain a deeper understanding of the genetic structure and evolutionary trajectory of populations.