Understanding Hardy-Weinberg equilibrium and applying chi-square tests are crucial skills in population genetics. This guide provides a comprehensive overview of both concepts, along with example problems and detailed solutions. We'll explore the principles, assumptions, and applications, making this a valuable resource for students and anyone interested in population genetics.
What is Hardy-Weinberg Equilibrium?
The Hardy-Weinberg principle states that the genetic variation in a population will remain constant from one generation to the next in the absence of disturbing factors. This principle describes a theoretical scenario where allele and genotype frequencies remain stable. It provides a baseline against which to compare real-world populations and identify evolutionary forces at play.
Assumptions of Hardy-Weinberg Equilibrium:
The principle rests on five key assumptions:
- No Mutation: The rate of mutation is negligible.
- Random Mating: Individuals mate randomly without any preference for specific genotypes.
- No Gene Flow: There is no migration of individuals into or out of the population.
- No Genetic Drift: The population is infinitely large, preventing random fluctuations in allele frequencies.
- No Natural Selection: All genotypes have equal survival and reproductive rates.
Hardy-Weinberg Equations:
The principle is expressed mathematically through two equations:
- p + q = 1: where 'p' represents the frequency of the dominant allele and 'q' represents the frequency of the recessive allele.
- p² + 2pq + q² = 1: where p² represents the frequency of the homozygous dominant genotype, 2pq represents the frequency of the heterozygous genotype, and q² represents the frequency of the homozygous recessive genotype.
Chi-Square (χ²) Goodness-of-Fit Test
The chi-square test is a statistical method used to determine if there is a significant difference between the observed and expected frequencies of genotypes in a population. It helps us determine if a population is actually in Hardy-Weinberg equilibrium or if evolutionary forces are acting upon it.
Calculating Chi-Square:
The chi-square statistic is calculated using the following formula:
χ² = Σ [(Observed - Expected)² / Expected]
Where:
- Observed = the number of individuals with a particular genotype in the sample.
- Expected = the number of individuals with that genotype expected under Hardy-Weinberg equilibrium.
Degrees of Freedom and p-value:
The degrees of freedom (df) for a Hardy-Weinberg chi-square test is typically 1 (because if you know the frequency of one allele, you can calculate the frequency of the other). The p-value is then determined using a chi-square distribution table based on the calculated χ² value and the degrees of freedom. A p-value less than a predetermined significance level (usually 0.05) indicates that the observed frequencies deviate significantly from the expected frequencies, suggesting that the population is not in Hardy-Weinberg equilibrium.
Example Problem and Solution:
Let's consider a population of wildflowers with two alleles for flower color: red (R) and white (r). In a sample of 100 wildflowers, 64 have red flowers (RR), 32 have pink flowers (Rr), and 4 have white flowers (rr). Is this population in Hardy-Weinberg equilibrium?
1. Calculate allele frequencies:
- Number of R alleles: (2 * 64) + 32 = 160
- Number of r alleles: (2 * 4) + 32 = 40
- Total number of alleles: 160 + 40 = 200
- p (frequency of R): 160/200 = 0.8
- q (frequency of r): 40/200 = 0.2
2. Calculate expected genotype frequencies under Hardy-Weinberg:
- p² (RR): (0.8)² = 0.64
- 2pq (Rr): 2 * (0.8) * (0.2) = 0.32
- q² (rr): (0.2)² = 0.04
3. Calculate expected genotype numbers:
- Expected RR: 0.64 * 100 = 64
- Expected Rr: 0.32 * 100 = 32
- Expected rr: 0.04 * 100 = 4
4. Perform the chi-square test:
| Genotype | Observed | Expected | (Observed - Expected)² | (Observed - Expected)² / Expected |
|---|---|---|---|---|
| RR | 64 | 64 | 0 | 0 |
| Rr | 32 | 32 | 0 | 0 |
| rr | 4 | 4 | 0 | 0 |
| Total | 100 | 100 | χ² = 0 |
5. Interpret the results:
The calculated χ² value is 0. With 1 degree of freedom, this corresponds to a p-value significantly greater than 0.05. Therefore, we fail to reject the null hypothesis that the population is in Hardy-Weinberg equilibrium.
This example demonstrates the application of the Hardy-Weinberg principle and chi-square analysis. Remember that deviations from expected values might indicate evolutionary pressures are at work. Further investigation would be necessary to pinpoint the specific forces influencing the population's genetic structure. Understanding these principles is essential for comprehending the dynamics of evolution and genetic variation within populations.
