How To Find Expected Frequency In Hardy Weinberg

Here's a comprehensive guide on calculating expected frequencies in Hardy-Weinberg equilibrium, designed to provide a deep understanding and practical application of the concepts.

Introduction

The Hardy-Weinberg principle is a cornerstone of population genetics, describing the conditions under which allele and genotype frequencies in a population will remain constant from generation to generation in the absence of other evolutionary influences. Understanding how to calculate expected frequencies under Hardy-Weinberg equilibrium is essential for determining whether a population is evolving at a particular locus. When observed genotype frequencies deviate significantly from expected frequencies, it suggests that factors such as natural selection, mutation, gene flow, genetic drift, or non-random mating are at play. Let’s delve into the methods for calculating expected frequencies and what they reveal about population dynamics.

The Hardy-Weinberg equilibrium serves as a null hypothesis. It provides a baseline against which to compare observed genotype frequencies. If a population is not in Hardy-Weinberg equilibrium, it indicates that evolutionary forces are acting upon it. Therefore, calculating expected frequencies is a critical step in assessing the genetic structure of a population and identifying the factors that may be driving evolutionary change.

Fundamentals of Hardy-Weinberg Equilibrium

The Hardy-Weinberg principle states that in a large, randomly mating population, the allele and genotype frequencies will remain constant from generation to generation if other evolutionary influences are not operating. This principle is based on several key assumptions:

• No Mutation: The rate of mutation is negligible.
• Random Mating: Individuals mate randomly, without any preference for certain genotypes.
• No Gene Flow: There is no migration of individuals into or out of the population.
• No Genetic Drift: The population is large enough to prevent random changes in allele frequencies due to chance.
• No Selection: All genotypes have equal survival and reproductive rates.

When these conditions are met, the population is said to be in Hardy-Weinberg equilibrium, and the allele and genotype frequencies can be predicted using the following equations:

• Allele Frequencies: p + q = 1
• Genotype Frequencies: p² + 2pq + q² = 1

Here, p represents the frequency of one allele (typically the dominant allele), and q represents the frequency of the other allele (typically the recessive allele). The terms p², 2pq, and q² represent the expected frequencies of the homozygous dominant, heterozygous, and homozygous recessive genotypes, respectively.

Step-by-Step Guide to Calculating Expected Frequencies

To determine the expected genotype frequencies in a population, follow these steps:

1. Determine the Observed Genotype Frequencies:

   • Collect data on the number of individuals with each genotype.
   • Calculate the observed frequency of each genotype by dividing the number of individuals with that genotype by the total number of individuals in the population.

2. Calculate the Allele Frequencies:

   • Use the observed genotype frequencies to estimate the allele frequencies, p and q.
   • If you know the frequency of the homozygous recessive genotype (q²), you can calculate q by taking the square root of q².
   • Then, calculate p using the equation p = 1 - q.

3. Calculate the Expected Genotype Frequencies:

   • Use the calculated allele frequencies (p and q) to determine the expected genotype frequencies using the Hardy-Weinberg equation: p² + 2pq + q² = 1.
   • Calculate the expected frequency of the homozygous dominant genotype as p².
   • Calculate the expected frequency of the heterozygous genotype as 2pq.
   • Calculate the expected frequency of the homozygous recessive genotype as q².

4. Calculate the Expected Number of Individuals for Each Genotype:

   • Multiply the expected genotype frequencies by the total number of individuals in the population to obtain the expected number of individuals for each genotype.

Detailed Examples

To illustrate these steps, let’s consider a few detailed examples:

Example 1: Simple Dominance

Suppose we have a population of 500 individuals with a simple dominant trait. We observe the following:

• 320 individuals with the dominant phenotype (AA or Aa)
• 180 individuals with the recessive phenotype (aa)

1. Observed Genotype Frequencies:

   • Frequency of aa (q²) = 180 / 500 = 0.36

2. Allele Frequencies:

   • q = √0.36 = 0.6
   • p = 1 - q = 1 - 0.6 = 0.4

3. Expected Genotype Frequencies:

   • Expected frequency of AA (p²) = (0.4)² = 0.16
   • Expected frequency of Aa (2pq) = 2 * 0.4 * 0.6 = 0.48
   • Expected frequency of aa (q²) = (0.6)² = 0.36

4. Expected Number of Individuals for Each Genotype:

   • Expected number of AA = 0.16 * 500 = 80
   • Expected number of Aa = 0.48 * 500 = 240
   • Expected number of aa = 0.36 * 500 = 180

In this example, we have calculated the expected number of individuals for each genotype under Hardy-Weinberg equilibrium. We can then compare these expected values with the observed values to determine if the population is in equilibrium.

Example 2: Codominance

In codominance, both alleles are expressed in the phenotype, allowing us to distinguish all three genotypes directly. Suppose we have a population of 1,000 individuals with the following genotype counts:

• AA: 490 individuals
• AB: 420 individuals
• BB: 90 individuals

1. Observed Genotype Frequencies:

   • Frequency of AA = 490 / 1000 = 0.49
   • Frequency of AB = 420 / 1000 = 0.42
   • Frequency of BB = 90 / 1000 = 0.09

2. Allele Frequencies:

   • To calculate the allele frequencies, we can count the number of A and B alleles in the population.
   • Number of A alleles = (2 * number of AA) + (number of AB) = (2 * 490) + 420 = 980 + 420 = 1400
   • Number of B alleles = (2 * number of BB) + (number of AB) = (2 * 90) + 420 = 180 + 420 = 600
   • Total number of alleles in the population = 2 * total number of individuals = 2 * 1000 = 2000
   • Frequency of A (p) = Number of A alleles / Total number of alleles = 1400 / 2000 = 0.7
   • Frequency of B (q) = Number of B alleles / Total number of alleles = 600 / 2000 = 0.3

3. Expected Genotype Frequencies:

   • Expected frequency of AA (p²) = (0.7)² = 0.49
   • Expected frequency of AB (2pq) = 2 * 0.7 * 0.3 = 0.42
   • Expected frequency of BB (q²) = (0.3)² = 0.09

4. Expected Number of Individuals for Each Genotype:

   • Expected number of AA = 0.49 * 1000 = 490
   • Expected number of AB = 0.42 * 1000 = 420
   • Expected number of BB = 0.09 * 1000 = 90

In this case, the observed and expected genotype frequencies are the same, indicating that the population is in Hardy-Weinberg equilibrium.

Example 3: Sex-Linked Traits

For sex-linked traits (genes located on the X chromosome), the calculations are slightly different, especially for males, who have only one X chromosome. Let’s consider a population of 2,000 individuals (1,000 males and 1,000 females) with a sex-linked trait. We observe the following in males:

• 600 males with the dominant phenotype (X^AY)
• 400 males with the recessive phenotype (X^aY)

1. Allele Frequencies in Males:

   • Frequency of X^A (p) = 600 / 1000 = 0.6
   • Frequency of X^a (q) = 400 / 1000 = 0.4

2. Expected Genotype Frequencies in Females:

   • Since females have two X chromosomes, we can use the standard Hardy-Weinberg equation.
   • Expected frequency of X^AX^A (p²) = (0.6)² = 0.36
   • Expected frequency of X^AX^a (2pq) = 2 * 0.6 * 0.4 = 0.48
   • Expected frequency of X^aX^a (q²) = (0.4)² = 0.16

3. Expected Number of Individuals for Each Genotype in Females:

   • Expected number of X^AX^A = 0.36 * 1000 = 360
   • Expected number of X^AX^a = 0.48 * 1000 = 480
   • Expected number of X^aX^a = 0.16 * 1000 = 160

4. Expected Number of Individuals for Each Genotype in Males:

   • Expected number of X^AY = 600
   • Expected number of X^aY = 400

By calculating these expected values, we can compare them with the observed values for females to determine if the female population is in Hardy-Weinberg equilibrium for this sex-linked trait.

Statistical Analysis: The Chi-Square Test

Once you have calculated the expected genotype frequencies, you can use a statistical test, such as the chi-square (χ²) test, to determine if the observed genotype frequencies significantly differ from the expected frequencies. The chi-square test helps determine if the deviations are due to chance or if there is a significant evolutionary force acting on the population.

The chi-square statistic is calculated as follows:

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

Where:

• Σ represents the sum across all genotypes.
• Observed is the observed number of individuals for each genotype.
• Expected is the expected number of individuals for each genotype.

After calculating the chi-square value, you need to determine the degrees of freedom (df). For a simple autosomal locus with two alleles, the degrees of freedom is typically calculated as:

df = Number of genotypes - Number of alleles + 1 df = 3 - 2 = 1

For a population in Hardy-Weinberg equilibrium, the degrees of freedom is usually 1.

Once you have the chi-square value and the degrees of freedom, you can compare the calculated chi-square value to a critical value from the chi-square distribution table. If the calculated chi-square value is greater than the critical value at a chosen significance level (e.g., p = 0.05), you reject the null hypothesis (that the population is in Hardy-Weinberg equilibrium) and conclude that there is a significant difference between the observed and expected frequencies. This suggests that the population is evolving or that one or more of the Hardy-Weinberg assumptions are being violated.

Factors Causing Deviations from Hardy-Weinberg Equilibrium

Deviations from Hardy-Weinberg equilibrium can indicate that evolutionary forces are at play. Here are some of the primary factors that can cause a population to deviate from equilibrium:

1. Natural Selection:

   • If certain genotypes have higher survival or reproductive rates than others, natural selection will alter allele and genotype frequencies.
   • For example, if a particular allele confers resistance to a disease, individuals with that allele may have higher survival rates, leading to an increase in the frequency of that allele over time.

2. Mutation:

   • Although mutation rates are generally low, mutations can introduce new alleles into a population, thereby altering allele and genotype frequencies.
   • If mutation rates are high enough, they can significantly affect the genetic structure of a population.

3. Gene Flow:

   • The migration of individuals between populations can introduce or remove alleles, leading to changes in allele and genotype frequencies.
   • Gene flow can homogenize allele frequencies between populations, reducing genetic differences.

4. Genetic Drift:

   • In small populations, random chance events can cause significant fluctuations in allele frequencies, a phenomenon known as genetic drift.
   • Genetic drift can lead to the loss of alleles or the fixation of alleles, reducing genetic diversity.

5. Non-Random Mating:

   • If individuals do not mate randomly, genotype frequencies can deviate from Hardy-Weinberg expectations.
   • For example, assortative mating, where individuals with similar phenotypes mate more frequently, can increase the frequency of homozygous genotypes.
   • Inbreeding, the mating of closely related individuals, also leads to an increase in homozygosity.

Real-World Applications and Significance

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

• Population Genetics: Assessing the genetic structure of populations and identifying evolutionary forces.
• Conservation Biology: Managing endangered species by monitoring genetic diversity and inbreeding.
• Human Genetics: Studying the inheritance of genetic disorders and predicting the risk of disease.
• Agriculture: Improving crop and livestock breeding by understanding genetic variation.

By understanding the Hardy-Weinberg principle and how to calculate expected frequencies, researchers and practitioners can gain valuable insights into the genetic dynamics of populations and make informed decisions in various fields.

FAQ (Frequently Asked Questions)

• Q: What is the significance of Hardy-Weinberg equilibrium?

  • A: Hardy-Weinberg equilibrium serves as a null hypothesis for assessing whether a population is evolving. Deviations from equilibrium indicate that evolutionary forces are at play.

• Q: What are the assumptions of Hardy-Weinberg equilibrium?

  • A: The assumptions include no mutation, random mating, no gene flow, no genetic drift, and no selection.

• Q: How do you calculate allele frequencies from genotype frequencies?

  • A: If you know the frequency of the homozygous recessive genotype (q²), you can calculate q by taking the square root of q², and then calculate p using the equation p = 1 - q.

• Q: What does it mean if a population is not in Hardy-Weinberg equilibrium?

  • A: It means that one or more of the Hardy-Weinberg assumptions are being violated, and evolutionary forces are acting on the population.

• Q: How is the chi-square test used in Hardy-Weinberg analysis?

  • A: The chi-square test is used to compare the observed and expected genotype frequencies to determine if the deviations are statistically significant.

Conclusion

Calculating expected frequencies in Hardy-Weinberg equilibrium is a fundamental tool in population genetics. By comparing observed genotype frequencies with expected frequencies, we can gain insights into the evolutionary forces acting on a population. The Hardy-Weinberg principle provides a valuable framework for understanding genetic variation and the mechanisms of evolutionary change. Whether you are studying natural populations, managing endangered species, or investigating human genetic disorders, a solid understanding of Hardy-Weinberg equilibrium is essential.

How do you plan to apply these calculations in your own research or studies? What fascinating populations might you analyze using these principles?