No Migration - Biology

1. Fundamental Concepts

Definition: The Hardy-Weinberg equilibrium is a principle in population genetics that states the allele and genotype frequencies in a population will remain constant from generation to generation in the absence of other evolutionary influences.
Assumption: No migration means there is no movement of alleles into or out of the population, maintaining genetic stability.

2. Key Concepts

Basic Rule: $p^2 + 2pq + q^2 = 1$

Degree Preservation: The Hardy-Weinberg equation remains valid as long as the assumptions are met, including no migration.

Application: Used to predict the frequency of alleles and genotypes in a population under ideal conditions.

3. Examples

Example 1 (Basic)

Problem: In a population where the frequency of the dominant allele $A$ is $0.7$, calculate the expected frequency of the homozygous recessive genotype $aa$.

Step-by-Step Solution:

Identify the frequency of the recessive allele $q$: $q = 1 - p = 1 - 0.7 = 0.3$.
Calculate the frequency of the homozygous recessive genotype $aa$: $q^2 = 0.3^2 = 0.09$.

Validation: Given $p = 0.7$, $q = 0.3$. Using the formula $q^2$, we get $0.3^2 = 0.09$. ✓

Example 2 (Intermediate)

Problem: A population is in Hardy-Weinberg equilibrium with respect to a gene locus with two alleles, $A$ and $a$. If the frequency of the heterozygous genotype $Aa$ is $0.6$, what are the frequencies of the homozygous genotypes $AA$ and $aa$?

Step-by-Step Solution:

Given $2pq = 0.6$, solve for $pq$: $pq = 0.3$.
Let $p + q = 1$. Since $pq = 0.3$, we can use the quadratic equation $x^2 - x + 0.3 = 0$ to find $p$ and $q$.
Solving the quadratic equation gives $p \approx 0.7$ and $q \approx 0.3$.
Calculate the frequencies of the homozygous genotypes: $AA = p^2 = 0.7^2 = 0.49$ and $aa = q^2 = 0.3^2 = 0.09$.

Validation: Given $2pq = 0.6$, solving for $p$ and $q$ yields $p \approx 0.7$ and $q \approx 0.3$. Using these values, $AA = 0.7^2 = 0.49$ and $aa = 0.3^2 = 0.09$. ✓

4. Problem-Solving Techniques

Visual Strategy: Use Punnett squares to visualize allele combinations.
Error-Proofing: Double-check calculations by ensuring the sum of all genotype frequencies equals 1.
Concept Reinforcement: Practice applying the Hardy-Weinberg equation to different scenarios to reinforce understanding.