Hardy-Weinberg and Population Genetics

Understanding Hardy‑Weinberg Equilibrium

The Hardy‑Weinberg principle provides a mathematical baseline for studying how allele and genotype frequencies behave in an idealized population. By mastering this concept, you can detect evolutionary forces such as selection, migration, mutation, and genetic drift.

Key Terms and Symbols

• p: frequency of the dominant allele (A)
• q: frequency of the recessive allele (a), where q = 1 – p
• p²: expected frequency of the homozygous dominant genotype (AA)
• 2pq: expected frequency of the heterozygous genotype (Aa)
• q²: expected frequency of the homozygous recessive genotype (aa)
• Hardy‑Weinberg equilibrium: a state in which allele and genotype frequencies remain constant from generation to generation, provided that five assumptions are met.

When Does a Genotype Have the Smallest Frequency?

Consider a large, randomly mating population with two alleles, A (frequency p) and a (frequency q). If p = 0.9, then q = 0.1. The genotype frequencies are calculated as follows:

• AA: p² = 0.9² = 0.81
• Aa: 2pq = 2 × 0.9 × 0.1 = 0.18
• aa: q² = 0.1² = 0.01

Because q² is the square of the smallest allele frequency, it is the smallest genotype frequency. Mnemonic: “Q² is tiny because Q is already tiny.”

Estimating Allele Frequencies from Observed Genotypes

When genotype proportions are known, you can back‑calculate the allele frequency. For a sample showing 40% AA, 35% Aa, and 25% aa, the frequency of allele A (p) is:

• Count the A alleles: AA contributes 2 × 0.40 = 0.80, and Aa contributes 1 × 0.35 = 0.35.
• Total A alleles = 0.80 + 0.35 = 1.15.
• Divide by 2 (because each individual has two alleles): p = 1.15 / 2 = 0.575.

This calculation assumes the population is in Hardy‑Weinberg equilibrium, which allows the use of genotype proportions to infer allele frequencies.

Factors That Disturb Hardy‑Weinberg Equilibrium

Five classic forces can disrupt the equilibrium:

• Non‑random mating (e.g., inbreeding or assortative mating)
• Mutation introducing new alleles
• Natural selection favoring certain genotypes
• Genetic drift, especially in small populations
• Gene flow (migration) between populations

Among the answer choices, large population size does not disturb equilibrium; it actually helps maintain it by reducing the impact of genetic drift.

Carrier Frequency for Recessive Diseases

When a recessive disease has a known prevalence (q²), the carrier frequency (heterozygotes) can be estimated using the Hardy‑Weinberg formula 2pq. Example:

• Given q² = 1/2500, then q = 1/50 = 0.02.
• Thus, p = 1 – q = 0.98.
• Carrier frequency = 2pq = 2 × 0.98 × 0.02 ≈ 0.0392 ≈ 1/25.

Therefore, about 1 in 25 individuals are carriers.

Inbreeding and Excess Homozygosity

When a small, isolated community shows more homozygotes than expected, the most likely explanation is inbreeding. Inbreeding increases the probability that two alleles are identical by descent, raising homozygote frequencies and reducing heterozygosity.

X‑Linked Recessive Traits

For X‑linked traits, allele frequencies differ between sexes because males have only one X chromosome. If the recessive allele frequency in males is q = 1/12:

• Male phenotype frequency = q = 1/12.
• Female carrier frequency (heterozygous) = 2pq = 2 × (11/12) × (1/12) = 22/144.

This demonstrates how the same allele frequency yields different genotype expectations across sexes.

Impact of Migration (Gene Flow)

Migration introduces new alleles or changes existing frequencies. The correct statement is:

• It can change allele frequencies if migrant and resident populations differ.

Gene flow can either increase or decrease genetic variation within a population, depending on the direction and magnitude of the movement.

Testing Hardy‑Weinberg with the Chi‑Square Test

To assess whether observed genotype counts fit Hardy‑Weinberg expectations, calculate the chi‑square statistic:

• Observed counts: ++ = 16, +- = 28, -- = 20 (total = 64).
• First, estimate allele frequencies: p = (2×16 + 28) / (2×64) = 0.5625, q = 1 – p = 0.4375.
• Expected genotype frequencies: p² ≈ 0.316, 2pq ≈ 0.492, q² ≈ 0.191.
• Expected counts: ++ = 0.316×64 ≈ 20.2, +- = 0.492×64 ≈ 31.5, -- = 0.191×64 ≈ 12.2.
• Chi‑square = Σ[(O‑E)² / E] ≈ 0.955.

A chi‑square value of 0.955 is well below the critical value (≈3.84 for df = 1, α = 0.05), indicating the population does not deviate significantly from Hardy‑Weinberg equilibrium.

Summary of Core Concepts

• Hardy‑Weinberg provides expected genotype frequencies: p², 2pq, q².
• Allele frequencies can be derived from observed genotype proportions when equilibrium is assumed.
• Large population size maintains equilibrium; small size, non‑random mating, mutation, selection, and migration can disturb it.
• Carrier frequencies for recessive traits are calculated with 2pq.
• Inbreeding leads to excess homozygosity, while gene flow can shift allele frequencies.
• The chi‑square test evaluates the fit between observed and expected genotype distributions.

Further Reading and Practice

To deepen your understanding, explore these topics:

• Mathematical derivation of Hardy‑Weinberg equilibrium.
• Case studies of population bottlenecks and founder effects.
• Software tools for chi‑square goodness‑of‑fit tests (e.g., R, Python).
• Clinical implications of carrier screening for recessive diseases.