Concepts of Population Genetics & Hardy–Weinberg Law

Introduction

Population genetics studies how allele and genotype frequencies change over time within populations. It unites Mendelian inheritance with evolutionary theory by describing how evolutionary forces — mutation, natural selection, genetic drift, migration, and non-random mating — alter genetic variation. While Mendel described inheritance in individuals, population genetics addresses the dynamics of genes in groups of interbreeding individuals (the population).

Key Concepts

Population

A population is a set of individuals of the same species that live in a particular geographic area and interbreed. Evolution is a property of populations: allele frequencies change across generations in the population, not in individual organisms.

Gene pool

The gene pool is the sum of all alleles present in a population at a given locus. Every copy of a gene present in every individual contributes to the gene pool. The gene pool is the basis from which allele frequencies are computed.

Allele frequency

For a locus with two alleles (A and a) in a diploid population of size N, the total number of alleles = 2N. Allele frequencies are defined as:

p = (number of A alleles) / (2N)
q = (number of a alleles) / (2N)
where p + q = 1

Example: In a population of 50 diploid plants (2N = 100 alleles), if there are 70 A alleles and 30 a alleles, then p = 0.70 and q = 0.30.

Genotype frequency

Genotype frequencies are the proportion of individuals with each genotype:

f(AA) = (number of AA individuals) / N
f(Aa) = (number of Aa individuals) / N
f(aa) = (number of aa individuals) / N
and f(AA) + f(Aa) + f(aa) = 1

Forces that Change Allele Frequencies

In nature, allele frequencies change due to several evolutionary forces. Understanding these forces helps explain why populations deviate from the Hardy–Weinberg expectations.

Mutation

Mutation introduces new alleles into the population. While the per-locus mutation rate is usually low, over long timescales mutation supplies raw material for evolution.

Natural selection

Selection increases the frequency of alleles that enhance reproductive success. Selection can be directional, stabilizing, or disruptive and may act on genotypes or phenotypes.

Genetic drift

Genetic drift refers to random changes in allele frequencies, especially important in small populations (founder events, population bottlenecks). Drift can lead to loss or fixation of alleles by chance.

Migration (gene flow)

Movement of individuals (and their alleles) between populations tends to homogenize allele frequencies across populations and introduce new alleles to recipient populations.

Non-random mating

Non-random mating (e.g., inbreeding or assortative mating) changes genotype proportions without altering allele frequencies directly. Inbreeding increases homozygosity and can expose recessive deleterious alleles.

Hardy–Weinberg Law: Statement and Assumptions

The Hardy–Weinberg Law (Hardy, 1908; Weinberg, 1908) provides a null model: in an idealized population (no mutation, no migration, no selection, infinitely large size, and random mating), allele and genotype frequencies remain constant across generations. This equilibrium state is useful because deviations indicate the action of evolutionary forces.

Assumptions

The HWE model requires:

• A very large (effectively infinite) population size
• Random mating with respect to the locus in question
• No mutation introducing or removing alleles
• No migration into or out of the population (closed population)
• No natural selection affecting survival or reproduction of genotypes

Mathematical Formulation

Consider a single locus with two alleles A and a. Let p be the frequency of A and q the frequency of a. Then p + q = 1. Under random mating, gametes combine according to allele frequencies, producing genotype probabilities:

P(AA) = p * p = p^2
P(Aa) = p * q + q * p = 2pq
P(aa) = q * q = q^2
So: p^2 + 2pq + q^2 = 1

Interpretation

If a population is in HWE, genotype frequencies are entirely predictable from allele frequencies. Conversely, if genotype frequencies differ from p^2 : 2pq : q^2, one or more HWE assumptions are violated (selection, migration, drift, mutation, or non-random mating).

Worked Example — Checking HWE

A population of 1,000 diploid individuals shows the following genotype counts at a locus:

• AA = 360 individuals
• Aa = 480 individuals
• aa = 160 individuals

Determine whether the population is in Hardy–Weinberg equilibrium.

Step 1: Calculate allele frequencies

Total alleles = 2N = 2000. Number of A alleles = (2 × 360) + 480 = 720 + 480 = 1200. Number of a alleles = (2 × 160) + 480 = 320 + 480 = 800. So p = 1200/2000 = 0.6 and q = 800/2000 = 0.4.

Step 2: Calculate expected genotype frequencies under HWE

Expected proportions:

f(AA) = p^2 = 0.6^2 = 0.36
f(Aa) = 2pq = 2 × 0.6 × 0.4 = 0.48
f(aa) = q^2 = 0.4^2 = 0.16

Multiply by N = 1000 individuals to get expected counts: AA = 360, Aa = 480, aa = 160.

Step 3: Compare observed vs. expected

Observed genotype counts exactly match expected counts; therefore, the population is consistent with Hardy–Weinberg equilibrium at this locus.

Estimating Carrier Frequency for a Recessive Disease

HWE is particularly useful in medical genetics to estimate the fraction of carriers (heterozygotes) for autosomal recessive disorders when disease prevalence is known.

Example — Sickle-cell anemia: Suppose the frequency of affected individuals (homozygous recessive, aa) in a population is 1 in 10,000 (q^2 = 1/10,000 = 0.0001). Find carrier frequency (2pq).

Step 1: q = sqrt(q^2) = sqrt(0.0001) = 0.01 Step 2: p = 1 - q = 0.99 Step 3: Carrier frequency ≈ 2pq = 2 × 0.99 × 0.01 ≈ 0.0198 (≈ 1.98% or ~2%).

When and Why Populations Deviate from HWE

Real populations often violate HWE assumptions. Deviations point to evolutionary processes:

• Selection: Differential survival or reproduction alters genotype frequencies. Example: a recessive allele confers disease—selection reduces q over generations.
• Mutation: Constant input of new alleles can shift p and q (though mutation rates are typically low).
• Migration: Gene flow can introduce alleles from other populations and shift frequencies.
• Genetic drift: Random fluctuations in small populations can fix or eliminate alleles by chance.
• Non-random mating: Inbreeding increases homozygosity and can reveal recessive deleterious alleles.
• Assortative mating: Mating based on phenotype can increase or decrease heterozygosity at correlated loci.

Practical Considerations and Applications

HWE is used across fields:

• Conservation genetics: Track loss of genetic diversity in small or endangered populations.
• Plant and animal breeding: Predict genotype proportions and maintain desirable traits.
• Population screening: Estimate carrier frequencies for inherited disorders.
• Forensic genetics and ancestry inference: Underlying allele frequency models inform match probabilities.

Worked Numerical Problems (Practice)

1) In a population of 200 individuals you observe 98 AA, 84 Aa and 18 aa. Are these counts in HWE? Show calculations.

(Hint: compute p and q from allele counts, derive expected genotype counts p^2, 2pq, q^2 and compare with observed; you may use a chi-square test for formal assessment.)

2) A plant breeder finds that 9% of progeny show a recessive phenotype (aa). Estimate p, q and the expected heterozygote frequency.

Summary

Population genetics quantifies genetic variation in populations and identifies the forces that change allele frequencies. The Hardy–Weinberg law establishes a null expectation (p^2 : 2pq : q^2) under ideal conditions. Deviations from HWE indicate evolutionary processes such as selection, drift, migration, mutation, or non-random mating. HWE is a simple but powerful tool used in evolutionary biology, medicine, conservation, and breeding.