Why do some genetic diseases persist in populations for generations despite reducing survival? Why do rare recessive conditions appear more often in isolated communities? Population genetics — the study of how allele frequencies change over time — answers these questions with elegant mathematics. This guide covers the core ideas, starting from first principles.
Alleles, Genotypes, and Phenotypes
Every gene in a diploid organism exists in two copies (alleles), one inherited from each parent. If we label two versions of a gene A (dominant) and a (recessive):
- AA — homozygous dominant
- Aa — heterozygous (carrier)
- aa — homozygous recessive
The genotype (which alleles are present) determines the phenotype (what is actually expressed). If A is fully dominant, then AA and Aa have the same appearance; only aa individuals express the recessive trait.
Allele frequency is the proportion of each allele in the gene pool:
- p = frequency of A allele
- q = frequency of a allele
- p + q = 1 (all alleles must add up to 100%)
If a population of 100 individuals has 120 A alleles out of 200 total alleles, then p = 0.6 and q = 0.4.
Hardy-Weinberg Equilibrium
In 1908, mathematician G.H. Hardy and physician Wilhelm Weinberg independently showed that, in the absence of evolutionary forces, allele frequencies and genotype frequencies remain constant across generations.
The Hardy-Weinberg equation predicts genotype frequencies from allele frequencies:
p² + 2pq + q² = 1
Where:
- p² = frequency of AA
- 2pq = frequency of Aa (heterozygotes)
- q² = frequency of aa
Example: If p = 0.6 (A) and q = 0.4 (a):
- AA frequency: 0.6² = 0.36 (36%)
- Aa frequency: 2 × 0.6 × 0.4 = 0.48 (48%)
- aa frequency: 0.4² = 0.16 (16%)
These proportions arise naturally when individuals mate randomly — each allele is drawn independently from the gene pool, so frequencies multiply like independent probabilities.
The Five Conditions for Equilibrium
Hardy-Weinberg equilibrium only holds when five conditions are met:
- Random mating — individuals pair without preference for genotype
- No mutation — alleles don't change from one form to another
- No migration — no individuals entering or leaving the population
- Infinite population size — no random fluctuations
- No natural selection — all genotypes have equal fitness
In practice, none of these are perfectly met. The value of Hardy-Weinberg isn't as a description of reality — it's as a null model. Deviations from expected frequencies tell you which forces are at work.
Using Hardy-Weinberg in Practice
Hardy-Weinberg lets you estimate allele frequencies from observable phenotype counts:
Problem: 1 in 10,000 people has a recessive genetic disease. What fraction are carriers?
- Disease frequency = q² = 1/10,000 = 0.0001
- Therefore q = √0.0001 = 0.01
- And p = 1 − 0.01 = 0.99
- Carrier frequency = 2pq = 2 × 0.99 × 0.01 = 1.98% ≈ 1 in 50
This is a striking result: for every person with the disease, there are roughly 200 carriers — nearly invisible but carrying one copy of the allele.
Genetic Drift: Random Allele Frequency Change
Even without selection, mutation, or migration, allele frequencies change by chance in finite populations. A small population might, by luck, have slightly more A alleles reproduced in one generation. This is genetic drift.
The variance in allele frequency change per generation is:
Var(Δp) = p(1-p) / 2N
Where N is population size. In a population of 50, the standard deviation is √(p×q/100) — if p = q = 0.5, that's ±5% per generation by chance alone.
Consequences of genetic drift:
- Small populations lose genetic diversity rapidly
- Alleles can reach fixation (p = 1) or be lost (p = 0) by chance, regardless of fitness
- Isolated populations diverge genetically even without selection
Founder Effect and Bottlenecks
The founder effect occurs when a small group colonises a new area. The founders carry only a subset of the original population's alleles, so the new population starts with reduced diversity and skewed frequencies.
The Old Order Amish in Pennsylvania are a striking example: several rare genetic disorders — including Ellis-van Creveld syndrome (extra fingers plus heart defects) — appear at frequencies 10–100 times higher than the global average, traceable to a handful of 18th-century founders.
A population bottleneck is a drastic, temporary reduction in population size (through disease, disaster, or hunting). The surviving gene pool may not represent the original allele frequencies, and genetic diversity is permanently reduced.
Natural Selection
Natural selection changes allele frequencies systematically — not randomly like drift. The selection coefficient (s) measures the fitness disadvantage of a genotype:
If the most fit genotype has relative fitness 1, a disadvantaged genotype has fitness (1 − s). When s = 1, the allele is lethal.
The change in frequency of a recessive allele per generation under selection against aa:
Δq ≈ -sq²p / (1 - sq²)
Selection against recessive alleles is slow when rare — most copies hide in carriers (Aa) where they're invisible to selection. This is why genetic diseases don't disappear even with strong selection against the aa phenotype.
Balanced Polymorphism: Sickle Cell Anemia
The classic example of heterozygote advantage: sickle cell anemia is caused by a recessive allele (HbS). Homozygous (HbS HbS) individuals have severe anemia; the allele clearly reduces fitness. So why does it persist at high frequencies (up to 25%) in sub-Saharan Africa?
Because Aa carriers (HbA HbS) are more resistant to malaria than normal individuals (HbA HbA). In malaria-endemic regions, carriers have higher fitness than either homozygote — this maintains both alleles in the population through balancing selection.
The stable equilibrium frequency is:
q_eq = s₁ / (s₁ + s₂)
Where s₁ is the disadvantage of AA (normal) and s₂ is the disadvantage of aa (full sickle cell). In regions without malaria, s₁ ≈ 0 and the allele drifts downward — exactly what we observe in African-descended populations outside malaria zones.
Mutation Rate
New alleles enter the population through mutation. The human germline mutation rate is approximately 1.1 × 10⁻⁸ per base pair per generation — about 33 new mutations per person.
For a gene locus:
μ = new mutations / (2N × generations)
Mutation rate is low enough that it barely shifts allele frequencies in any single generation (unlike selection or drift). But over thousands of generations, mutation-selection balance determines the steady-state frequency of deleterious alleles in the population.
Biodiversity: Measuring What's There
Population genetics also gives us tools to measure biodiversity. The Shannon-Wiener diversity index H' quantifies species diversity:
H' = -Σ(pᵢ × ln pᵢ)
Where pᵢ is the proportion of each species. A community with 10 species all equally abundant has higher H' than one where 90% of individuals belong to a single species.
Evenness (J) = H' / H'max measures how equally individuals are distributed among species, independently of richness. J = 1 means perfectly even; J close to 0 means one species dominates.
These metrics are used in conservation biology to assess ecosystem health, plan protected areas, and track the effects of habitat loss over time.
From Population Genetics to Evolution
Population genetics provides the mathematical framework that connects Darwinian evolution (survival of the fittest) to Mendelian genetics (inheritance of alleles). The four forces — selection, drift, mutation, migration — act on allele frequencies, and over sufficient time, their cumulative effects produce speciation.
Use our Hardy-Weinberg Calculator, Allele Frequency Calculator, Population Growth Calculator, Genetic Drift Calculator, and Biodiversity Index Calculator to explore these models with your own values.