Hardy Weinberg Equilibrium

01 / UNDERSTAND & APPLY

1. A null model separates inheritance from evolutionary forces

Population genetics studies how inherited variants are distributed within and among populations and how those distributions change. A population’s gene pool includes the alleles present in its reproducing members. The unit counted must be explicit: individuals, genotypes, allele copies and phenotypes have different denominators.

The Hardy–Weinberg model provides a baseline for a diploid, autosomal locus. Random union of gametes with the same allele frequencies in the two gamete pools produces predictable zygote genotype proportions. Under additional conditions that preserve those frequencies, the proportions can remain stable across generations.

This is a model, not a claim that real populations are literally infinite or free of every evolutionary process. Its value is that it gives a specified expectation against which observations can be compared. Agreement is compatible with the model but does not prove that every assumption is exactly true.

Distinguish genotype proportions at one life-cycle stage from allele-frequency change through time. A population can depart from Hardy–Weinberg genotype proportions without changing its allele frequencies, and a locus can display Hardy–Weinberg proportions in each generation while its allele frequency changes between generations.

02 / UNDERSTAND & APPLY

2. Count copies directly before using an equilibrium equation

For N diploid individuals with genotypes AA, Aa and aa, there are 2N allele copies at the autosomal locus. Each AA contributes two A copies, each Aa one A and one a, and each aa two a copies. Thus p = (2nAA + nAa)/(2N), while q = (2naa + nAa)/(2N).

These counting equations do not require Hardy–Weinberg equilibrium. They are bookkeeping identities for the stated diploid genotypes. Dividing the number of AA individuals by N gives the AA genotype frequency, not the A allele frequency.

If A and a are the only alleles, p + q = 1. The labels p and q do not mean common and rare, beneficial and harmful, or dominant and recessive. Dominance describes a heterozygote phenotype and does not dictate an allele’s population frequency.

The factor 2N depends on ploidy and locus. A haploid sample has one copy per individual, and a sex-linked locus can require separate counting by sex-chromosome constitution. Do not automatically apply the autosomal two-copy denominator to every biological sample.

03 / UNDERSTAND & APPLY

3. Derive the square from independent gamete draws

Let the frequency of A in each gamete pool be p and that of a be q. Random union gives AA with probability p × p = p² and aa with probability q × q = q². A heterozygote can arise in two distinguishable parental orders: A from one pool and a from the other, or the reverse.

Those two heterozygote routes have probabilities pq and qp, so their combined frequency is 2pq. Adding all zygote outcomes gives p² + 2pq + q² = (p + q)² = 1. The two in 2pq is not because a heterozygote contains two copies of the same allele.

For p = 0.6 and q = 0.4, the expected proportions are 0.36 AA, 0.48 Aa and 0.16 aa. In a model cohort of 500 zygotes, the expected counts are 180, 240 and 80. Expected counts need not be whole numbers for every chosen cohort size; real samples can fluctuate.

One generation of random union can produce these zygote proportions even if parental genotypes did not follow them, provided the stated gamete-pool conditions hold. Maintaining the same proportions at later stages and generations additionally requires that other processes do not change the relevant contributions.

04 / UNDERSTAND & APPLY

4. State the assumptions and the stage being sampled

The standard equilibrium model assumes random mating with respect to the locus, no mutation, no gene flow, no differential reproductive success among the genotypes, and a sufficiently large population that drift is negligible. Ordinary Mendelian segregation and equal allele frequencies in the contributing gamete pools are also part of the simple derivation.

Random mating concerns the pairing process. It does not mean every organism has the same phenotype or that mating is random with respect to every possible trait. Conversely, assortative mating or inbreeding can alter genotype proportions at a locus without requiring a new allele to appear.

Life-cycle timing matters. Random mating may establish Hardy–Weinberg proportions among zygotes, after which genotype-dependent survival changes the proportions among adults. Comparing adult data with a zygote expectation without considering selection can obscure the process under study.

A sample that departs from expectation may reflect selection, population subdivision, nonrandom mating, genotyping errors or sampling variation. The departure identifies a question to investigate, not a unique cause. Statistical evaluation and study design are needed before treating a small numerical difference as meaningful.

05 / UNDERSTAND & APPLY

5. Taking a square root requires a phenotype model

If a fully penetrant autosomal recessive phenotype occurs only in aa individuals, and the population satisfies the relevant Hardy–Weinberg assumptions, its frequency can be identified with q². Taking the positive square root then estimates q, after which p = 1 − q and the heterozygote frequency is 2pq.

Without those conditions, observed phenotype prevalence is not automatically q². Incomplete penetrance, multiple genetic causes, environmental phenocopies, age-dependent expression and population structure can break the shortcut. These classroom calculations are not personal genetic-risk assessments or diagnostic methods.

For a fictional recessive phenotype frequency of 0.09, q = 0.3 and p = 0.7. The model heterozygote frequency is 0.42 and the dominant-phenotype frequency is p² + 2pq = 0.91. The dominant phenotype is not p² alone because heterozygotes also display it under complete dominance.

If the question asks for the probability of being a carrier among unaffected individuals, condition on that group: 2pq/(1 − q²). In this example the result is 0.42/0.91 = 6/13, approximately 46.15%. It differs from the 42% carrier frequency in the entire population because the denominator changed.

06 / UNDERSTAND & APPLY

6. Heterozygosity is a diversity measure with a stated model

At a biallelic locus under random union, expected heterozygosity is H = 2pq = 2p(1 − p). It is largest at p = q = 0.5, where H = 0.5. If either allele is fixed, expected heterozygosity is zero.

This maximum applies to the biallelic model. With three or more alleles, heterozygosity can exceed one-half. If allele frequencies are p1, p2 and so on, expected homozygosity is the sum of their squared frequencies, and expected heterozygosity is 1 minus that sum.

For three alleles at frequencies 0.5, 0.3 and 0.2, expected homozygosity is 0.25 + 0.09 + 0.04 = 0.38, so heterozygosity is 0.62. Individual heterozygous genotype frequencies still combine two parental orders, giving 2p1p2, 2p1p3 and 2p2p3.

Observed heterozygosity is the fraction of sampled individuals actually heterozygous. It can differ from the random-union expectation. Neither measure alone describes all genome-wide diversity, all adaptive potential or the health of every organism in the population.

07 / UNDERSTAND & APPLY

7. Drift is inherited sampling change, not purposeful improvement

Genetic drift arises because a finite set of allele copies contributes to later generations. Even when variants have equal expected reproductive success, the realised sample can contain more or fewer copies of one allele. The direction is not chosen to improve adaptation.

Imagine a source gamete pool with A frequency 0.5. A small next generation can contain 6 A copies among 10 sampled copies, giving 0.6, while another replicate contains 4 of 10, giving 0.4. The same starting probability can produce different outcomes. An expectation of 0.5 does not require every replicate to equal 0.5.

Drift generally has stronger effects when effective population size is small. Effective size summarises aspects of reproductive sampling and can differ from a census count because reproductive success, sex ratios and demographic history are uneven. A large head count after a recent severe bottleneck does not instantly restore lost variants.

Drift can eliminate or fix an allele, including variants that are not advantageous. Selection can operate simultaneously, so a real allele’s fate need not be assigned exclusively to one process. A favourable variant can be lost while rare, and a frequency increase does not by itself prove an adaptive benefit.

08 / UNDERSTAND & APPLY

8. Founder events and bottlenecks change the sampling context

A founder event occurs when a new population is established by a limited subset of a source population. The founders may carry allele proportions that differ from the source simply because they are a small sample. Their descendants inherit that starting composition unless later processes modify it.

A bottleneck sharply reduces a population’s size. Surviving lineages can represent only part of the earlier genetic variation. Recovery in numbers increases the population’s size, but an allele absent from every survivor cannot reappear merely because those survivors reproduce; mutation or immigration would be needed to introduce it again.

Not every founder sample must have a different frequency, and not every bottleneck must lose every rare allele. These are probabilistic risks rather than guaranteed outcomes. A rare allele has fewer copies available and can be particularly vulnerable to loss in a small sample.

The demographic event can also involve selection. If a storm disproportionately removes organisms lacking a heritable anchoring trait, survival is not random with respect to that trait. It is useful to distinguish the change in population size from the mechanisms determining which variants survive.

09 / UNDERSTAND & APPLY

9. Repeated drift can separate populations while reducing diversity within them

In a closed neutral finite-population model with no mutation, repeated drift eventually leads to fixation or loss at a biallelic locus. Fixation means one allele has frequency one within that population; it does not mean the allele is dominant, beneficial or fixed across the species.

Under that ideal neutral model, the probability that an allele eventually fixes equals its current frequency. An allele at 0.2 has a fixation probability of 0.2 across many comparable replicate histories, not a promise that it will reach fixation after a particular number of generations.

Independent populations can fix different alleles. Average variation within each population can decrease while differences among populations increase. Pooling the populations can therefore conceal the loss of local diversity and create a misleading picture of one freely mating gene pool.

Mutation and gene flow can replenish variation, and selection can change fixation probabilities. The neutral prediction is a conditional baseline. It must not be applied unchanged to a strongly selected allele, a changing migration system or a population with a different inheritance model.

10 / UNDERSTAND & APPLY

10. Mixture and mating structure require separate accounting

For a simple mixture of genetic contributions, let a fraction m come from a source with allele frequency pm and the remaining fraction 1 − m from residents with frequency pr. The mixed allele frequency is p′ = (1 − m)pr + m pm. Here m describes the relevant genetic contribution, not automatically the fraction of visitors seen in the habitat.

If 10% of the contributing pool has A frequency 0.8 and 90% has A frequency 0.2, the mixture has p′ = 0.9 × 0.2 + 0.1 × 0.8 = 0.26. This changes allele frequency without requiring mutation or a selective advantage for A.

Population structure can also alter pooled genotype expectations. Two equal-sized subpopulations with A frequencies 0.2 and 0.8 can each have heterozygosity 0.32 under their own Hardy–Weinberg model. Pooled A frequency is 0.5, whose single-population expectation would be 0.5, yet the actual pooled heterozygosity remains 0.32.

This heterozygote deficit from pooling differentiated groups is called the Wahlund effect. It illustrates why a combined sample is not necessarily one random-mating population. The deficit does not require heterozygotes to have lower survival, and it cannot identify individual ancestry or fitness by itself.

11 / UNDERSTAND & APPLY

11. Use a consistent sequence for population questions

Begin by identifying ploidy, locus type and what was actually measured. If genotypes are known, count alleles directly. If only a recessive phenotype is supplied, state the inheritance and equilibrium assumptions before taking a square root. Then calculate genotype expectations and convert to counts only after the probabilities are clear.

Check that allele frequencies sum to one, genotype frequencies sum to one, and expected counts sum to the stated population size. Compare observed and expected values at the same life-cycle stage. A percentage-point change and a relative percentage change are different quantities.

Finally, separate calculation from interpretation. A correct p value is not evidence that mating was random. A heterozygote deficit does not uniquely prove inbreeding. A rise in an allele does not uniquely prove selection. Mechanistic claims require information beyond the arithmetic.

Hardy–Weinberg reasoning and drift reasoning complement one another: the former gives a deterministic random-union baseline, while the latter explains finite-sampling departures through time. Knowing why a model is useful includes knowing the conditions under which its simplest shortcuts fail.

12 / UNDERSTAND & APPLY

12. Worked population-genetic cases

All datasets are original and fictional. Phenotype calculations are model exercises, not personal genetic counselling.

13 / UNDERSTAND & APPLY

13. Original practice with explained answers

Keep allele, genotype and conditional-probability denominators separate.