Genetics and Molecular Biology · Unit 6: Populations and complex traits

Population Genetics: Hardy-Weinberg and Evolutionary Forces

Why does a dominant allele not take over a population, and what does it take to change how common an allele is? This lesson derives the Hardy-Weinberg proportions, tests a real-looking sample against them with a chi-square, and then sets each evolutionary force (selection, drift, migration, mutation and non-random mating) against that null model. Afterwards you can turn genotype counts into allele frequencies, estimate carrier frequency from the incidence of a recessive condition, and explain how heterozygote advantage keeps the sickle allele in malaria-exposed populations.

Watch first Why Dominant Alleles Do Not Spread · 2:52

If an allele is dominant, it shows up in heterozygotes as well as homozygotes. So why does a dominant allele not spread, generation by generation, until almost everyone shows it?

That was the question put to the mathematician G. H. Hardy in 1908. The answer is short, and it is the starting point for all of population genetics: dominance describes how alleles act inside one person, and says nothing about how common they become. To change how common an allele is, something has to act on the population. This lesson builds the baseline in which nothing acts, then adds each force in turn.

The idea: with nothing acting, frequencies stay put

A population, here, is a group of individuals that mate with one another. At a locus (a place on a chromosome) with two alleles, A and a, call the fraction of all allele copies that are A p, and the fraction that are a q. With only two alleles, p + q = 1.

Now picture every egg and sperm in the population thrown into one pool, and zygotes made by drawing one of each at random. The chance of drawing A from both is p x p. The chance of a heterozygote is pq for A from the egg and a from the sperm, plus qp for the reverse. That gives the Hardy-Weinberg proportions:

AA = p^2 Aa = 2pq aa = q^2 (and p^2 + 2pq + q^2 = 1)

This is the product and sum rule from probability and crosses, applied to a whole population instead of one cross. Count the A copies in the next generation and you get p^2 + pq = p(p + q) = p. The allele frequency has not moved, and one round of random mating restores the proportions whatever they were before.

Wilhelm Weinberg, a physician in Stuttgart, derived the same result independently in a lecture in January 1908, so the principle carries both names.

The assumptions are the whole point. The population is large, mating is random with respect to this locus, allele frequencies are the same in both sexes, and there is no selection, mutation, migration or drift. Real populations break these. Hardy-Weinberg is a null model: a statement of what you would see if nothing were happening, against which you detect what is.

Counting alleles and testing equilibrium

Worked problem, part 1. In a sample of 1,000 people typed at a codominant locus (each genotype distinguishable), there are MM 298, MN 489 and NN 213. Assume unrelated individuals and accurate typing.

  1. Count alleles, not people. Each MM person carries two M copies and each MN person one. M copies = 2 x 298 + 489 = 1,085, out of 2,000. So p = 0.5425, and q = 1 - p = 0.4575. Count first; never take a square root of a homozygote frequency to find p before you know the sample is at equilibrium.
  2. Expected counts. MM = 1,000 x 0.5425^2 = 294.3. MN = 1,000 x 2 x 0.5425 x 0.4575 = 496.4. NN = 1,000 x 0.4575^2 = 209.3. They add to 1,000.
  3. Chi-square. (298 - 294.3)^2 / 294.3 = 0.046; (489 - 496.4)^2 / 496.4 = 0.110; (213 - 209.3)^2 / 209.3 = 0.065. The total is 0.22.
  4. Degrees of freedom. Three classes, minus 1 because the counts must add to 1,000, minus 1 more because p was estimated from these same counts: 1, not 2.
  5. Decide. The critical value at P = 0.05 for one degree of freedom is 3.84. Since 0.22 is far below it (P is about 0.64), there is no evidence of departure.

That last sentence is all you can say. A good fit does not prove random mating or the absence of selection. A poor fit does not prove selection either, because inbreeding, a sample that mixes two populations, and genotyping errors can all produce one.

Worked problem, part 2. A recessive condition affects 1 in 2,500 newborns in a hypothetical population. Assume the population is at equilibrium at this locus. Affected people are aa, so q^2 = 1/2,500 and q = 0.02, p = 0.98. Carriers are 2pq = 2 x 0.98 x 0.02 = 0.039, about 1 in 26.

Carriers are about 100 times as common as affected people (0.0392 / 0.0004 = 98). The ratio is 2pq / q^2 = 2p/q, close to 2/q when q is small. A rare allele sits almost entirely in heterozygotes: here 98% of a copies are in carriers. This is how families meet a recessive condition with no affected relatives in the pedigree.

The forces that move frequencies

Selection means genotypes differ consistently in survival or reproduction. It changes allele frequencies, and its speed depends on which genotypes it can see. Against a rare recessive it is slow, because selection acts only on aa, and almost every a copy is hiding in a carrier. If aa individuals never reproduced, q would fall as q_n = q_0 / (1 + n q_0). Starting at 0.02, halving q to 0.01 would take 50 generations.

Migration (gene flow) mixes gene pools. If a fraction m of the next generation’s allele copies comes from migrants with frequency p_m:

p’ = (1 - m) p + m p_m

With p = 0.2, p_m = 0.6 and m = 0.1, p’ = 0.18 + 0.06 = 0.24. Migration pulls populations toward each other.

Mutation is the source of new alleles (see mutation) but a weak force on frequency. In human parent-offspring trios, the TOPMed study estimated about 1.2 x 10^-8 new single-base changes per base pair per generation, a genome average. Over a few generations, that barely moves an allele frequency.

Non-random mating is the odd one out. Inbreeding, mating between relatives, and assortative mating for a trait influenced by the locus, make homozygotes more common and heterozygotes rarer. They pair the same copies differently, so on their own they change genotype frequencies and leave p and q where they were. They can still expose recessive alleles to selection.

Drift is change by chance.

Drift and founders: chance in small populations

Each generation, only a sample of the parents’ allele copies makes it into the offspring. In a small population that sample can stray far from the parents’ frequency. Under the simplest model, with a constant effective population size Ne (the number of breeders in an idealized population that would drift at the same rate), the spread of the change in one generation is:

variance of the change in p = p(1 - p) / (2Ne)

With Ne = 50 and p = 0.5, the variance is 0.0025, a standard deviation of 0.05 in a single generation. Drift has no direction, and in a small population a frequency wanders until it hits 0 (the allele is lost) or 1 (it is fixed).

Drift and selection: replicate populations⚙ original · interactive

Expected beside observed
ExpectedObserved

With no systematic force acting, the expected p stays put; drift scatters small populations around that line without a direction, and selection gives the line one. This is a sketch of the logic: constant Ne, random mating, selection against aa only. Viability selection with wAA = wAa = 1 and waa = 1 - s; each generation draws 2Ne allele copies at random (binomial sampling); 30 replicates, 50 generations. Here s is the cost to aa, not the s of q = s/(s + t).

Peter Buri saw this in 1956 with 107 laboratory populations of Drosophila, each kept at 8 males and 8 females and each started at a frequency of one half for an eye-colour allele. Over 19 generations the populations spread apart, and many lost or fixed the allele, while the average across them stayed near where it began.

A founder effect is drift at the start of a population: a small group carries, by chance, a different set of alleles than the population it left. In a 2020 exome study of 5,521 Lancaster County Old Order Amish participants, 124 carried the KCNQ1 variant p.T224M, about 1 in 45, and none was homozygous. That is an allele frequency of 124 / (2 x 5,521), about 0.011: a carrier frequency of 1 in 45 is not an allele frequency of 1/45. The authors note that pathogenic variants may occur at much higher frequency in founder populations, and they attribute this variant’s frequency to a founder effect and drift. Tay-Sachs disease among Ashkenazi Jewish people, and the group of rare diseases called the Finnish disease heritage, are other examples often taught this way. These are features of particular populations and their histories, not of races: most human genetic variation lies within populations, not between them.

Balanced polymorphism: sickle cell and malaria

The sickle allele of the β-globin gene (HBB) changes glutamate to valine at position 6 of the β chain, and deoxygenated HbS polymerises (see protein binding and allostery). Sickle-cell disease (SS) is autosomal recessive. In conditions where the disease shortens life, selection should remove S. Yet in regions with intense falciparum malaria the allele is common.

The reason is heterozygote advantage. Sickle-cell trait (AS) does not give immunity to malaria. In two cohorts of children on the Kenyan coast it made no difference to symptomless infection, but it was associated with about 50% protection against mild malaria, 75% against hospital admission and almost 90% against severe disease. AS outsurvives AA through malaria and SS through the disease, and selection then holds both alleles. If the cost to AA is s and the cost to SS is t, the equilibrium is:

q = s / (s + t)

Design problem

Interpret these illustrative data (not measurements) for a malaria-endemic population. Newborns are AA 0.81, AS 0.18 and SS 0.01, so q = 0.10, at Hardy-Weinberg proportions. Relative survival to adulthood is AA 0.88, AS 1.00 and SS 0.20. What maintains the S allele, where should its frequency settle, and what would adults look like? What other data would you need to rule out migration or drift?

One way to do it

Heterozygotes survive best, so this is balancing selection. The fitness costs are s = 1 - 0.88 = 0.12 for AA and t = 1 - 0.20 = 0.80 for SS. Assuming a large, randomly mating population with survival as the only force, the equilibrium is q = 0.12 / 0.92 = 0.13. Since newborns start at 0.10, q should rise slightly over generations.

Check one generation. Survivors are AA 0.81 x 0.88 = 0.713, AS 0.18, SS 0.01 x 0.20 = 0.002, out of 0.895. As fractions of adults: AA 0.797, AS 0.201, SS 0.002. Adult q = 0.201/2 + 0.002 = 0.103. Under Hardy-Weinberg, q = 0.103 predicts 0.184 heterozygotes, but adults have 0.201. Adults show a heterozygote excess, even though their newborns fit. When to sample matters.

To separate selection from the alternatives, show that the allele’s frequency follows the historical intensity of malaria across regions, that AS children are protected in follow-up studies, and that newborns fit equilibrium while adults show the excess. Migration from a high-S population would raise q without producing that newborn-to-adult change.

The next lesson uses the same allele counts, across hundreds of thousands of loci at once, to look for variants associated with complex traits.

How we measure it

Gene counting and the Hardy-Weinberg chi-square test

Each genotype is counted, and allele frequencies are found by counting allele copies, with no assumption about equilibrium. Expected counts under Hardy-Weinberg proportions are then compared with the observed ones, by a chi-square with one degree of freedom or by an exact test when an allele is rare.

Hemoglobin electrophoresis

Hemoglobin from a blood sample is run through a gel in an electric field. HbS has lost one negative charge on each beta chain, so it moves differently from HbA, and carriers (AS) show both bands. This lets a survey count AA, AS and SS individuals directly.

Replicate small populations

Many populations are started from the same allele frequency and kept at a fixed small size. Chance acts differently in each, so the spread of frequencies across populations shows drift directly, which a single population cannot.

Exome sequencing in a founder population

The protein-coding parts of the genome are sequenced in thousands of members of a population descended from few founders. Variants that drift made common can be counted and tested for association with a trait, and then tested for function in cells.

Finish line

Test yourself

8 questions

Practice and MCAT-style, with explained answers.

Next · Lesson 26

Complex Traits, Heritability and GWAS

If height is 80 percent heritable, is 80 percent of your height genetic? No, and the reason is the whole lesson.

Continue →

← Lesson 24 · Forward and Reverse Genetics: Screens, RNAi and CRISPR

Deoxyhemoglobin A — the normal protein
Deoxyhemoglobin A — the normal protein ● LIVE · RCSB PDB 2HHB
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Loading live structure 2HHB from RCSB PDB…

Two alpha and two beta chains, each holding a heme. Find residue 6 near the start of each beta chain, on the surface: in HbA it is glutamate.

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Deoxyhemoglobin S — the sickle protein
Deoxyhemoglobin S — the sickle protein ● LIVE · RCSB PDB 2HBS
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Loading live structure 2HBS from RCSB PDB…

The same fold with valine at beta 6. In the crystal, that valine sits against a pocket on a neighbouring tetramer, the contact that lets deoxy HbS form long fibres in red cells.

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Oxyhemoglobin A — the shape that does not polymerise
Oxyhemoglobin A — the shape that does not polymerise ● LIVE · RCSB PDB 1HHO
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Loading live structure 1HHO from RCSB PDB…

Compare the arrangement of the chains with the deoxy form. HbS fibres form from the deoxy state, which is why sickling follows loss of oxygen and oxygenated HbS stays in solution.

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G. H. Hardy

G. H. Hardy

Showed in a short 1908 note that under random mating the genotype proportions at a locus settle at p^2, 2pq and q^2 and stay there, so dominance by itself does not make an allele spread.

Hardy was a pure mathematician at Trinity College, Cambridge, and his only well-known contribution to biology is a note of about a page. He wrote it, the note explains, because R. C. Punnett had drawn his attention to remarks by the statistician Udny Yule. The puzzle was this: if a trait such as short fingers is dominant, should it not spread until three people in four show it? Hardy's answer, published in Science on 10 July 1908, was no. He assumed a large population, random mating, equal fertility of the genotypes and the same genotype distribution in both sexes. Under those conditions he showed that the genotype proportions reached after one generation of random mating stay fixed, whatever they are, and that a dominant allele has no built-in tendency to increase. The calculation needs only multiplication. What it made possible was a baseline: once you know what happens when nothing acts on a population, any departure from it becomes something to measure and explain.

✓ verbatim Hardy, Mendelian proportions in a mixed population, Science 28:49 (1908).
Wilhelm Weinberg

Wilhelm Weinberg

A physician in Stuttgart who derived the same equilibrium independently in 1908 while working on statistical problems of human heredity.

Weinberg practised medicine in Stuttgart and studied heredity in his own time, using the records of human families. Human data are hard to read. Families are small, matings cannot be arranged, and the families that come to notice are often those with an affected member, which inflates the apparent proportion of affected children. Weinberg worked on corrections for problems of this kind. According to the geneticist Curt Stern, writing in Science in 1943, Weinberg gave a lecture on 13 January 1908 to the natural history society of Württemberg in which he derived the same genotype proportions that Hardy published in July of that year, and the lecture was printed in the society's yearbook. For decades English-speaking geneticists called the result Hardy's law. Stern's short note argued that Weinberg had reached it independently, and the double name has been standard since. The credit matters for a practical reason: Weinberg came to the principle as a tool for analysing human populations, which is how it is used in medical genetics today.

✓ verbatim Stern, The Hardy-Weinberg law, Science 97:137 (1943); Weinberg, Jahreshefte des Vereins für vaterländische Naturkunde in Württemberg 64 (1908).
Anthony Allison

Anthony Allison

Linked the high frequency of sickle-cell trait in parts of East Africa to malaria, and reported in 1954 that carriers of the trait were protected against falciparum malaria.

Sickle-cell disease was a puzzle for population genetics. People with two copies of the sickle allele were often severely ill and, in the conditions of the time, many died young, so selection should have removed the allele. Yet in parts of Africa the trait, one copy, was common. J. B. S. Haldane had suggested in 1949 that some inherited red-cell disorders might persist because they protect against malaria. Allison, working in East Africa, compared the frequency of sickle-cell trait across regions and found it highest where falciparum malaria was intense. In 1954, in the British Medical Journal, he reported that carriers of the trait had lower malaria parasite counts, and he described infecting adult volunteers with falciparum malaria, an experiment that would not be approved today, in which carriers of the trait developed infection less often. The paper made the sickle allele the standard example of a balanced polymorphism, an allele held in a population because heterozygotes survive best.

✓ verbatim Allison, Protection afforded by sickle-cell trait against subtertian malarial infection, British Medical Journal 1:290 (1954).
Peter Buri

Peter Buri

Watched genetic drift happen in 107 small laboratory populations of Drosophila, all started at the same allele frequency, which spread apart by chance until many lines had lost or fixed the allele.

Drift is easy to state and hard to see, because one population follows one path and you cannot tell chance from cause. Buri's solution was to run the same history many times. He set up 107 populations of the fruit fly Drosophila melanogaster, each founded by flies heterozygous for bw75, an allele of the brown gene that affects eye colour, so that every population began with the allele at a frequency of one half. In each generation he kept eight males and eight females from each population to breed the next, and he scored eye colour to count alleles, for 19 generations. The populations did not stay near one half. Their frequencies spread out generation by generation, and by the end many populations had lost bw75 entirely and many had fixed it, while the average across all of them stayed close to where it began. The experiment turned a mathematical prediction about small populations into a picture students can read off a set of histograms.

✓ verbatim Buri, Gene frequency in small populations of mutant Drosophila, Evolution 10:367 (1956).
Test yourself
Question 1 of 8 Score 0 / 0

In a sample of 100 people typed at a two-allele locus, 36 are AA, 48 are Aa and 16 are aa. What is the frequency of allele A?
Sources & Integrity

Every structure, slide, simulation, and clip on this page is listed below with the source it came from. None of the 3D structures is drawn by hand or by AI: each one loads by accession ID from RCSB PDB or PubChem. This panel is generated from the lesson's own data file, so what it lists is what the page loads.

Molecular structures live data

  • Deoxyhemoglobin A — the normal protein — RCSB PDB 2HHB · coordinates CC0 · Human deoxyhemoglobin, X-ray crystal structure (Fermi et al., J. Mol. Biol., 1984)
  • Deoxyhemoglobin S — the sickle protein — RCSB PDB 2HBS · coordinates CC0 · Human deoxy sickle-cell hemoglobin, X-ray crystal structure (Harrington et al., J. Mol. Biol., 1997)
  • Oxyhemoglobin A — the shape that does not polymerise — RCSB PDB 1HHO · coordinates CC0 · Human oxyhemoglobin, X-ray crystal structure (Shaanan, J. Mol. Biol., 1983)

Simulations

  • Drift and selection: replicate populations · Original interactive — Biochemistrypedia (Deppmann Lab).

Scientist profiles & quotes

  • G. H. Hardy — Hardy, Mendelian proportions in a mixed population, Science 28:49 (1908).
  • Wilhelm Weinberg — Stern, The Hardy-Weinberg law, Science 97:137 (1943); Weinberg, Jahreshefte des Vereins für vaterländische Naturkunde in Württemberg 64 (1908).
  • Anthony Allison — Allison, Protection afforded by sickle-cell trait against subtertian malarial infection, British Medical Journal 1:290 (1954).
  • Peter Buri — Buri, Gene frequency in small populations of mutant Drosophila, Evolution 10:367 (1956).

Quotes are verbatim from the cited source. Profile text is original to this site and was checked against the sources cited.

Animated explainer

Why Dominant Alleles Do Not Spread · narration: an AI narrator voice (Gemini, Charon) reading narration written from this lesson's text · animation: AI-generated in code (Manim) by Claude, a schematic with no drawn molecular structures, plots computed from their equations · captions from the narration script.

Images & figures

This lesson carries no raster figures. Every structure is rendered live from RCSB PDB or PubChem, and every plot is drawn from its equation in the browser.

integrity Machines check the mechanics. The build confirms that every local file exists, that every question's answer key points at a real choice, and that any AI-generated scientific figure names the person who checked it; a weekly job re-resolves every accession ID against its database. Questions are original; MCAT-style items are written for this site and are not real AAMC questions. Banned sources (the Tymoczko textbook, publisher decks, copyrighted figures) are not used.

Like I'm five

Imagine a big jar of red and blue marbles. Every time you make a new jar, you pull marbles out two at a time and copy them. If nothing picks one colour over the other, the jar keeps the same mix forever, even if red marbles are the bossy ones. The mix only changes if something removes some colours, adds new marbles, or the jar is so small that luck matters.
High school

Each person carries two copies of each gene, and the different versions are called alleles. If you know how common each allele is in a population (call them p and q), you can predict how common each pair will be: p times p, two times p times q, and q times q. This prediction holds when mating is random and nothing is adding, removing or favouring alleles. It is called the Hardy-Weinberg principle. Being dominant does not make an allele more common. Natural selection, chance in small populations, people moving between populations, and new mutations are what change allele frequencies.
College

For a diploid autosomal locus with alleles A and a at frequencies p and q = 1 - p, random union of gametes gives zygote frequencies p^2, 2pq and q^2, reached after one generation of random mating when allele frequencies are equal in the two sexes. The model assumes a large population with random mating and no selection, mutation, migration or drift, so it serves as a null model. Allele frequencies are estimated by gene counting, and a sample is tested against the expected counts by a chi-square with one degree of freedom. Selection, drift, migration and mutation change allele frequencies. Inbreeding and assortative mating change genotype frequencies, raising homozygosity, without by themselves changing allele frequencies. For a rare recessive, carriers (about 2q) far outnumber affected homozygotes (q^2), which is why selection against the recessive is slow. Heterozygote advantage, as with the sickle allele where falciparum malaria is common, holds both alleles at q = s/(s + t).
Graduate

Under a neutral Wright-Fisher model with constant effective size Ne, the variance of the one-generation change in allele frequency is p(1 - p)/(2Ne), so drift is strongest when Ne is small and p is intermediate. With viability selection on Hardy-Weinberg zygotes, p' = (p^2 wAA + pq wAa)/w-bar, and with overdominance the interior equilibrium is q = s/(s + t). For the sickle allele, a 2018 whole-genome haplotype analysis inferred a single origin and, under its model, an age of about 7,300 years, and its balancing-selection model estimated a heterozygote advantage of about 15%. Testing equilibrium in genomic data is complicated by technique: Pearson's chi-square can inflate false rejections when the minor allele is rare, exact tests condition on allele counts, and in 1000 Genomes sequence data disequilibrium was about 11 times more frequent in segmental duplications and simple tandem repeats.

An open question is how to separate the forces from the data alone. A heterozygote excess in adults could reflect overdominant survival, mixing of differentiated groups, or calls inflated at paralogous sites, and a single sample often cannot tell them apart without newborn versus adult sampling and independent genotyping.
A mathematician answers a biologist's puzzle

Hardy's note says that R. C. Punnett had brought to his attention some remarks of Udny Yule, who had suggested that a dominant trait such as brachydactyly, short fingers, ought to spread until it reached a ratio of three affected to one unaffected. Hardy, a pure mathematician, replied in Science with a few lines of algebra showing that there is no such tendency: whatever the genotype proportions are in one generation, random mating gives proportions that then stay the same from generation to generation. He added that the point seemed so simple he would have expected biologists to know it already.

— Hardy, Science 28:49 (1908).
Credit restored after 35 years

For more than three decades English-language textbooks attributed the equilibrium to Hardy alone. In 1943 Curt Stern published a short note in Science pointing out that Wilhelm Weinberg, a physician in Stuttgart, had derived it independently in a lecture given in January 1908, months before Hardy's note appeared, and had gone on to apply it to human inheritance. Stern proposed that the principle carry both names, and it has done so since. The episode is a reminder that priority can depend on which language a paper was printed in.

— Stern, Science 97:137 (1943).
Protection against severe disease, not against infection

Thomas Williams and colleagues followed two cohorts of children on the coast of Kenya and separated malaria outcomes by severity. Sickle-cell trait made no difference to how often children carried parasites without symptoms. It was associated with about 50% protection against mild clinical malaria, about 75% against admission to hospital and almost 90% against severe or complicated malaria, and with no significant protection against other childhood diseases. The protection that keeps the sickle allele common acts mainly against dying of malaria, not against being infected.

— Williams et al., Journal of Infectious Diseases 192:178 (2005).

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