19.02.01 · eco-evo-bio / pop-genetics

Hardy-Weinberg equilibrium

draft3 tiersLean: nonepending prereqs

Anchor (Master): Charlesworth & Charlesworth, *Elements of Evolutionary Genetics* (Roberts, 2010), Ch. 2 + Ch. 5; Crow & Kimura, *An Introduction to Population Genetics Theory* (Harper & Row, 1970), Ch. 1–3; Walsh & Lynch, *Evolution and Selection of Quantitative Traits* (Oxford, 2018), Ch. 1; Ewens, *Mathematical Population Genetics I* 2nd ed. (Springer, 2004), Ch. 1–3

Intuition [Beginner]

Two brown-eyed parents have a blue-eyed child. The grandparents on both sides have brown eyes too. Where did the blue eyes come from?

Mendel's answer, worked out from pea plants in the 1860s, is that each parent carries two copies of every gene and passes one — chosen at random — to the child. A version of a gene is called an allele. The brown allele is dominant: a single copy makes eyes brown. The blue allele is recessive: blue eyes appear only when a person carries two blue alleles, one from each parent. The brown-eyed parents were each carriers — one brown allele, one blue — and the child inherited the blue allele from both. The blue allele is visible only when it doubles up.

This explains the family, but it raises a population-level puzzle. If brown is dominant, shouldn't brown alleles take over the population eventually? Shouldn't the blue allele be slowly squeezed out? The intuition that "dominant alleles dominate" is wrong, and the way it is wrong is the heart of population genetics.

The Hardy-Weinberg principle says: under five assumptions — no selection, no mutation, no migration in or out, random mating, and a population large enough that chance fluctuations are negligible — the frequencies of the alleles in the population do not change from one generation to the next. Brown and blue stay at whatever ratio they started at, forever. Dominance has nothing to do with frequency. It only describes which allele you can see in a heterozygote.

The mathematics is simple. Let $p$ be the fraction of all the gene copies in the population that are the dominant allele (call it $A$ ), and $q = 1 - p$ the fraction that are the recessive allele ( $a$ ). Under random mating each child gets one allele drawn at random from the population's pool for each of their two slots. The chance of getting $AA$ is $p \times p = p^{2}$ . The chance of $aa$ is $q^{2}$ . The chance of getting one of each — $A a$ or $a A$ , either order — is $2 pq$ . These are the Hardy-Weinberg genotype frequencies:

P (AA) = p^{2}, P (A a) = 2 pq, P (aa) = q^{2}, p^{2} + 2 pq + q^{2} = 1.

Worked example. Cystic fibrosis is a recessive disease in humans; in many European-descended populations the disease frequency is about 1 in 2500. That means $q^{2} \approx 1/2500$ , so $q \approx 1/50 = 0.02$ , and $p \approx 0.98$ . The carrier frequency, the people who have one disease allele and one normal allele, is

2 pq \approx 2 \cdot 0.98 \cdot 0.02 \approx 0.039,

or about 1 person in 25. The disease itself shows up in 1 in 2500 births, but roughly 1 in 25 people walks around carrying the disease allele invisibly. The 100-fold gap between disease frequency and carrier frequency is a direct prediction of Hardy-Weinberg — a calculation that genetic counsellors use every day, derived from nothing more than "alleles pair up at random."

Why does the equilibrium matter? Two reasons that compound.

First, constant allele frequencies are the null hypothesis of evolution. If a real population's frequencies do not change, nothing interesting is happening; if they do change, some force — selection, drift, mutation, migration — is acting. Population genetics rests on diagnosing which.

Second, the genotype frequencies $p^{2} : 2 pq : q^{2}$ are the equilibrium pattern reached in a single generation of random mating, regardless of how the alleles were distributed in the parental generation. Hardy-Weinberg is not a slow asymptotic limit. It is a one-step reset to a fixed point. If you sample a population and find that the genotype counts differ noticeably from the Hardy-Weinberg prediction, you know the population is not the simple textbook one — assortative mating, recent migration, recent selection, or hidden subpopulation structure are diagnosable from a single snapshot.

The picture to hold in your head: a bag of marbles where $p$ are red and $q$ are blue. To make a child, reach in twice. The chance of two reds is $p^{2}$ ; two blues is $q^{2}$ ; one of each is $2 pq$ . Evolution is what happens when the bag's contents change between generations; Hardy-Weinberg is what holds when they don't.

Visual [Beginner]

The Hardy-Weinberg surface for two alleles is the simplest visual in all of population genetics: three curves on a single allele-frequency axis.

Plot the three genotype frequencies as functions of $p$ from $0$ to $1$ . The homozygote-dominant curve $P (AA) = p^{2}$ rises from $0$ at $p = 0$ to $1$ at $p = 1$ , accelerating. The homozygote-recessive curve $P (aa) = q^{2} = (1 - p)^{2}$ mirrors it, falling from $1$ at $p = 0$ to $0$ at $p = 1$ . The heterozygote curve $P (A a) = 2 pq = 2 p (1 - p)$ is a downward-opening parabola; it is zero at both endpoints and reaches its maximum of $1/2$ exactly at $p = 1/2$ .

This picture says three things at a glance. First, when one allele is rare, almost every copy of it sits in a heterozygote: for $q = 0.02$ , the ratio of $aa$ carriers ( $q^{2} = 0.0004$ ) to $A a$ carriers ( $2 pq \approx 0.039$ ) is about 1 to 100. The rare allele is almost never visible as a phenotype but persists because heterozygotes pass it on. Second, heterozygotes are most common at $p = 1/2$ ; populations with intermediate allele frequencies are the most genetically varied per locus. Third, rare recessive alleles cannot be eliminated by selection against affected homozygotes — almost all copies sit in heterozygotes, hidden from selection. This is why programmes targeting affected individuals are mathematically futile.

Worked example [Beginner]

A textbook population of 10000 individuals is surveyed for a single recessive trait — say, the inability to taste the chemical phenylthiocarbamide (PTC), where non-tasting is recessive. Of the 10000, 1600 are non-tasters. The other 8400 are tasters, but the survey cannot tell which tasters are heterozygotes ( $A a$ ) and which are homozygotes ( $AA$ ). Hardy-Weinberg gives the breakdown.

Step 1. Compute $q^{2}$ . The non-tasters are the $aa$ homozygotes, so $q^{2} = 1600/10000 = 0.16$ . The recessive allele frequency is $q = 0.16 = 0.4$ .

Step 2. Compute $p$ . $p = 1 - q = 0.6$ .

Step 3. Predict the genotype counts under Hardy-Weinberg.

P (AA) = p^{2} = 0.36, P (A a) = 2 pq = 0.48, P (aa) = q^{2} = 0.16.

Out of 10000 individuals, the predicted numbers are $3600$ homozygous tasters, $4800$ heterozygous tasters, and $1600$ non-tasters.

Step 4. The takeaway. Of the 8400 tasters, about $4800/8400 \approx 57%$ are heterozygotes — they look identical to homozygotes phenotypically, but each is carrying one copy of the non-taster allele. Nearly half the population are silent carriers. This is the qualitative shape of the prediction that genetic counsellors and human geneticists rely on every day, and Hardy-Weinberg is doing all the work.

Check your understanding [Beginner]

Exercise (easy, multiple choice).

A population is at Hardy-Weinberg equilibrium for a single locus. Which one of the following must be true?

A. The dominant allele is more common than the recessive allele. B. The heterozygote frequency equals $2 pq$ where $p$ and $q$ are the allele frequencies. C. The genotype frequencies are equal to $1/3$ each. D. The recessive phenotype is being selected against.

Hint

Hardy-Weinberg does not constrain which allele is more common, only the relationship between allele frequencies and genotype frequencies.

Answer

Option B.

Hardy-Weinberg says $P (AA) = p^{2}$ , $P (A a) = 2 pq$ , $P (aa) = q^{2}$ , with no constraint forcing $p > q$ (A is wrong), no equal-thirds requirement (C is wrong), and no claim about selection (D is wrong — Hardy-Weinberg assumes the absence of selection, not its presence). Only B follows from the definition.

Formal definition [Intermediate+]

Fix a single autosomal locus with two alleles $A$ and $a$ in a diploid population. Each individual carries two gene copies at the locus, drawn from ${A, a}$ , and has one of three genotypes ${AA, A a, aa}$ . Let $P, H, Q$ denote the population frequencies of $AA$ , $A a$ , $aa$ respectively, so $P + H + Q = 1$ . The allele frequencies are

p := P + \frac{1}{2} H, q := Q + \frac{1}{2} H, p + q = 1,

obtained by counting each homozygote as contributing two copies of its allele and each heterozygote as contributing one copy of each. The transformation $(P, H, Q) \mapsto (p, q)$ collapses three-dimensional genotype-frequency data onto a one-dimensional allele-frequency axis, losing the information of how the alleles are paired into genotypes.

The Hardy-Weinberg assumptions. A diploid population is said to satisfy the Hardy-Weinberg conditions for a locus if:

No selection. All three genotypes have equal viability and fertility — the genotype an individual carries does not affect the expected number of gene copies it transmits to the next generation.
No mutation. No allele converts to another during the reproductive cycle, on the timescale considered.
No migration. The population is closed: no individuals enter or leave.
Random mating (panmixia) with respect to the locus. Each mating pair is formed by drawing two individuals independently at random from the population; equivalently, each offspring gene copy is an independent draw from the population gene pool. Random mating may hold for one locus while failing for another (e.g., assortative mating for body size).
Infinite population size. Stochastic sampling fluctuations are negligible, so realised genotype frequencies coincide with expected frequencies. The finite-population correction is the subject of the Master tier.

Hardy-Weinberg theorem (two-allele single-locus case). Under assumptions 1–5, the genotype frequencies in the next generation are

P^{'} = p^{2}, H^{'} = 2 pq, Q^{'} = q^{2},

regardless of $(P, H, Q)$ , and the allele frequencies are unchanged: $p^{'} = p$ , $q^{'} = q$ . The equilibrium is attained in a single generation of random mating and is preserved indefinitely under the same conditions.

A point $(P, H, Q)$ on the Hardy-Weinberg curve in genotype-frequency space is one satisfying $H = 2 P Q$ , equivalently $H^{2} = 4 P Q$ — exactly the relation $P = p^{2}$ , $Q = q^{2}$ gives after eliminating $p, q$ . The Hardy-Weinberg curve is the one-parameter family of genotype-frequency triples parametrised by the single allele-frequency $p$ .

Counterexamples to common slips

Dominance is irrelevant to Hardy-Weinberg. The theorem makes no use of which allele is dominant or recessive. Hardy-Weinberg holds for codominant alleles (M and N blood-group antigens, microsatellite repeats), for incompletely dominant alleles, and for overdominant or underdominant alleles, provided assumptions 1–5 hold. Dominance affects only the visibility of the genotype as a phenotype.
Random mating $\neq =$ random anything else. Random mating with respect to the locus in question does not imply that the population is unstructured in space, age, or other loci. A population with strong spatial structure but locally random mating at the locus, no migration, no selection, no mutation can be at Hardy-Weinberg for that one locus.
One generation, not many. Even if the parental generation has heavily skewed genotype frequencies — e.g., all $AA$ and all $aa$ , no heterozygotes — one generation of random mating produces the Hardy-Weinberg proportions in the offspring. The "approach to equilibrium" is single-step, not asymptotic, as long as allele frequencies are well-defined and mating is random.
Hardy-Weinberg for sex-linked or multi-locus systems differs. X-linked loci approach Hardy-Weinberg over several generations because the male and female allele-frequency reservoirs are initially distinct. Two-locus genotypes approach the Hardy-Weinberg product slowly, controlled by the recombination rate — the linkage disequilibrium $D$ decays as $D^{'} = (1 - r) D$ each generation, not in one step. The single-locus single-generation statement is the simplest case; the Master tier develops the corrections.

Key theorem with proof [Intermediate+]

Theorem (Hardy-Weinberg, 1908, single-locus two-allele case). Let a diploid population have genotype frequencies $(P_{0}, H_{0}, Q_{0})$ with $P_{0} + H_{0} + Q_{0} = 1$ at generation $0$ . Let $p = P_{0} + \frac{1}{2} H_{0}$ and $q = 1 - p$ . Suppose mating is random with respect to the locus and that no selection, mutation, or migration occurs, and that the population is sufficiently large that sampling fluctuations are negligible. Then the genotype frequencies at generation $1$ are $(p^{2}, 2 pq, q^{2})$ , and the allele frequencies are $p_{1} = p$ , $q_{1} = q$ . Moreover, these frequencies are stationary: $(P_{n}, H_{n}, Q_{n}) = (p^{2}, 2 pq, q^{2})$ for all $n \geq 1$ .

Proof. Under random mating each offspring gene copy is an independent draw from the population gene pool. The gene pool at generation $0$ contains gene copies in the ratio $p : q$ between alleles $A$ and $a$ , by definition of $p$ and $q$ . An offspring is constructed by drawing two gene copies independently from the gene pool, one for each allele slot.

For the genotype probabilities at generation $1$ ,

P_{1} = Pr (both draws are A) = p \cdot p = p^{2},

Q_{1} = Pr (both draws are a) = q \cdot q = q^{2},

and the heterozygote case has two ordered realisations $A a$ and $a A$ , each with probability $pq$ , giving

H_{1} = Pr (one draw is A and the other is a) = 2 pq .

These three sum to $p^{2} + 2 pq + q^{2} = (p + q)^{2} = 1$ , consistent with $(P_{1}, H_{1}, Q_{1})$ being a probability vector.

For the allele frequency at generation $1$ ,

p_{1} = P_{1} + \frac{1}{2} H_{1} = p^{2} + \frac{1}{2} (2 pq) = p^{2} + pq = p (p + q) = p .

Symmetrically $q_{1} = q$ . The allele frequencies are unchanged from generation $0$ to generation $1$ .

For stationarity, observe that the argument above used only the allele frequencies at the parental generation, not the genotype frequencies. Since $p_{1} = p_{0}$ and the Hardy-Weinberg conditions hold again at generation $1$ , the same argument applied to generation $1$ produces $(P_{2}, H_{2}, Q_{2}) = (p_{1}^{2}, 2 p_{1} q_{1}, q_{1}^{2}) = (p^{2}, 2 pq, q^{2})$ , and so on by induction. The genotype-frequency vector is fixed for all $n \geq 1$ . $□$

Multi-allele extension. For a single locus with $k$ alleles $A_{1}, \dots, A_{k}$ of frequencies $p_{1}, \dots, p_{k}$ summing to $1$ , the same independence argument gives the multinomial-square expansion

P (A_{i} A_{i}) = p_{i}^{2}, P (A_{i} A_{j}) = 2 p_{i} p_{j} (i \neq = j),

so that the homozygote frequencies sum to $\sum_{i} p_{i}^{2}$ and the heterozygote frequencies to $\sum_{i \neq = j} p_{i} p_{j} = 1 - \sum_{i} p_{i}^{2}$ . The expression $\sum_{i} p_{i}^{2}$ is the homozygosity of the locus; its complement $1 - \sum_{i} p_{i}^{2}$ is the expected heterozygosity $H_{e}$ , the standard summary of allele-frequency diversity at a locus.

Diagnosing deviations from Hardy-Weinberg

A population whose observed genotype frequencies $(P_{obs}, H_{obs}, Q_{obs})$ depart significantly from the Hardy-Weinberg prediction $(p^{2}, 2 pq, q^{2})$ — with $p$ and $q$ computed from the same data — is not satisfying the five assumptions. The direction of the departure diagnoses the violation:

Deviation	Likely cause
$H_{obs} < 2 pq$ (heterozygote deficit)	Inbreeding, assortative mating for the phenotype, or pooling of subpopulations (Wahlund effect)
$H_{obs} > 2 pq$ (heterozygote excess)	Outbreeding, heterozygote-favouring selection (overdominance), or recent admixture between populations
Frequency change across generations	Selection, mutation pressure, migration, or finite-population drift

Chi-square test. Given observed counts $n_{AA}, n_{A a}, n_{aa}$ summing to $N$ , estimate $\overset{p}{^} = (2 n_{AA} + n_{A a}) / (2 N)$ and $\overset{q}{^} = 1 - \overset{p}{^}$ , compute expected counts $E_{AA} = N \overset{p}{^}^{2}$ , $E_{A a} = 2 N \overset{p}{^} \overset{q}{^}$ , $E_{aa} = N \overset{q}{^}^{2}$ , and form the statistic

χ^{2} = g \in {AA, A a, aa} \sum \frac{( n _{g} - E _{g} ) ^{2}}{E _{g}} .

Under the Hardy-Weinberg null, $χ^{2}$ has approximately one degree of freedom (three categories minus one constraint from $N$ minus one estimated parameter $\overset{p}{^}$ ), and $χ^{2} > 3.84$ rejects Hardy-Weinberg at the $5%$ level. Worked example: a sample of $200$ humans with MN blood-group counts $n_{M M} = 80$ , $n_{M N} = 100$ , $n_{N N} = 20$ gives $\overset{p}{^}_{M} = (160 + 100) /400 = 0.65$ , $\overset{p}{^}_{N} = 0.35$ , expected counts $(84.5, 91, 24.5)$ , and $χ^{2} = 0.24 + 0.89 + 0.83 = 1.96$ , well below $3.84$ — consistent with Hardy-Weinberg.

Inbreeding and the coefficient $F$

The inbreeding coefficient $F$ of an individual is the probability that the two gene copies it carries at a randomly chosen locus are identical by descent (copies of the same ancestral allele). Under random mating $F = 0$ ; under inbreeding $F > 0$ . The genotype frequencies in an inbred population at allele-frequency $p$ are

P (AA) = p^{2} + F pq, P (A a) = 2 pq (1 - F), P (aa) = q^{2} + F pq,

interpolating between Hardy-Weinberg ( $F = 0$ ) and complete inbreeding ( $F = 1$ , no heterozygotes). The heterozygote deficit is exactly $2 pq \cdot F$ , providing a direct estimator $\hat{F} = 1 - H_{obs} / (2 \overset{p}{^} \overset{q}{^})$ — Wright's $F_{I S}$ in the within-population case.

Two-locus Hardy-Weinberg and linkage disequilibrium

For two loci with alleles $A / a$ at locus $1$ and $B / b$ at locus $2$ , the four gamete frequencies $p_{A B}, p_{A b}, p_{a B}, p_{ab}$ satisfy the Hardy-Weinberg-like product condition $p_{A B} = p_{A} p_{B}$ (and analogues) only at multi-locus equilibrium. The departure is captured by the linkage disequilibrium coefficient

D := p_{A B} - p_{A} p_{B} = p_{ab} - p_{a} p_{b} = - (p_{A b} - p_{A} p_{b}) = - (p_{a B} - p_{a} p_{B}),

a single number that quantifies the four-way correlation. Under random mating with recombination rate $r$ between the two loci, $D$ decays geometrically each generation: $D_{n + 1} = (1 - r) D_{n}$ , so $D_{n} = (1 - r)^{n} D_{0}$ . The single-generation reset of single-locus Hardy-Weinberg is replaced by a multi-generation approach at rate $r$ for two loci; the unlinked case $r = 1/2$ gives a halving each generation and approximate equilibrium within a handful of generations, while tightly linked loci ( $r ≪ 1$ ) can preserve disequilibrium for many generations and form the substrate for genetic-mapping methods.

Population substructure — the Wahlund effect

If a sample is pooled from two subpopulations with different allele frequencies $p_{1}$ and $p_{2}$ but each subpopulation is internally at Hardy-Weinberg, the pooled sample shows an apparent heterozygote deficit. With equal sample sizes the pooled allele frequency is $\overset{p}{ˉ} = (p_{1} + p_{2}) /2$ , the expected heterozygote frequency under naive Hardy-Weinberg is $2 \overset{p}{ˉ} \overset{q}{ˉ}$ , but the actual pooled heterozygote frequency is $\overset{ˉ}{H} = (2 p_{1} q_{1} + 2 p_{2} q_{2}) /2$ . The difference is

2 \overset{p}{ˉ} \overset{q}{ˉ} - \overset{ˉ}{H} = \frac{1}{2} (p_{1} - p_{2})^{2} \geq 0,

a nonnegative deficit that vanishes only when $p_{1} = p_{2}$ . This is the Wahlund effect (Wahlund 1928): pooling subpopulations with differentiated allele frequencies always looks like inbreeding in the apparent-heterozygote-deficit sense, even when each subpopulation is panmictic. Inbreeding and substructure produce numerically identical signatures at the pooled level; distinguishing them requires sampling design or auxiliary marker data.

Exercises [Intermediate+]

Exercise 2 (medium, numeric).

In a sample of $1000$ individuals at a codominant locus with alleles $M$ and $N$ , the observed genotype counts are $n_{M M} = 360$ , $n_{M N} = 520$ , $n_{N N} = 120$ . Test the Hardy-Weinberg hypothesis by chi-square with one degree of freedom.

Hint

Estimate $\overset{p}{^}_{M}$ first from the counts, then compute expected counts and the chi-square statistic. Compare to the critical value $3.84$ for $p = 0.05$ .

Answer

Allele-frequency estimate: $\overset{p}{^}_{M} = (2 \cdot 360 + 520) / (2 \cdot 1000) = 1240/2000 = 0.62$ , $\overset{p}{^}_{N} = 0.38$ . Expected counts: $E_{M M} = 1000 \cdot 0.6 2^{2} = 384.4$ , $E_{M N} = 1000 \cdot 2 \cdot 0.62 \cdot 0.38 = 471.2$ , $E_{N N} = 1000 \cdot 0.3 8^{2} = 144.4$ . Chi-square:

χ^{2} = \frac{( 360 - 384.4 ) ^{2}}{384.4} + \frac{( 520 - 471.2 ) ^{2}}{471.2} + \frac{( 120 - 144.4 ) ^{2}}{144.4} \approx 1.55 + 5.06 + 4.12 \approx 10.7.

This exceeds $3.84$ at one degree of freedom, rejecting Hardy-Weinberg at $p < 0.005$ . The direction is a heterozygote excess ( $520$ observed vs $471.2$ expected), consistent with overdominant selection or recent admixture; the observed homozygote counts are both below their expectations.

Exercise 3 (medium, numeric).

Two subpopulations each at Hardy-Weinberg have allele frequencies $p_{1} = 0.2$ and $p_{2} = 0.8$ at a locus. Equal numbers from each are pooled into one sample. Compute (a) the pooled allele frequency $\overset{p}{ˉ}$ , (b) the actual pooled heterozygote frequency $\overset{ˉ}{H}$ , (c) the heterozygote frequency that naive Hardy-Weinberg would predict from $\overset{p}{ˉ}$ , and (d) the Wahlund deficit.

Hint

Average the per-subpopulation $2 pq$ for the actual pooled heterozygote frequency; compute $2 \overset{p}{ˉ} \overset{q}{ˉ}$ separately for the naive prediction.

Answer

(a) $\overset{p}{ˉ} = 0.5$ , $\overset{q}{ˉ} = 0.5$ . (b) Per-subpopulation heterozygote frequencies are $2 \cdot 0.2 \cdot 0.8 = 0.32$ in each, so $\overset{ˉ}{H} = 0.32$ . (c) Naive prediction $2 \overset{p}{ˉ} \overset{q}{ˉ} = 0.5$ . (d) Wahlund deficit $= 0.5 - 0.32 = 0.18 = \frac{1}{2} (p_{1} - p_{2})^{2} = \frac{1}{2} (0.6)^{2}$ . The pooled sample looks like a Hardy-Weinberg population with $F = 0.18/0.5 = 0.36$ , a strong apparent inbreeding signature produced by subdivision alone.

Exercise 4 (medium, symbolic).

For two loci with recombination rate $r$ and initial linkage disequilibrium $D_{0}$ , derive the recursion $D_{n + 1} = (1 - r) D_{n}$ under random mating with no selection and no mutation. State the implied half-life of disequilibrium in generations.

Hint

A gamete in generation $n + 1$ is either non-recombinant (probability $1 - r$ , drawn from a single parental gamete) or recombinant (probability $r$ , drawn from two parental gametes independently at the two loci).

Answer

A randomly chosen offspring gamete in generation $n + 1$ is non-recombinant with probability $1 - r$ , in which case its joint allele state at the two loci is distributed as a single parental gamete and contributes $p_{A B, n}$ to $p_{A B, n + 1}$ . With probability $r$ it is recombinant, in which case its two loci are independent draws, contributing $p_{A, n} p_{B, n}$ to $p_{A B, n + 1}$ . Therefore

p_{A B, n + 1} = (1 - r) p_{A B, n} + r p_{A, n} p_{B, n} .

Allele frequencies are unchanged under random mating: $p_{A, n + 1} = p_{A, n}$ and $p_{B, n + 1} = p_{B, n}$ . Subtracting $p_{A, n + 1} p_{B, n + 1} = p_{A, n} p_{B, n}$ from both sides,

D_{n + 1} = p_{A B, n + 1} - p_{A, n + 1} p_{B, n + 1} = (1 - r) (p_{A B, n} - p_{A, n} p_{B, n}) = (1 - r) D_{n} .

Iterating, $D_{n} = (1 - r)^{n} D_{0}$ . The half-life is $n_{1/2} = lo g 2/ lo g (1/ (1 - r))$ , approximately $0.69/ r$ for small $r$ . Unlinked loci ( $r = 0.5$ ) lose half their disequilibrium per generation; tightly linked loci ( $r = 0.01$ , say) require about $70$ generations for the disequilibrium to halve. This is why association-mapping studies for common diseases can resolve a causal variant to a region of typical width $\sim 1/ (4 N_{e} r)$ — the linkage-disequilibrium block structure of the genome.

Exercise 6 (hard, symbolic).

Show that for a sex-linked locus with two alleles at frequencies $p$ in females and $p_{m}$ in males in generation $0$ , the male and female allele frequencies oscillate toward a common limit and approach Hardy-Weinberg over several generations rather than in one step.

Hint

Each generation, a male inherits his single X allele from his mother; a female inherits one X allele from each parent. Track $p_{f}$ and $p_{m}$ separately.

Answer

Let $p_{f}^{(n)}, p_{m}^{(n)}$ denote the female and male allele frequencies at the locus in generation $n$ . A male's single X comes from his mother, so $p_{m}^{(n + 1)} = p_{f}^{(n)}$ . A female's two X alleles come one from each parent, so $p_{f}^{(n + 1)} = (p_{f}^{(n)} + p_{m}^{(n)}) /2$ . In vector form,

(p_{f}^{(n + 1)} p_{m}^{(n + 1)}) = (1/2 1 1/2 0) (p_{f}^{(n)} p_{m}^{(n)}) .

The matrix has eigenvalues $1$ and $- 1/2$ , with the $λ = 1$ eigenvector $(1, 1)^{T}$ (common allele frequency in both sexes) and the $λ = - 1/2$ eigenvector capturing the deviation. The mean allele frequency $\overset{p}{ˉ} = (2 p_{f} + p_{m}) /3$ is preserved exactly (the $λ = 1$ mode), while the sex difference $p_{f} - p_{m}$ decays by a factor $- 1/2$ each generation, oscillating in sign and shrinking to zero. After $\sim 5$ generations the sex difference is $< 1%$ of its initial value. Hardy-Weinberg genotype frequencies in females approach $\overset{p}{ˉ}^{2} : 2 \overset{p}{ˉ} \overset{q}{ˉ} : \overset{q}{ˉ}^{2}$ on the same timescale; males remain hemizygous with frequencies $\overset{p}{ˉ} : \overset{q}{ˉ}$ for $A : a$ . This is the canonical X-linked correction to one-generation Hardy-Weinberg.

Exercise 7 (hard, numeric).

A continent is divided into four equal-sized subpopulations with frequencies of the dominant allele $p_{1} = 0.1, p_{2} = 0.3, p_{3} = 0.5, p_{4} = 0.7$ . Each subpopulation is internally at Hardy-Weinberg. Compute $F_{S T}$ for this metapopulation, where $F_{S T} := (H_{T} - H_{S}) / H_{T}$ with $H_{S}$ the average within-subpopulation heterozygosity and $H_{T}$ the total heterozygosity under naive pooling.

Hint

$H_{S}$ is the mean of $2 p_{i} q_{i}$ across subpopulations; $H_{T}$ is $2 \overset{p}{ˉ} \overset{q}{ˉ}$ with $\overset{p}{ˉ}$ the mean.

Answer

$\overset{p}{ˉ} = (0.1 + 0.3 + 0.5 + 0.7) /4 = 0.4$ , $\overset{q}{ˉ} = 0.6$ , $H_{T} = 2 \cdot 0.4 \cdot 0.6 = 0.48$ . Within-subpopulation heterozygosities: $2 \cdot 0.1 \cdot 0.9 = 0.18$ , $2 \cdot 0.3 \cdot 0.7 = 0.42$ , $2 \cdot 0.5 \cdot 0.5 = 0.5$ , $2 \cdot 0.7 \cdot 0.3 = 0.42$ . Mean $H_{S} = (0.18 + 0.42 + 0.5 + 0.42) /4 = 0.38$ . Thus $F_{S T} = (0.48 - 0.38) /0.48 = 0.10/0.48 \approx 0.208$ . A value of $F_{S T}$ around $0.2$ is in the range typical for strongly structured continental populations of vertebrates; human populations between continents typically show $F_{S T}$ of $0.10$ – $0.15$ at autosomal loci.

Exercise 8 (hard, symbolic).

A two-allele locus is in Hardy-Weinberg equilibrium with allele frequencies $p$ and $q$ . A genotyping assay misclassifies heterozygotes as homozygous-dominant with probability $ϵ$ (so a true $A a$ is recorded as $AA$ with probability $ϵ$ and correctly as $A a$ with probability $1 - ϵ$ ). Derive the expected genotype counts under the misclassification, and the apparent $F$ value an investigator would estimate from the observed (mis-classified) heterozygote deficit.

Hint

The misclassified- $AA$ fraction is $p^{2} + ϵ \cdot 2 pq$ . The apparent allele frequency from the observed data is $\overset{p}{^}_{obs} = P_{obs} + H_{obs} /2$ .

Answer

Observed genotype frequencies:

P_{obs} = p^{2} + 2 pq ϵ, H_{obs} = 2 pq (1 - ϵ), Q_{obs} = q^{2} .

Apparent allele-frequency estimate from the observed data: $\overset{p}{^}_{obs} = P_{obs} + H_{obs} /2 = p^{2} + 2 pq ϵ + pq (1 - ϵ) = p^{2} + pq + pq ϵ = p + pq ϵ$ , and $\overset{q}{^}_{obs} = q - pq ϵ$ . The naive Hardy-Weinberg heterozygote prediction from the observed data is $2 \overset{p}{^}_{obs} \overset{q}{^}_{obs}$ , and the heterozygote deficit is

2 \overset{p}{^}_{obs} \overset{q}{^}_{obs} - H_{obs} = 2 (p + pq ϵ) (q - pq ϵ) - 2 pq (1 - ϵ) .

To leading order in small $ϵ$ , $2 \overset{p}{^}_{obs} \overset{q}{^}_{obs} \approx 2 pq + 2 pq (q - p) ϵ + O (ϵ^{2})$ and $H_{obs} = 2 pq - 2 pq ϵ$ , so the deficit is $2 pq [1 + (q - p) ϵ] ϵ + O (ϵ^{2})$ and the apparent inbreeding coefficient is

\hat{F} = 1 - H_{obs} / (2 \overset{p}{^}_{obs} \overset{q}{^}_{obs}) \approx ϵ + (q - p) ϵ^{2} /2 + O (ϵ^{3}) .

So a $5%$ heterozygote-miscalling rate produces an apparent $\hat{F} \approx 0.05$ , indistinguishable from real inbreeding by single-locus analysis. Multi-locus consistency tests (a real $F$ is consistent across loci; a miscalling rate is locus-specific) are the standard control.

Wright-Fisher drift and effective population size [Master]

Removing assumption 5 (infinite population size) replaces deterministic genotype recurrences with a stochastic process. The simplest formal model is Wright-Fisher: a population of $N$ diploid individuals at a single locus, with $2 N$ gene copies per generation. Each gene copy in generation $n + 1$ is an independent draw, with replacement, from the gene pool at generation $n$ — equivalently, generation $n + 1$ inherits $2 N$ gene copies sampled multinomially from the parental gene pool. Selection, mutation, and migration are all switched off.

Let $X_{n}$ denote the number of $A$ alleles at generation $n$ ; the state space is ${0, 1, \dots, 2 N}$ . The transition law is

Pr (X_{n + 1} = k ∣ X_{n} = j) = (k 2 N) π_{j}^{k} (1 - π_{j})^{2 N - k}, π_{j} = j / (2 N) .

That is, conditional on $X_{n} = j$ , the next generation's $A$ -count is binomial $(2 N, π_{j})$ . The expected allele frequency is unchanged: $E [X_{n + 1} ∣ X_{n} = j] = 2 N π_{j} = j$ , so allele frequencies are a martingale. However, the variance per generation is $Var (X_{n + 1} / (2 N) ∣ X_{n} = j) = π_{j} (1 - π_{j}) / (2 N)$ , which is nonzero for $0 < π_{j} < 1$ . Iterated for $n$ generations, the variance accumulates at rate $\sim 1/ (2 N)$ per generation while the mean stays fixed — the allele frequency wanders, eventually hitting an absorbing boundary.

Absorption. The states $X = 0$ (the $A$ allele is lost) and $X = 2 N$ (the $A$ allele is fixed) are absorbing. The martingale stopping theorem gives the fixation probability of allele $A$ starting from frequency $π_{0}$ :

Pr (fixation of A) = π_{0} .

That is, a neutral allele's probability of eventually fixing equals its current frequency — a new mutation at frequency $1/ (2 N)$ has probability $1/ (2 N)$ of eventually replacing all others. Expected time to fixation for a new mutation that eventually fixes is approximately $4 N$ generations (Kimura-Ohta 1969); expected time to loss for one that eventually loses is approximately $2 ln (2 N)$ generations.

Effective population size. Real populations rarely satisfy the Wright-Fisher idealisations — sex ratios are unequal, family sizes vary, generations overlap. The effective population size $N_{e}$ is the size of an idealised Wright-Fisher population that exhibits the same rate of drift as the real population, and admits three distinct operational definitions:

Variance effective size $N_{e}^{(v)}$ : defined so that $Var (Δ p) = p (1 - p) / (2 N_{e}^{(v)})$ in the real population matches the Wright-Fisher form.
Inbreeding effective size $N_{e}^{(i)}$ : defined so that the rate of increase of inbreeding $Δ F = 1/ (2 N_{e}^{(i)})$ per generation matches.
Eigenvalue effective size $N_{e}^{(λ)}$ : defined so that the decay rate of heterozygosity matches the Wright-Fisher eigenvalue $λ = 1 - 1/ (2 N_{e}^{(λ)})$ .

These three definitions coincide for a textbook Wright-Fisher population but diverge under unequal sex ratios, variance in offspring number, or fluctuating population size. The harmonic-mean formula $N_{e} = t / \sum_{i = 1}^{t} (1/ N_{i})$ for a population fluctuating across $t$ generations of sizes $N_{i}$ illustrates the typical pattern: a single bottleneck of size $N_{m i n}$ dominates $N_{e}$ regardless of the long-term mean.

Selection on a Hardy-Weinberg background [Master]

Selection enters by relaxing assumption 1. Let $w_{AA}, w_{A a}, w_{aa}$ be the relative fitnesses of the three genotypes — the expected number of surviving offspring per individual of each genotype, in units where the maximum fitness is $1$ . If Hardy-Weinberg proportions $p^{2}, 2 pq, q^{2}$ describe the zygote frequencies but selection acts before reproduction, the post-selection genotype frequencies are proportional to $p^{2} w_{AA}, 2 pq w_{A a}, q^{2} w_{aa}$ , renormalised by

\overset{w}{ˉ} := p^{2} w_{AA} + 2 pq w_{A a} + q^{2} w_{aa},

the population mean fitness. The post-selection allele frequency is

p^{'} = \frac{p ^{2} w _{AA} + pq w _{A a}}{w ˉ} = \frac{p ( p w _{AA} + q w _{A a} )}{w ˉ},

and the per-generation change is

Δ p = p^{'} - p = \frac{pq [ p ( w _{AA} - w _{A a} ) + q ( w _{A a} - w _{aa} ) ]}{w ˉ} .

Writing $w_{AA} = 1, w_{A a} = 1 - h s, w_{aa} = 1 - s$ in the standard selection-coefficient parametrisation (where $s$ is the selection coefficient against the recessive homozygote and $h$ is the dominance coefficient, $h = 0$ for fully recessive disadvantage, $h = 1$ for fully dominant, $h = 1/2$ for additive),

Δ p = \frac{pq [ h s p + ( 1 - h ) s q ]}{w ˉ} = \frac{pq s [ h p + ( 1 - h ) q ]}{1 - 2 pq h s - q ^{2} s},

which for small $s$ becomes the familiar weak-selection approximation $Δ p \approx pq s [h p + (1 - h) q]$ .

Fisher's fundamental theorem (1930): the rate of change of mean fitness due to natural selection equals the additive genetic variance in fitness divided by $\overset{w}{ˉ}$ . In the single-locus case, $d \overset{w}{ˉ} / d t \approx V_{A} (w) / \overset{w}{ˉ}$ where $V_{A} (w)$ is the variance in fitness attributable to additive allelic effects. The theorem is a near-tautology after careful accounting (Price 1972; Edwards 1994), but it remains the cleanest expression of the link between population genetics and quantitative-trait response to selection.

Mutation-selection balance. A recurrent deleterious mutation at rate $μ$ per generation, opposed by selection of coefficient $s$ against the homozygous-recessive genotype with full recessivity ( $h = 0$ ), settles to an equilibrium allele frequency $\overset{q}{^} \approx μ / s$ . The corresponding equilibrium disease frequency is $\overset{q}{^}^{2} \approx μ / s$ , independent of scaling — the genetic load $μ / s$ at the population level equals the mutation rate divided by the selection coefficient, regardless of how the load is distributed across genotypes. For partial dominance ( $h > 0$ ) the balance shifts to $\overset{q}{^} \approx μ / (h s)$ , much smaller — even a small dominance coefficient drastically reduces the equilibrium frequency by exposing the allele to selection in heterozygotes.

The diffusion limit and the coalescent [Master]

For large $N$ and small allele-frequency changes per generation, the Wright-Fisher chain on ${0, 1, \dots, 2 N}$ admits a diffusion approximation. Let $p (t)$ be the continuous-time allele frequency on $[0, 1]$ measured in units of $2 N$ generations. Under selection coefficient $s$ and mutation rates $μ, ν$ , the density $ϕ (p, t)$ of the allele frequency satisfies the Kolmogorov forward (Fokker-Planck) equation

\frac{\partial ϕ}{\partial t} = - \frac{\partial}{\partial p} [M (p) ϕ] + \frac{1}{2} \frac{\partial ^{2}}{\partial p ^{2}} [V (p) ϕ],

with drift coefficient $M (p) = s p (1 - p) - μ p + ν (1 - p)$ and variance coefficient $V (p) = p (1 - p)$ . The diffusion limit is the route to closed-form expressions for fixation probabilities under selection (Kimura 1962): for a new mutation of selection coefficient $s$ in a population of effective size $N_{e}$ starting at frequency $1/ (2 N)$ ,

Pr (fixation) = \frac{1 - e ^{- 2 s}}{1 - e ^{- 4 N_{e} s}} \approx {2 s 1/ (2 N) if 4 N_{e} s ≫ 1, if ∣4 N_{e} s ∣ ≪ 1.

The crossover at $∣4 N_{e} s ∣ \sim 1$ is the nearly-neutral threshold (Ohta 1973): alleles with selection coefficient $∣ s ∣ ≪ 1/ (4 N_{e})$ behave essentially as neutral and obey the $1/ (2 N)$ neutral fixation probability, while those with $∣ s ∣ ≫ 1/ (4 N_{e})$ behave as effectively deterministic and obey Haldane's $2 s$ rule for positive selection.

The coalescent (Kingman 1982). Reversing the Wright-Fisher chain in time gives the genealogy of a sample of $n$ gene copies from a present-day population. Tracing each gene copy backward, two copies coalesce into a common ancestor at rate $(2 k) / (2 N)$ per generation when $k$ copies remain. Rescaling time in units of $2 N$ generations, the resulting continuous-time process — the Kingman coalescent — is a Markov process on partitions of ${1, \dots, n}$ in which each pair of currently-distinct lineages coalesces at exponential rate $1$ . The expected time for $k$ lineages to reduce to $k - 1$ is $2/ (k (k - 1))$ time units; the expected time to the most recent common ancestor of a sample of $n$ is $\sum_{k = 2}^{n} 2/ (k (k - 1)) = 2 (1 - 1/ n)$ , approaching $2$ time units (i.e., $4 N$ generations) for large $n$ .

Watterson's estimator and the neutral theory. Under the infinite-sites neutral model, mutations arise at rate $μ$ per generation along each lineage of the coalescent tree, and the expected number of segregating sites in a sample of $n$ is $θ \cdot a_{n}$ where $θ = 4 N_{e} μ$ and $a_{n} = \sum_{k = 1}^{n - 1} 1/ k$ . The estimator $\hat{θ} = S / a_{n}$ , where $S$ is the observed number of segregating sites, is Watterson's estimator — the cleanest expression of the coalescent's predictive content. The neutral theory of molecular evolution (Kimura 1968, 1983) takes the coalescent as its formal substrate: nucleotide substitutions accumulate at rate equal to the per-lineage mutation rate independent of population size, giving the molecular clock, while polymorphism within populations is the residue of mutations that have not yet fixed or lost. Hardy-Weinberg, in this view, is the deterministic null inside a stochastic universe whose finite-population deviations are precisely what the coalescent describes.

Connections [Master]

Hardy-Weinberg is the entry point to single-locus population genetics; every subsequent topic in $§19$ is constructed by relaxing one of the five assumptions and quantifying the resulting deviation. The selection equation $Δ p = pq [\cdot] / \overset{w}{ˉ}$ above is the foundation of the natural-selection unit ( $§19.03$ ). The Wright-Fisher chain and effective population size feed the genetic-drift and neutral-theory unit ( $§19.04$ ). Quantitative-genetic decompositions into additive, dominance, and epistatic variance ( $§19.05$ , planned) rest on Hardy-Weinberg as the null genotype-frequency reference. Speciation theory ( $§19.06$ ) uses Hardy-Weinberg as the within-population baseline against which reproductive isolation between populations is measured. Phylogenetics ( $§19.07$ ) uses the coalescent introduced here as its genealogical substrate. The Lotka-Volterra and life-history models of $§19.09$ – $11$ are population-level analogues of the same allele-frequency dynamics on demographic rather than genetic state. Cross-domain: the diffusion-limit Fokker-Planck equation is the same machinery used in Langevin / Brownian-motion treatments in physics $§11$ (stat mech) and in stochastic ODE work in math $§02$ . The coalescent is a random tree; its connection to phylogenetic graph theory and to random-graph theory is the cross-direction hook into math (probability, combinatorics, graph theory).

Hardy-Weinberg is the deterministic infinite-population limit; the stochastic finite-population generalisation is the Wright-Fisher model and its diffusion approximation 19.02.05. The Master sub-section above on Wright-Fisher drift and effective population size sketches the chain; the dedicated unit develops the full discrete chain, the diffusion limit derived from the Kolmogorov-forward equation, Kimura's fixation-probability formula $u (p) = (1 - e^{- 2 N s p}) / (1 - e^{- 2 N s})$ , the heterozygosity-decay rate $1/ (2 N)$ , and the Kingman coalescent as the dual genealogical process. Every quantitative test of Hardy-Weinberg implicitly compares observed allele-frequency data against the Wright-Fisher null with finite $N_{e}$ , and the foundational reason "deviation from Hardy-Weinberg" is the operational signature of every evolutionary force is precisely that the Wright-Fisher chain with $s = 0$ , $μ = 0$ , $m = 0$ converges to the Hardy-Weinberg fixed point on the deterministic timescale.

Historical & philosophical context [Master]

Hardy-Weinberg was named for two independent discoverers in 1908 — G. H. Hardy, an English mathematician known for his work in analytic number theory, and Wilhelm Weinberg, a German physician — neither of whom worked in evolutionary biology as a primary discipline. Hardy's paper in Science (vol. 28, 49–50) was written in response to a misconception, raised by R. C. Punnett, that a dominant allele would inevitably increase in frequency in a population. Hardy's algebra refuted the misconception in two pages. Weinberg's paper, in Jahreshefte des Vereins für vaterländische Naturkunde in Württemberg, contained the same result derived independently and developed at greater length, with applications to the inheritance of twinning and other human traits; it remained largely unknown to the English-speaking community until Curt Stern's 1943 review (Stern, Science 97, 137–138) restored Weinberg's priority. The asymmetric attribution — "Hardy-Weinberg" rather than "Hardy" or "Weinberg-Hardy" — reflects the linguistic structure of mid-20th-century population genetics, dominated by English-language texts.

The principle's deeper role emerged in the Modern Synthesis of the 1930s–40s. R. A. Fisher's 1930 Genetical Theory of Natural Selection and Sewall Wright's 1931 Evolution in Mendelian Populations and J. B. S. Haldane's 1924–1932 papers on the mathematical theory of natural selection together demonstrated that Mendelian inheritance and Darwinian selection, far from contradicting each other as some early geneticists had argued, combine into a quantitatively predictive theory of evolutionary change. Hardy-Weinberg is the equilibrium baseline of that theory: every quantitative claim about selection, drift, mutation, or migration is a statement about the rate or direction of deviation from this fixed point. Wright's $F$ -statistics, the Fisher fundamental theorem, the Wright-Fisher diffusion, and the Kimura-Kingman coalescent are all built on the same single-locus single-generation reset.

Philosophically, Hardy-Weinberg is the model case of a null hypothesis in biology — a reference state defined by what it excludes (selection, mutation, drift, migration, non-random mating) rather than by what it asserts. As such it has been a recurring touchstone in the philosophy-of-biology literature on idealisation, model-target relations, and the role of equilibrium thinking in evolutionary explanation (Sober 1984 The Nature of Selection; Beatty 1987 on chance vs design in evolutionary biology; Brandon-Carson 1996 on the indeterminism issue; Walsh 2007 on the propensity interpretation of fitness). The unit-of-selection problem — whether selection acts on genes, individuals, groups, or species — frames itself in terms of where and at what frequency-level Hardy-Weinberg breaks down. The neutral theory of Kimura (1968, 1983) is, at its core, the claim that for molecular polymorphism the Hardy-Weinberg-plus-drift baseline is not a null hypothesis to be rejected but the empirically correct first-order theory, with selection a perturbation of secondary importance at the sequence level. The continued use of Hardy-Weinberg as both the textbook entry point and the operational baseline of every population-genetic test reflects its dual role as a mathematical fixed point and an epistemological zero.

Bibliography [Master]

@article{Hardy1908,
  author    = {Hardy, G. H.},
  title     = {Mendelian Proportions in a Mixed Population},
  journal   = {Science},
  year      = {1908},
  volume    = {28},
  pages     = {49--50},
}

@article{Weinberg1908,
  author    = {Weinberg, W.},
  title     = {{\"U}ber den Nachweis der Vererbung beim Menschen},
  journal   = {Jahreshefte des Vereins f{\"u}r vaterl{\"a}ndische Naturkunde in W{\"u}rttemberg},
  year      = {1908},
  volume    = {64},
  pages     = {369--382},
}

@book{Fisher1930,
  author    = {Fisher, R. A.},
  title     = {The Genetical Theory of Natural Selection},
  publisher = {Clarendon Press},
  year      = {1930},
  address   = {Oxford},
}

@article{Wright1931,
  author    = {Wright, S.},
  title     = {Evolution in {M}endelian Populations},
  journal   = {Genetics},
  year      = {1931},
  volume    = {16},
  pages     = {97--159},
}

@article{Wahlund1928,
  author    = {Wahlund, S.},
  title     = {Zusammensetzung von Populationen und Korrelationserscheinungen vom Standpunkt der Vererbungslehre aus betrachtet},
  journal   = {Hereditas},
  year      = {1928},
  volume    = {11},
  pages     = {65--106},
}

@article{Kimura1968,
  author    = {Kimura, M.},
  title     = {Evolutionary Rate at the Molecular Level},
  journal   = {Nature},
  year      = {1968},
  volume    = {217},
  pages     = {624--626},
}

@book{Kimura1983,
  author    = {Kimura, M.},
  title     = {The Neutral Theory of Molecular Evolution},
  publisher = {Cambridge University Press},
  year      = {1983},
}

@article{Kingman1982a,
  author    = {Kingman, J. F. C.},
  title     = {The Coalescent},
  journal   = {Stochastic Processes and their Applications},
  year      = {1982},
  volume    = {13},
  pages     = {235--248},
}

@article{Kingman1982b,
  author    = {Kingman, J. F. C.},
  title     = {On the Genealogy of Large Populations},
  journal   = {Journal of Applied Probability},
  year      = {1982},
  volume    = {19A},
  pages     = {27--43},
}

@article{Stern1943,
  author    = {Stern, C.},
  title     = {The {H}ardy-{W}einberg Law},
  journal   = {Science},
  year      = {1943},
  volume    = {97},
  pages     = {137--138},
}

@book{HartlClark2007,
  author    = {Hartl, D. L. and Clark, A. G.},
  title     = {Principles of Population Genetics},
  publisher = {Sinauer Associates},
  edition   = {4},
  year      = {2007},
}

@book{CrowKimura1970,
  author    = {Crow, J. F. and Kimura, M.},
  title     = {An Introduction to Population Genetics Theory},
  publisher = {Harper and Row},
  year      = {1970},
}

@book{Ewens2004,
  author    = {Ewens, W. J.},
  title     = {Mathematical Population Genetics {I}: Theoretical Introduction},
  publisher = {Springer},
  edition   = {2},
  year      = {2004},
}

@book{CharlesworthCharlesworth2010,
  author    = {Charlesworth, B. and Charlesworth, D.},
  title     = {Elements of Evolutionary Genetics},
  publisher = {Roberts and Company},
  year      = {2010},
}

@book{WalshLynch2018,
  author    = {Walsh, B. and Lynch, M.},
  title     = {Evolution and Selection of Quantitative Traits},
  publisher = {Oxford University Press},
  year      = {2018},
}

@book{Futuyma2017,
  author    = {Futuyma, D. J.},
  title     = {Evolution},
  publisher = {Sinauer Associates},
  edition   = {4},
  year      = {2017},
}

@book{Gillespie2004,
  author    = {Gillespie, J. H.},
  title     = {Population Genetics: A Concise Guide},
  publisher = {Johns Hopkins University Press},
  edition   = {2},
  year      = {2004},
}

@article{Kimura1962,
  author    = {Kimura, M.},
  title     = {On the Probability of Fixation of Mutant Genes in a Population},
  journal   = {Genetics},
  year      = {1962},
  volume    = {47},
  pages     = {713--719},
}

@article{Ohta1973,
  author    = {Ohta, T.},
  title     = {Slightly Deleterious Mutant Substitutions in Evolution},
  journal   = {Nature},
  year      = {1973},
  volume    = {246},
  pages     = {96--98},
}

@book{Sober1984,
  author    = {Sober, E.},
  title     = {The Nature of Selection},
  publisher = {MIT Press},
  year      = {1984},
}

Prerequisites

02.13.01 pending
19.01.01 pending

Tier anchors

beginner: Coyne, *Why Evolution Is True* (Viking, 2009), Ch. 1 + Ch. 5; Crash Course Biology, *Population Genetics* episode; Hartl & Clark, *Principles of Population Genetics* 4th ed. (Sinauer, 2007), Ch. 1 introductory sections
intermediate: Hartl & Clark, *Principles of Population Genetics* 4th ed. Ch. 2–3; Futuyma, *Evolution* 4th ed. (Sinauer, 2017), Ch. 8 (variation in populations); Gillespie, *Population Genetics: A Concise Guide* 2nd ed. (Johns Hopkins, 2004), Ch. 1–2
master: Charlesworth & Charlesworth, *Elements of Evolutionary Genetics* (Roberts, 2010), Ch. 2 + Ch. 5; Crow & Kimura, *An Introduction to Population Genetics Theory* (Harper & Row, 1970), Ch. 1–3; Walsh & Lynch, *Evolution and Selection of Quantitative Traits* (Oxford, 2018), Ch. 1; Ewens, *Mathematical Population Genetics I* 2nd ed. (Springer, 2004), Ch. 1–3

References

TODO_REF
Hardy, G. H. — *Science* 28, 49–50 (1908) · Originator paper, *Mendelian proportions in a mixed population* — the algebra of $p^2 + 2pq + q^2$
TODO_REF
Weinberg, W. — *Jahreshefte des Vereins für vaterländische Naturkunde in Württemberg* 64, 369–382 (1908) · Independent simultaneous originator paper, *Über den Nachweis der Vererbung beim Menschen*
TODO_REF
Hartl, D. L. & Clark, A. G. — *Principles of Population Genetics*, 4th ed. (Sinauer Associates, 2007) · Ch. 2 organising principles of population genetics; Ch. 3 causes of evolution
TODO_REF
Futuyma, D. J. — *Evolution*, 4th ed. (Sinauer Associates, 2017) · Ch. 8 variation in populations; Ch. 9–12 selection, drift, and population structure
TODO_REF
Charlesworth, B. & Charlesworth, D. — *Elements of Evolutionary Genetics* (Roberts and Company, 2010) · Ch. 2 single-locus theory; Ch. 5 effective population size and the coalescent
TODO_REF
Crow, J. F. & Kimura, M. — *An Introduction to Population Genetics Theory* (Harper & Row, 1970; reprinted Blackburn Press, 2009) · Ch. 1 models of population growth and gene frequency change; Ch. 3 random genetic drift
TODO_REF
Wright, S. — *Genetics* 16, 97–159 (1931) · Originator paper, *Evolution in Mendelian populations* — drift, $F$-statistics, the shifting balance
TODO_REF
Fisher, R. A. — *The Genetical Theory of Natural Selection* (Clarendon Press, 1930) · Originator monograph — selection on Mendelian factors, fundamental theorem of natural selection
TODO_REF
Kimura, M. — *Nature* 217, 624–626 (1968); *The Neutral Theory of Molecular Evolution* (Cambridge, 1983) · Originator paper and monograph on the neutral theory
TODO_REF
Kingman, J. F. C. — *Stochastic Processes and their Applications* 13, 235–248 (1982); *Journal of Applied Probability* 19A, 27–43 (1982) · Originator papers on the coalescent process
TODO_REF
Ewens, W. J. — *Mathematical Population Genetics I: Theoretical Introduction*, 2nd ed. (Springer, 2004) · Ch. 1–3 deterministic and stochastic single-locus theory; Ch. 7 the coalescent
TODO_REF
Gillespie, J. H. — *Population Genetics: A Concise Guide*, 2nd ed. (Johns Hopkins, 2004) · Ch. 1–2 single-locus theory; Ch. 3 genetic drift
TODO_REF
Walsh, B. & Lynch, M. — *Evolution and Selection of Quantitative Traits* (Oxford, 2018) · Ch. 1 quantitative-genetic foundations grounded in single-locus pop-gen

Reviewer

Tyler (pending external evolutionary-biology reviewer per BIOLOGY_PLAN §7 — top recruitment priority; bio-side population-genetics specialist needed for master-tier sign-off)

Estimated time

beginner: 14m
intermediate: 35m
master: 60m

Intuition [Beginner]

Visual [Beginner]

Worked example [Beginner]

Check your understanding [Beginner]

Formal definition [Intermediate+]

Counterexamples to common slips

Key theorem with proof [Intermediate+]

Diagnosing deviations from Hardy-Weinberg

Inbreeding and the coefficient F

Two-locus Hardy-Weinberg and linkage disequilibrium

Population substructure — the Wahlund effect

Exercises [Intermediate+]

Wright-Fisher drift and effective population size [Master]

Selection on a Hardy-Weinberg background [Master]

The diffusion limit and the coalescent [Master]

Connections [Master]

Historical & philosophical context [Master]

Bibliography [Master]

Inbreeding and the coefficient $F$