Hardy–Weinberg Equilibrium As A Population Model

Introduction

Genotype frequencies describe how common each genotype is in a population. At a biallelic locus, the Hardy–Weinberg model connects those frequencies to the frequencies of the two alleles. It provides a simple population model: when its conditions hold, the allele frequencies determine the genotype probabilities.

The model is useful as a reference, not as a claim that every real population follows it exactly. Its assumptions concern how individuals mate, how gametes contribute alleles to offspring, and whether allele frequencies change across generations. Keeping those conditions distinct helps clarify what the model predicts and when that prediction is plausible.

Definition: At a biallelic autosomal locus with alleles \(A\) and \(a\), let \(p\) be the population frequency of \(A\) and \(q\) the population frequency of \(a\), so \(p+q=1\). The Hardy–Weinberg equilibrium (HWE) model assigns genotype probabilities \(p^2\) to \(AA\), \(2pq\) to \(Aa\), and \(q^2\) to \(aa\). These are model probabilities for genotypes in the population, not necessarily the exact proportions in a finite sample.

What the Model Says

The three probabilities in the definition sum to one: \(p^2+2pq+q^2=(p+q)^2=1\). The model therefore allocates the whole population across the three possible genotypes. The two homozygote probabilities depend on the square of the corresponding allele frequency; the heterozygote probability includes both ways the two different alleles can occur together.

HWE is a statement about genotype proportions in a population, conditional on its allele frequencies. It does not say that \(A\) and \(a\) are equally frequent, that every genotype is equally common, or that observed sample counts must match the probabilities exactly. If \(p\) is small, for example, the model assigns a small probability to \(AA\), but it does not guarantee that a particular sample will contain a particular number of \(AA\) individuals.

The equilibrium language refers to the relationship between allele and genotype frequencies under a population model. It is useful to distinguish a population that has HWE genotype proportions at one time from a population whose allele frequencies remain unchanged across generations. The latter is a condition supporting a persistent equilibrium, but the two statements are not identical: a population can have HWE genotype proportions at one time and then experience a change in allele frequencies.

Worked Example: Translating Allele Frequencies Into Model Proportions

Suppose a hypothetical population has \(p=0.70\) for \(A\) and \(q=0.30\) for \(a\). If the HWE model is appropriate, find the genotype probabilities and the expected genotype counts in a population of 1,200 individuals.

The allele frequencies sum to one: \(0.70+0.30=1.00\). The model probabilities are

P(AA)=p^2=(0.70)^2=0.49,\qquad P(Aa)=2pq=2(0.70)(0.30)=0.42,\qquad P(aa)=q^2=(0.30)^2=0.09.

Multiplying each probability by 1,200 gives the expected counts:

1{,}200(0.49)=588\ AA,\qquad 1{,}200(0.42)=504\ Aa,\qquad 1{,}200(0.09)=108\ aa.

These counts total \(588+504+108=1{,}200\). They are expected counts under the model, not a promise that a sample or finite population will have exactly these counts. The corresponding proportions are 49%, 42%, and 9%.

How Population Conditions Support HWE

The model is associated with a set of simplifying conditions. For a biallelic autosomal locus, the central ideas are random mating with respect to that locus, random sampling of gametes from the population gene pool, and stable allele frequencies across generations. The classical idealization also assumes a sufficiently large population and no systematic forces such as selection, mutation, or migration that change the allele frequencies.

Random mating: mating partners pair independently of their genotypes at the locus. This rules out systematic preference for a partner with a particular genotype at that locus. It does not mean that every individual has the same number of offspring, or that mating is random with respect to every characteristic.
Random sampling of gametes: the alleles transmitted to offspring are sampled from the population’s gamete pool according to its allele frequencies. In the idealized model, there is no genotype-specific transmission advantage, and the two gametes forming an offspring genotype are independent draws from that pool.
Stable allele frequencies: the values of \(p\) and \(q\) remain the same across generations. In the idealized population, this is supported by the absence of evolutionary forces that systematically alter allele frequencies. In a finite population, random sampling can cause allele frequencies to fluctuate, so exact stability is an idealization.

These ideas have related but distinct roles. Random mating and random union of gametes explain why genotype probabilities are expressed in terms of the gamete allele frequencies. Stability explains why those same allele frequencies can continue to describe the population over generations. If a force changes \(p\), the HWE proportions for the new generation, if the other conditions hold, must be calculated using the new \(p\), not the earlier value.

A Short Check on Allele-Frequency Stability

The HWE genotype proportions preserve the allele frequencies they are based on. This can be checked by counting the \(A\) allele copies represented in those proportions. Each \(AA\) individual contributes two copies of \(A\), each \(Aa\) individual contributes one, and each \(aa\) individual contributes none. Thus the allele frequency implied by the model is

p_{\text{implied}}=P(AA)+\frac{1}{2}P(Aa) =p^2+\frac{1}{2}(2pq) =p^2+pq =p(p+q) =p.

Likewise, the implied frequency of \(a\) is \(P(aa)+\frac{1}{2}P(Aa)=q^2+pq=q(p+q)=q\). This algebra shows why HWE proportions and their allele frequencies are internally consistent. It does not prove that allele frequencies will remain stable if selection, migration, mutation, or other changes act on the population.

Worked Example: Checking Allele Frequencies Implied by HWE

In a hypothetical population, suppose HWE genotype probabilities are \(P(AA)=0.36\), \(P(Aa)=0.48\), and \(P(aa)=0.16\). Check the allele frequencies implied by these proportions.

An \(AA\) genotype contributes two \(A\) copies, an \(Aa\) genotype contributes one, and an \(aa\) genotype contributes none. The frequency of \(A\) is therefore

p=0.36+\frac{1}{2}(0.48)=0.36+0.24=0.60.

Similarly, the frequency of \(a\) is

q=0.16+\frac{1}{2}(0.48)=0.16+0.24=0.40.

The frequencies sum to \(0.60+0.40=1.00\), as required. The calculation verifies that the given genotype proportions are consistent with allele frequencies \(p=0.60\) and \(q=0.40\). It does not, by itself, establish that the population’s mating or evolutionary conditions satisfy the HWE assumptions.

When Allele Frequencies Change

A population may begin with HWE genotype proportions and still move away from the same equilibrium if a process changes the allele frequencies. For example, if one genotype has higher survival or reproductive success, its alleles may be overrepresented among the next generation’s gametes. Random mating among those gametes cannot restore the original allele frequencies; it can only pair the gametes available in that generation’s gene pool.

Worked Example: Selection Breaks Allele-Frequency Stability

Consider a hypothetical population with initial HWE proportions \(0.25\) \(AA\), \(0.50\) \(Aa\), and \(0.25\) \(aa\), so \(p=0.50\). Suppose relative survival is 1 for \(AA\), 1 for \(Aa\), and 0.5 for \(aa\). Calculate the \(A\) frequency among survivors and explain the implication for stable allele frequencies.

Before survival, the genotype proportions sum to \(0.25+0.50+0.25=1.00\). After applying the relative survival values, the survivor weights are \(0.25(1)=0.25\) for \(AA\), \(0.50(1)=0.50\) for \(Aa\), and \(0.25(0.5)=0.125\) for \(aa\). Their total is \(0.875\).

The weighted number of \(A\) copies per original individual is \(2(0.25)+1(0.50)+0(0.125)=1.00\). The total number of allele copies per original individual among survivors is \(2(0.875)=1.75\). Consequently,

p_{\text{survivors}}=\frac{1.00}{1.75}\approx 0.5714.

The \(A\) frequency among survivors is about 0.5714, rather than the initial 0.50. Selection has changed the allele frequency, so the assumption of stable allele frequencies across this transition does not hold. Random mating among the survivors would not undo this change; any subsequent HWE proportions would be based on the survivor gamete pool’s allele frequencies, if the other model conditions were appropriate.

Interpreting HWE as a Reference Model

HWE gives a baseline relationship between allele frequencies and genotype probabilities. It can help organize expectations and motivate later comparisons between observed genotype frequencies and model frequencies. But computing the HWE probabilities alone does not show that a particular population satisfies the assumptions. Nor does a close match between observed and model proportions prove that every assumption holds: different processes can have similar consequences for the observed genotype counts.

For any application, first specify the population and locus, then state the allele frequencies being used and the model conditions that make the HWE interpretation reasonable. When counts come from a sample, sampling variation also means observed proportions need not equal model probabilities exactly. The model describes population probabilities; the data are a finite realization of those probabilities.

Common Mistakes

Treating the formula as a universal identity. The proportions \(p^2\), \(2pq\), and \(q^2\) are HWE model probabilities, not genotype frequencies that must hold in every population.
Equating random mating with stable allele frequencies. Random mating concerns how partners pair. Selection or migration can still alter allele frequencies, even if mating is random.
Assuming a sample must match the model exactly. Expected counts need not be observed counts. Finite samples and populations can show fluctuations around model probabilities.
Confusing a single generation with long-run stability. A population can have HWE proportions at one time and subsequently change its allele frequencies. Stability is a condition about change across generations.
Interpreting HWE as evidence of no evolutionary process whatsoever. HWE is a model reference. Agreement with its proportions does not, by itself, establish the absence of every process that could affect the population.

Key takeaway: At a biallelic autosomal locus, HWE relates allele frequencies \(p\) and \(q\) to genotype probabilities \(p^2\), \(2pq\), and \(q^2\). Random mating and random sampling of gametes support that genotype relationship; stable allele frequencies support its persistence across generations.

Check Your Understanding

Use the model conditions and definitions to answer the following questions.

If \(p=0.40\) and \(q=0.60\), what genotype probabilities does the HWE model assign?
In your own words, distinguish random mating from stable allele frequencies.
Why are HWE probabilities not necessarily identical to observed proportions in a finite sample?
If selection changes the allele frequency from one generation to the next, which HWE condition is violated?
Do genotype proportions that match \(p^2\), \(2pq\), and \(q^2\) prove that all HWE assumptions hold? Explain.