Deriving Genotype Frequencies Under Hardy-Weinberg Equilibrium

Why Derive Genotype Frequencies?

A PBPK virtual cohort may need genotypes as well as age, organ size, and other physiological attributes. If allele frequencies are known, Hardy–Weinberg equilibrium (HWE) provides a simple way to calculate the genotype frequencies expected under random mating. These are population expectations, not guarantees about the exact composition of a particular cohort. The derivation matters because it shows both what information the calculation uses and which assumptions connect allele frequencies to genotype frequencies.

This tutorial considers one autosomal locus with two alleles, denoted \(A\) and \(a\), in a diploid population. It does not assign functional effects to either allele. The calculation is about the expected distribution of genotypes, not about enzyme activity or drug disposition.

Deriving the Three Frequencies

The key assumption for the one-generation derivation is random union of gametes with respect to this locus: an allele in one gamete pairs independently of the allele in the other gamete. Assume that both gamete pools have the same allele frequencies, \(p\) for \(A\) and \(q\) for \(a\). Each offspring receives one allele from each gamete.

There are four possible ordered pairings. An \(AA\) offspring requires an \(A\) from each gamete, so its probability is \(p \times p\). An \(aa\) offspring requires \(a\) from each, with probability \(q \times q\). A heterozygote can arise in either order: \(A\) from the first gamete and \(a\) from the second, or \(a\) from the first and \(A\) from the second. Each order has probability \(pq\), so the total heterozygote probability is \(pq+qp=2pq\).

The probabilities sum to one because \(p+q=1\):

This identity is a useful arithmetic check. The same derivation can be written as \((p+q)^2\): the two same-allele terms correspond to the homozygotes, and the two cross-products combine to give the heterozygote frequency.

From Frequencies to Expected Counts

For a population or sample of \(N\) individuals, multiply each expected genotype frequency by \(N\). The resulting values are expected counts; they need not be integers, and the observed counts in a finite sample need not equal them exactly.

Inferring Allele Frequency From a Hypothetical Genotype Table

Sometimes a genotype table is available and allele frequencies must first be calculated. Each homozygote contributes two copies of its allele, and each heterozygote contributes one copy of each allele. For \(N\) diploid individuals, there are \(2N\) allele copies. The sample frequency of \(A\) is therefore the number of \(A\) copies divided by \(2N\). Substituting this estimate into the HWE expressions gives expected counts conditional on that estimated allele frequency.

What If the Gamete Pools Differ?

The standard \(p^2,2pq,q^2\) expressions require the same allele frequencies in the two gamete pools. If the frequency of \(A\) is \(p_1\) in one pool and \(p_2\) in the other, the pairing probabilities instead follow directly from those two probabilities. The probability of a heterozygote is the sum of the two ways to form one.

Assumptions, Diagnostics, and Alternatives

The algebra is exact under its probability assumptions, but applying it to a real or simulated cohort requires care. The locus must be treated as diploid and biallelic for these three genotype categories to suffice. The allele frequencies must refer to a clearly defined population, and the gametes must pair independently with respect to the locus. If the population is expected to maintain the same allele frequencies over generations, additional evolutionary conditions matter: selection, mutation, migration, and random drift can change allele frequencies.

When observed genotype counts differ from HWE expectations, consider both biological and data-related explanations. Nonrandom mating, population structure, relatedness, selection, and unequal allele frequencies across mating groups can alter genotype proportions. Genotyping errors, sample selection, and small-sample variation can also create discrepancies. A deviation does not uniquely identify its cause, and matching HWE expectations does not prove that every assumption holds.

A useful diagnostic sequence is to verify the genotype definitions and sample size, recalculate allele frequencies from allele copies, check that the expected frequencies sum to one, and then compare observed counts with expected counts. Examine whether the sample represents one coherent population and whether there are reasons to expect nonrandom mating or allele-dependent survival. If the goal is simply to describe the cohort, observed genotype proportions are a direct alternative: report the observed counts and divide by \(N\), without imposing HWE. If different mating pools or subgroups are known, calculate expectations within those groups rather than treating them as one homogeneous pool.

HWE expectations are useful as a transparent baseline, including when constructing a hypothetical population from allele frequencies. They are not a substitute for measured genotype data when those data are available, nor do the expected genotype frequencies alone specify how genotypes affect a drug-related parameter. That requires a separate biological model.

Common Mistakes

• Forgetting the two ways to form a heterozygote. Using \(pq\) for \(Aa\) counts only one ordering. The correct frequency is \(pq+qp=2pq\).
• Treating expected counts as observed facts. \(Np^2\) is an expectation under the model; finite samples can differ from it.
• Using the wrong allele-frequency denominator. In \(N\) diploid individuals there are \(2N\) allele copies, not \(N\).
• Assuming a discrepancy proves a particular biological mechanism. Several biological, sampling, and measurement factors can produce differences from expected counts.
• Combining distinct gamete pools without checking them. The simple expressions assume the same \(p\) and \(q\) in both pools; otherwise use the separate-pool probabilities.