Genotype Frequencies And Their Allele Marginals

From Genotypes to Allele Copies

A genotype frequency describes the proportion of individuals with a particular genotype. An allele frequency describes the proportion of allele copies of a particular type. These are related quantities, but they do not have the same denominator: each diploid individual has two allele copies at a locus. To move from genotype frequencies to allele frequencies, count the copies contributed by each genotype.

This calculation is a probability marginal. Imagine selecting an individual at random and then selecting one of that individual's two allele copies at random. The probability that the selected copy is a particular allele is its allele frequency. A homozygote contributes two copies of the same allele; a heterozygote contributes one copy of each of its two alleles.

For a locus with \(k\) possible alleles, a homozygote \(A_iA_i\) contributes two copies of \(A_i\), while each heterozygote \(A_iA_j\) contributes one. In frequency form, this gives

The factor \(1/2\) appears because each genotype frequency is a proportion of individuals, and each individual has two allele copies. A heterozygote containing \(A_i\) contributes one of its two copies to \(A_i\). The homozygote frequency \(f_{ii}\) enters without a factor of \(1/2\), because both copies are \(A_i\).

Biallelic Loci

For a locus with alleles \(A\) and \(a\), the possible genotypes are \(AA\), \(Aa\), and \(aa\). Write their frequencies as \(f_{AA}\), \(f_{Aa}\), and \(f_{aa}\). The two allele frequencies are

The genotype frequencies must satisfy \(f_{AA}+f_{Aa}+f_{aa}=1\). The allele frequencies must also sum to 1, since every allele copy is either \(A\) or \(a\). The relationship follows directly from the formulas:

Allele Frequencies as Marginal Probabilities

The word marginal means that we focus on one component of a joint probability distribution while summing over the possible values of the other component. A diploid genotype describes the pair of allele copies in an individual. To obtain the marginal frequency of \(A_i\), count its share of the copies in every genotype, then add those contributions.

One way to formalize this is to define \(C_i\) as the number of \(A_i\) copies in an individual's genotype. For a homozygote \(A_iA_i\), \(C_i=2\); for a heterozygote containing \(A_i\), \(C_i=1\); and for other genotypes, \(C_i=0\). The average number of \(A_i\) copies per individual is therefore \(2f_{ii}+\sum_{j\ne i}f_{ij}\). Dividing by two copies per individual gives the allele frequency.

This argument also establishes the normalization identity for any number of alleles. Each genotype has exactly two copies in total, so the sum of the allele-copy counts across all allele types is 2 for every individual. The expected total number of copies is therefore 2; dividing by 2 means the allele frequencies sum to 1.

The identity requires a complete genotype distribution for diploid individuals: every individual must have exactly two allele copies at the locus, and the listed genotypes must cover all possibilities. It does not require a particular mating pattern or a special relationship among genotype frequencies.

More Than Two Alleles

The same copy-counting rule applies when there are three or more alleles. For each allele, include its homozygote frequency in full and each heterozygote frequency containing it at half weight. Genotype frequencies still describe individuals, while allele frequencies describe copies.

Checking Frequencies and Understanding the Reverse Calculation

There are two useful checks before interpreting an allele-frequency calculation. First, genotype frequencies must be nonnegative and sum to 1 if they describe a complete distribution. Second, the calculated allele frequencies must be nonnegative and sum to 1. These checks are necessary, though a sum of 1 alone does not guarantee that the frequencies were calculated correctly.

The operation is not generally reversible. Genotype frequencies determine allele frequencies, but allele frequencies alone do not usually determine genotype frequencies. Allele frequencies retain the overall proportions of copies while discarding information about how those copies are paired within individuals.

Common Mistakes and Why the Distinction Matters

• Using genotype frequencies as allele frequencies. A genotype frequency has individuals as its denominator; an allele frequency has allele copies as its denominator. A heterozygote contributes one copy of each allele, not a whole copy of each genotype frequency.
• Forgetting to halve heterozygote frequencies. A heterozygote's frequency is divided equally between its two allele types when calculating their copy proportions.
• Halving a homozygote frequency. Every copy in a homozygote is the same allele, so its full frequency contributes to that allele's frequency.
• Assuming allele frequencies uniquely give genotype frequencies. The reverse is not determined by copy proportions alone. As the example shows, distinct genotype distributions can have identical allele frequencies.
• Skipping the sum check. If genotype frequencies do not sum to 1, they may be incomplete or incorrectly recorded. If the resulting allele frequencies do not sum to 1, check the genotype list and the copy weights.

Allele marginals matter because they provide a compact summary of the allele-copy composition at a locus. They are calculated directly from genotype frequencies, but they should not be mistaken for a full description of genotype composition. Keeping those two levels distinct makes probability calculations easier to interpret and helps identify what the available frequencies can, and cannot, establish.