The Hardy-Weinberg principle is a foundational concept in population genetics, acting as a null hypothesis that describes a non-evolving population. It uses algebraic symbols to represent the frequencies of alleles and genotypes within a theoretical gene pool that is not being acted upon by evolutionary forces like mutation, migration, or natural selection. The model relies on two interconnected equations: one for allele frequencies and a second for genotype frequencies.
The Definition of ‘q’ and ‘p’ (Allele Frequencies)
The letter ‘q’ represents the relative frequency of the recessive allele (e.g., ‘a’) for a specific gene within a population’s gene pool. Its frequency is expressed as a proportion between 0 and 1. ‘q’ is intrinsically linked to ‘p’, which represents the frequency of the corresponding dominant allele (e.g., ‘A’).
These two variables form the basis of the Hardy-Weinberg allele frequency equation: \(p + q = 1\). This equation states that the frequencies of the dominant allele (‘p’) and the recessive allele (‘q’) must account for 100% of all alleles for that gene in the population.
Calculating ‘q’ from Observed Phenotypes (\(q^2\))
To determine the numerical value of ‘q’, geneticists use the second Hardy-Weinberg equation, which describes genotype frequencies: \(p^2 + 2pq + q^2 = 1\). In this equation, \(q^2\) represents the frequency of the homozygous recessive genotype, typically ‘aa’. This genotypic frequency is the starting point for calculating ‘q’ because the homozygous recessive phenotype is the only one whose genotype is unambiguously known from observation.
Individuals expressing the recessive trait must have inherited two copies of the recessive allele, meaning their genotype must be ‘aa’ or \(q^2\). In contrast, individuals showing the dominant trait could have either the homozygous dominant (\(p^2\)) or the heterozygous (\(2pq\)) genotype, making their exact genetic makeup impossible to determine by appearance alone. By counting the number of individuals with the recessive phenotype and dividing that by the total population size, one obtains the value for \(q^2\).
The frequency of the recessive allele ‘q’ is then found by taking the square root of the homozygous recessive genotype frequency, \(q^2\). For example, if a population study found that 16% of individuals showed the recessive trait, then \(q^2\) would be 0.16. Taking the square root of 0.16 yields a value of 0.4, meaning the frequency of the recessive allele ‘q’ is 0.4. This step provides the first allele frequency, which is necessary to calculate all other unknown frequencies in the model.
Using ‘q’ to Determine Population Genotypes (\(2pq\) and \(p^2\))
Once the numerical value for ‘q’ has been calculated from the observed recessive trait, it is possible to determine the frequency of the dominant allele ‘p’ using the relationship \(p + q = 1\). If ‘q’ is 0.4, then ‘p’ must be \(1 – 0.4\), which equals 0.6. With both allele frequencies known, the full genotypic structure of the population can be determined by solving the components of the second equation.
The frequency of the homozygous dominant genotype, ‘AA’ or \(p^2\), is found by squaring the value of ‘p’. In this example, \(p^2\) would be \(0.6 \times 0.6\), resulting in a frequency of 0.36. This figure represents the proportion of individuals in the population that are genetically ‘AA’.
The frequency of the heterozygous genotype, ‘Aa’ or \(2pq\), is calculated by multiplying \(2 \times p \times q\). Using the calculated values, this becomes \(2 \times 0.6 \times 0.4\), which yields a frequency of 0.48. This \(2pq\) value estimates the proportion of the population who are carriers for a recessive condition without expressing the trait themselves.
Finally, the accuracy of all calculations can be verified by summing the three genotype frequencies: \(p^2 (0.36) + 2pq (0.48) + q^2 (0.16)\). If the total equals 1.0, the frequencies are in Hardy-Weinberg proportions.