The Hardy-Weinberg Equilibrium (HWE) is a foundational concept in population genetics that provides a baseline for understanding how gene frequencies behave in a non-evolving population. It acts as a null hypothesis, offering a standard against which scientists can measure genetic change in real-world populations. When a population deviates from this equilibrium, it signals that an evolutionary force, such as natural selection or migration, is acting on the gene pool. Calculating HWE allows researchers to determine if allele and genotype frequencies remain constant across generations, defining genetic equilibrium.
Foundational Principles of Hardy-Weinberg Equilibrium
The principle describes a hypothetical state where a population’s genetic structure remains static because specific conditions prevent changes in allele frequencies. These conditions are rarely met in nature, which is why the model is valuable for detecting the forces of evolution. The model assumes there is no gene flow (no migration into or out of the population). It also assumes that no mutations are introducing new alleles or changing existing ones.
The model requires that mating within the population is completely random, ensuring individuals are not choosing mates based on specific genotypes. For the equilibrium to hold, the population size must be very large, ideally infinite, to prevent random shifts in allele frequencies known as genetic drift. Finally, the model assumes there is no natural selection, meaning all genotypes have equal survival and reproductive rates.
The algebraic notation for the two-allele system uses \(p\) and \(q\) to represent the frequencies of the two alleles (dominant and recessive, respectively). Since only two alleles are present, their frequencies must sum to one: \(p + q = 1\). The expected frequencies of the three possible genotypes are described by the genotype frequency equation, the binomial expansion of \((p+q)^2\): \(p^2 + 2pq + q^2 = 1\). Here, \(p^2\) represents the frequency of the homozygous dominant genotype, \(q^2\) represents the frequency of the homozygous recessive genotype, and \(2pq\) represents the frequency of the heterozygous genotype.
Step-by-Step Calculation for Two Alleles
Calculation of HWE for a two-allele system often begins by observing the frequency of the homozygous recessive phenotype. This phenotype is directly linked to the \(q^2\) term, as only individuals with two recessive alleles express the trait. Consider a hypothetical population of 1,000 individuals where 160 exhibit the recessive phenotype.
The first step is to determine the frequency of the homozygous recessive genotype, \(q^2\), by dividing 160 by 1,000, yielding \(q^2 = 0.16\). Step two involves calculating the frequency of the recessive allele, \(q\), by taking the square root of \(q^2\): \(q = \sqrt{0.16}\), or \(0.4\). This value represents the recessive allele frequency in the gene pool.
The third step uses the allele frequency equation, \(p + q = 1\), to find the frequency of the dominant allele, \(p\). Substituting \(q\): \(p + 0.4 = 1\), which simplifies to \(p = 0.6\). With both allele frequencies determined, the final step is to calculate the expected frequencies of the remaining two genotypes using the genotype frequency equation.
The frequency of the homozygous dominant genotype, \(p^2\), is \(0.6^2\), resulting in \(0.36\). The frequency of the heterozygous genotype, \(2pq\), is \(2 \times 0.6 \times 0.4\), which equals \(0.48\). To verify the calculation, the sum of all three genotype frequencies must equal one: \(0.36 (p^2) + 0.48 (2pq) + 0.16 (q^2) = 1\).
Extending the Calculation to Three Alleles
The standard two-allele model is inadequate for traits controlled by multiple alleles, such as the human ABO blood group system, which involves three distinct alleles: \(I^A, I^B,\) and \(i\). To accommodate this complexity, the Hardy-Weinberg principle is extended to include a third allele frequency, denoted as \(r\). The expanded allele frequency equation becomes \(p + q + r = 1\), where \(p, q\), and \(r\) are the frequencies of the \(I^A, I^B\), and \(i\) alleles, respectively.
The expanded genotype frequency equation is the trinomial expansion of \((p + q + r)^2\): \(p^2 + q^2 + r^2 + 2pq + 2pr + 2qr = 1\). This equation accounts for all six possible genotypes. The terms \(p^2, q^2\), and \(r^2\) represent the frequencies of the three homozygous genotypes, while \(2pq, 2pr\), and \(2qr\) represent the frequencies of the three heterozygous genotypes.
Calculation in a tri-allelic system begins by finding the frequency of the most easily identifiable homozygous genotype, which often corresponds to the recessive phenotype. For the ABO system, the O blood type (\(ii\)) corresponds to the \(r^2\) term. Taking the square root of the observed frequency of the O blood type determines the value for \(r\).
Once \(r\) is known, the frequencies of the dominant alleles, \(p\) and \(q\), can be calculated using the observed frequencies of the A and B blood types. The A blood type phenotype is the sum of two genotypes (\(I^A I^A\) and \(I^A i\)), corresponding to \(p^2 + 2pr\). The B blood type phenotype is \(q^2 + 2qr\). Algebraic manipulation, often relying on the relationship that \(p+r\) is the square root of the combined frequency of A and O phenotypes, allows for the determination of all three allele frequencies.