The Hardy-Weinberg principle is a fundamental mathematical model used in population genetics. This principle describes a theoretical state where allele and genotype frequencies within a population remain constant across generations. It serves as a baseline for understanding how genetic variation is maintained in a population that is not undergoing evolutionary change. By comparing real-world data to this ideal model, scientists can identify and measure the forces that cause genetic change, which is the definition of evolution.
Understanding the Two Core Equations
The Hardy-Weinberg model relies on two distinct but related mathematical expressions. The first, the Allele Frequency Equation, is represented as \(p+q=1\). Here, \(p\) symbolizes the proportion of the dominant allele, while \(q\) represents the proportion of the recessive allele. Since these are the only two alleles considered for a single gene locus, their frequencies must sum to one (100%).
The second expression is the Genotype Frequency Equation, written as \(p^2+2pq+q^2=1\). This equation is the mathematical expansion derived from squaring the allele frequency equation \((p+q)\). In this formula, \(p^2\) is the predicted frequency of individuals with the homozygous dominant genotype. Similarly, \(q^2\) represents the frequency of the homozygous recessive genotype. The term \(2pq\) represents the frequency of heterozygous individuals, who carry both the dominant and recessive alleles.
Solving for Allele Frequencies
Solving Hardy-Weinberg problems begins with calculating the allele frequencies, \(p\) and \(q\). This step relies on identifying the frequency of the homozygous recessive genotype, \(q^2\). The recessive phenotype is the only one directly observable that reveals the underlying genotype. Since the dominant phenotype can belong to either homozygous dominant (\(p^2\)) or heterozygous (\(2pq\)) individuals, the recessive trait provides the only unambiguous starting point.
To find the recessive allele frequency, \(q\), take the square root of the calculated \(q^2\) value. For example, if 4% of individuals exhibit a specific recessive trait, the frequency \(q^2\) is \(0.04\). Taking the square root of \(0.04\) yields a recessive allele frequency (\(q\)) of \(0.2\). This \(q\) value represents the proportion of recessive alleles in the gene pool.
Once \(q\) is established, the first Hardy-Weinberg equation, \(p+q=1\), is used to determine \(p\), the frequency of the dominant allele. Since the sum of the two allele frequencies must equal one, \(p\) is calculated as \(1 – q\). Continuing the example, \(p\) would be \(1 – 0.2\), resulting in a dominant allele frequency (\(p\)) of \(0.8\). This methodology translates the observed frequency of a physical trait into the underlying genetic frequencies of the population’s alleles.
Solving for Genotype Frequencies
After determining the frequencies of both the dominant (\(p\)) and recessive (\(q\)) alleles, the next step is to calculate the three individual genotype frequencies using the second Hardy-Weinberg equation. This calculation allows researchers to predict the precise genetic makeup of the population based on the established allele proportions. The frequency of homozygous dominant individuals (\(p^2\)) is determined by squaring the \(p\) value.
In the running example, where \(p=0.8\) and \(q=0.2\), the frequency of the homozygous dominant genotype (\(p^2\)) is \(0.8^2\), which equals \(0.64\). The frequency of the homozygous recessive genotype (\(q^2\)) is calculated as \(0.2^2\), resulting in \(0.04\). This \(q^2\) value should align with the initial observed frequency used to begin the calculation.
The most frequent application of this equation is finding the frequency of heterozygous carriers, represented by the term \(2pq\). This value is found by multiplying \(2 \times p \times q\). Using the sample values, the frequency of heterozygous carriers is \(2 \times 0.8 \times 0.2\), yielding \(0.32\). This indicates that 32% of the population are carriers, a proportion often substantially larger than the individuals who express the recessive trait. To ensure accuracy, all three genotype frequencies—\(p^2\), \(2pq\), and \(q^2\)—must sum precisely to \(1.0\).
Conditions for Hardy-Weinberg Equilibrium
The Hardy-Weinberg model relies on five specific theoretical conditions to maintain equilibrium.
Conditions for Equilibrium
- Absence of gene mutations.
- Completely random mating within the population.
- An infinitely large population to prevent genetic drift.
- No gene flow (no migration of individuals into or out of the population).
- Absence of natural selection, ensuring every genotype has equal survival and reproductive success.
Because all real populations violate at least one assumption, the principle acts as a baseline, or null hypothesis, for measuring the evolutionary forces at play.