Q.(p+q)² = p² + 2pq + q² = 1 represents an equation used in:
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Start your 14-day free trial to unlock the full solution →The equation $(p+q)^2 = p^2 + 2pq + q^2 = 1$ represents the Hardy-Weinberg principle, a fundamental concept in population genetics used to describe allele and genotype frequencies in a stable population.
The equation $(p+q)^2 = p^2 + 2pq + q^2 = 1$ is a mathematical expression of the Hardy-Weinberg principle, a cornerstone concept in the field of population genetics. To understand its significance, we first need to grasp what population genetics entails.
Population genetics is the study of genetic variation within populations, focusing on how allele and genotype frequencies change over time. It examines the gene pool, which is the total of all genes and their alleles in a population. Evolution, at its most fundamental level, can be defined as a change in these allele frequencies in a population across generations. The Hardy-Weinberg principle provides a mathematical model for a hypothetical population that is not evolving, serving as a baseline against which real populations can be compared to detect evolutionary change.
Let's break down the equation:
Consider a gene with two alleles, typically represented as a dominant allele (e.g., 'A') and a recessive allele (e.g., 'a').
- $p$ represents the frequency of the dominant allele ('A') in the population.
- $q$ represents the frequency of the recessive allele ('a') in the population. Since these are the only two alleles for this gene, their frequencies must sum to 1, meaning $p + q = 1$. This equation describes the allele frequencies in the gene pool.
When individuals in a population mate randomly, the probability of forming different genotypes in the next generation can be predicted using these allele frequencies. The expansion of $(p+q)^2$ gives us the frequencies of the three possible genotypes:
- $p^2$ represents the frequency of the homozygous dominant genotype (AA). This is the probability of an individual inheriting an 'A' allele from both parents ($p \times p$).
- $q^2$ represents the frequency of the homozygous recessive genotype (aa). This is the probability of an individual inheriting an 'a' allele from both parents ($q \times q$).
- $2pq$ represents the frequency of the heterozygous genotype (Aa). This accounts for two possibilities: inheriting 'A' from one parent and 'a' from the other ($p \times q$), or 'a' from the first parent and 'A' from the second ($q \times p$), hence $2pq$.
The sum of all these genotype frequencies must also equal 1, representing 100% of the population:
$$p^2 + 2pq + q^2 = 1$$
This equation describes the genotype frequencies in the population.
The Hardy-Weinberg principle states that allele frequencies in a population remain constant from generation to generation in the absence of evolutionary influences. This theoretical state of equilibrium is maintained only if five specific conditions are met:
- No gene mutation
- No gene flow (migration of individuals into or out of the population)
- Random mating among individuals
- No natural selection (all genotypes have equal survival and reproductive rates)
- A very large population size (to avoid genetic drift, which is random fluctuation in allele frequencies) …
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