According to the Hardy-Weinberg theorem, p + q = 1 and p2 + 2pq +q2 = 1. What does each of these formulas mean (how are they derived)
The Correct Answer and Explanation is:
The Hardy-Weinberg theorem is a principle in population genetics that describes how gene frequencies remain constant from generation to generation under certain ideal conditions. The two key equations involved are:
- p + q = 1
- p² + 2pq + q² = 1
In these formulas, p represents the frequency of the dominant allele, and q represents the frequency of the recessive allele in a population. Together, they describe the genetic variation of a trait governed by a single gene with two alleles.
Explanation and Derivation:
The first equation, p + q = 1, is a statement of total allele frequency. Since there are only two alleles in the population for a given gene, the sum of their frequencies must equal 1. For example, if 60 percent of the alleles in the population are dominant (p = 0.6), then the remaining 40 percent must be recessive (q = 0.4).
The second equation, p² + 2pq + q² = 1, describes the expected frequencies of the three possible genotypes in the next generation, assuming random mating and no evolutionary forces acting on the population. These three genotypes are:
- p²: the frequency of individuals who are homozygous dominant (carry two dominant alleles)
- 2pq: the frequency of heterozygous individuals (carry one dominant and one recessive allele)
- q²: the frequency of homozygous recessive individuals (carry two recessive alleles)
This equation comes from applying the binomial expansion of (p + q)², which represents all the possible combinations of alleles during fertilization. When you square the sum of allele frequencies, you model how alleles combine to form genotypes. Therefore, the Hardy-Weinberg equilibrium shows how allele and genotype frequencies remain stable over generations if no mutation, selection, migration, or genetic drift occurs. It serves as a null model in population genetics.
