use the Hardy–Weinberg principle to calculate allele and genotype frequencies in populations and state the conditions when this principle can be applied (the two equations for the Hardy–Weinberg principle will be provided, as shown in the Mathematica

Natural and Artificial Selection

Selection is the process by which certain genotypes become more common in a population because they confer a reproductive advantage. Natural selection occurs in the wild, whereas artificial selection is driven by human choice (e.g., breeding of crops or pets).

Hardy–Weinberg Principle

The Hardy–Weinberg principle provides a mathematical model for a population that is not evolving. It allows us to calculate allele and genotype frequencies and to test whether a population is in equilibrium.

Key Equations

For a gene with two alleles, A (dominant) and a (recessive), let:

• \$p\$ = frequency of allele A
• \$q\$ = frequency of allele a

Because there are only two alleles:

\$p + q = 1\$

The expected genotype frequencies under Hardy–Weinberg equilibrium are:

\$\$\begin{aligned}

\text{Frequency of } AA &= p^{2} \\

\text{Frequency of } Aa &= 2pq \\

\text{Frequency of } aa &= q^{2}

\end{aligned}\$\$

Conditions for Hardy–Weinberg Equilibrium

The principle applies only when all of the following conditions are met:

1. Very large (effectively infinite) population size – eliminates genetic drift.
2. No migration (gene flow) into or out of the population.
3. No mutation introducing new alleles.
4. Random mating – individuals pair without regard to genotype.
5. No natural or artificial selection – all genotypes have equal reproductive success.

Calculating Allele and Genotype Frequencies

Example 1 – Using Phenotype Data

In a population of 200 pea plants, the flower colour is determined by a single gene with alleles P (purple, dominant) and p (white, recessive). The observed phenotypes are:

• Purple flowers (genotypes PP or Pp): 170 plants
• White flowers (genotype pp): 30 plants

Step‑by‑step calculation:

1. Calculate \$q\$ from the recessive phenotype:

   \$q^{2} = \frac{30}{200} = 0.15\$

   \$q = \sqrt{0.15} \approx 0.387\$

2. Find \$p\$ using \$p + q = 1\$:

   \$p = 1 - 0.387 \approx 0.613\$

3. Predict genotype frequencies:

   • \$p^{2} = (0.613)^{2} \approx 0.376\$ (PP)
   • \$2pq = 2(0.613)(0.387) \approx 0.474\$ (Pp)
   • \$q^{2} = 0.150\$ (pp)

4. Convert to expected numbers (multiply by total population 200):

   • PP ≈ 0.376 × 200 ≈ 75
   • Pp ≈ 0.474 × 200 ≈ 95
   • pp = 30 (matches observed)

Example 2 – Testing for Evolution

Suppose a second generation of the same pea plants is sampled and the genotype counts are:

Because the observed frequencies are close to the expected Hardy–Weinberg frequencies, the population can be considered to be in equilibrium (no detectable evolution). Significant deviations would indicate that one or more of the equilibrium conditions are not being met, implying natural or artificial selection, migration, drift, etc.

Linking Selection to Hardy–Weinberg

When selection acts on a trait, the genotype frequencies change from the Hardy–Weinberg expectations. For example, if individuals with genotype aa have lower survival, the frequency of allele a (\$q\$) will decrease over generations, and the population will no longer satisfy the equilibrium equations.

Artificial Selection Example

Selective breeding of dogs for a particular coat colour can dramatically increase the frequency of the allele responsible for that colour. By repeatedly choosing only carriers of the desired allele as breeding stock, the population moves away from Hardy–Weinberg equilibrium, demonstrating the effect of human‑driven selection.

Summary

• The Hardy–Weinberg principle provides a null model for a non‑evolving population.
• Allele frequencies (\$p\$, \$q\$) and genotype frequencies (\$p^{2}\$, \$2pq\$, \$q^{2}\$) are calculated using the two core equations.
• All five equilibrium conditions must be met for the principle to apply.
• Deviations from expected frequencies indicate that forces such as natural or artificial selection, migration, mutation, drift, or non‑random mating are acting on the population.