Welcome to the Math of Evolution!

When we think of evolution, we often think of giant dinosaurs or dramatic changes over millions of years. But at its core, evolution is actually much simpler: it is a change in the genetic makeup of a population over time. To study this, scientists use population genetics and a special mathematical "baseline" called the Hardy-Weinberg Equilibrium.

If you aren't a "math person," don't worry! We are going to break these formulas down into simple steps. Think of Hardy-Weinberg as a "null hypothesis"—it describes a population that is not evolving. By seeing how a real population differs from this model, we can prove that evolution is happening.


Topic 7.4: Population Genetics

A population is a group of individuals of the same species that live in the same area and interbreed. The gene pool consists of all the copies of every type of allele at every locus in all members of the population.

Key Concept: Evolution is occurring if the frequencies of alleles in a gene pool change from one generation to the next. If the frequencies stay the same, the population is in genetic equilibrium.

Did you know? Even if an individual has "better" genes, the individual doesn't evolve. Only populations evolve over time as those better genes become more common in the group.


Topic 7.5: Hardy-Weinberg Equilibrium

The Hardy-Weinberg (H-W) Equilibrium is a model used to describe populations that are NOT evolving. For a population to stay in this state of "genetic stasis," five very specific conditions must be met.

The 5 Conditions for Hardy-Weinberg Equilibrium

If any of these conditions are not met, the allele frequencies will change, and the population will evolve:

  1. No Mutations: No new alleles can be added to the gene pool.
  2. Random Mating: Individuals do not choose mates based on specific genotypes.
  3. No Natural Selection: All individuals have an equal chance of survival and reproduction.
  4. Extremely Large Population Size: This prevents genetic drift (random changes in small groups).
  5. No Gene Flow: No migration of individuals (and their genes) in or out of the population.

Memory Trick: Use the mnemonic "Many Rabbits Never Lose Grass" to remember the conditions: Mutations (none), Random mating, Natural selection (none), Large population, Gene flow (none).


The Hardy-Weinberg Equations

To calculate whether a population is evolving, we use two main formulas. You will find these on your AP Biology Equations and Formulas sheet!

1. The Allele Frequency Formula:

\(p + q = 1\)

  • \(p\) = Frequency of the dominant allele (e.g., \(A\))
  • \(q\) = Frequency of the recessive allele (e.g., \(a\))

2. The Genotype Frequency Formula:

\(p^2 + 2pq + q^2 = 1\)

  • \(p^2\) = Frequency of the homozygous dominant genotype (\(AA\))
  • \(2pq\) = Frequency of the heterozygous genotype (\(Aa\))
  • \(q^2\) = Frequency of the homozygous recessive genotype (\(aa\))

Step-by-Step: Solving H-W Problems

Most AP Biology problems will give you the number of individuals with a certain trait. Follow these steps to find the rest:

Step 1: Find \(q^2\). This is the easiest place to start because the recessive phenotype (the "look" of the organism) always reveals its genotype (\(aa\)). Divide the number of recessive individuals by the total population.

Step 2: Find \(q\). Take the square root of \(q^2\). Now you have the frequency of the recessive allele.

Step 3: Find \(p\). Since \(p + q = 1\), simply subtract \(q\) from \(1\) (\(p = 1 - q\)).

Step 4: Find the other genotypes. Square \(p\) to get \(p^2\). Multiply \(2 \cdot p \cdot q\) to get \(2pq\).

Quick Tip: Always check your work by making sure \(p^2 + 2pq + q^2\) adds up to \(1\) (or 100%).


Common Pitfalls to Avoid

  • Confusing "Allele" vs. "Genotype": If the question asks for the allele frequency, they want \(p\) or \(q\). If they ask for the genotype frequency or percentage of the population, they want \(p^2\), \(2pq\), or \(q^2\).
  • Starting with \(p\): Never assume the dominant phenotype frequency is \(p^2\). Why? Because the dominant phenotype includes both \(p^2\) (homozygous) and \(2pq\) (heterozygous) individuals. Always start with \(q^2\)!

Example Walkthrough

Scenario: In a field of 100 flowers, 16 are white (recessive, \(aa\)) and 84 are red (dominant, \(AA\) or \(Aa\)).

  1. Find \(q^2\): \(16 / 100 = 0.16\)
  2. Find \(q\): \(\sqrt{0.16} = 0.4\)
  3. Find \(p\): \(1 - 0.4 = 0.6\)
  4. Find \(p^2\) (Homozygous Dominant): \(0.6^2 = 0.36\) (or 36%)
  5. Find \(2pq\) (Heterozygotes): \(2 \cdot 0.6 \cdot 0.4 = 0.48\) (or 48%)

Check: \(0.36 + 0.48 + 0.16 = 1.0\). Perfect!


Summary Key Takeaways

- Population genetics tracks the health and evolution of a group by looking at its gene pool.

- Hardy-Weinberg Equilibrium describes a non-evolving population where allele frequencies stay constant.

- The 5 conditions (Large population, no migration, no mutation, random mating, no selection) are rarely met in nature, which is why evolution is so common.

- The equations \(p + q = 1\) and \(p^2 + 2pq + q^2 = 1\) allow us to quantify changes and predict genotypes.

Next Chapter Preview: We will see how disruptions to this equilibrium—like natural selection and evidence from the fossil record—provide proof of the Evidence of Evolution!