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

Cambridge International AS & A Level Biology (9700) – Selection, Evolution and Population Genetics

1. Syllabus coverage at a glance

Syllabus codeTopic (AS 1‑11, A 12‑19)Key ideas covered in these notes
AS 1‑11Cell structure, biomolecules, enzymes, membranes, transport, cell cycle, nucleic acids, plant & mammal transport, gas exchange, infectious disease, immunityNot covered – these notes focus on Topic 17 (Selection, Evolution and Population Genetics). See separate modules for the other units.
A 12‑19Energy & respiration, photosynthesis, homeostasis, control & coordination, inheritance, classification, genetic technology, selection & evolutionInheritance, classification and genetic technology are referenced only where needed for population genetics. The main emphasis is on Topic 17 and the quantitative genetics component (Hardy–Weinberg).

2. Selection – core concepts and mechanisms

2.1 Core concepts (AO1 – required by the syllabus)

  • Variation – genetic (different alleles) and phenotypic (different traits) variation exists within every natural population.

  • Fitness (w) – the ability of an individual to survive and produce viable offspring; measured as relative fitness.

  • Differential reproductive success – individuals with higher fitness leave more offspring, causing allele‑frequency change over generations.

2.2 Mechanisms of natural selection (AO2)

MechanismPattern of phenotypic changeTypical Cambridge example
Directional selectionOne extreme phenotype is favoured; the distribution shifts toward that extreme.Increase in beak size of Galápagos finches during a drought (past paper 9700/20/03).
Stabilising selectionIntermediate phenotypes are favoured; extremes are selected against.Human birth weight – very low or very high weights have reduced survival.
Disruptive (diversifying) selectionBoth extremes are favoured; the intermediate phenotype has lower fitness.Colour morphs of the African seedcracker exploiting different seed sizes.
Frequency‑dependent selectionFitness of a phenotype depends on its frequency in the population.Predator–prey “search image” – rare prey morphs have higher survival.
Sexual selectionTraits that increase mating success (often without improving survival) become more common.Elaborate peacock tail; larger antlers in red‑deer.

2.3 Artificial selection (AO1)

  • Humans impose a selective pressure by choosing breeding stock.

  • Strong, usually directional, pressure can cause rapid changes in allele frequencies.

  • Examples: pedigree dogs, high‑yield wheat varieties, laboratory Drosophila experiments.

3. Speciation and evolutionary trees (AO1–AO2)

  • Reproductive isolation – mechanisms that prevent gene flow between diverging populations.

    • Pre‑zygotic barriers: habitat, temporal, behavioural, mechanical, gametic.

    • Post‑zygotic barriers: hybrid inviability, hybrid sterility, hybrid breakdown.

  • Modes of speciation

    • Allopatric – geographic separation (e.g., island colonisation).

    • Sympatric – speciation without physical separation (e.g., polyploidy in plants, niche differentiation).

  • Phylogenetic trees (cladograms)

    • Nodes represent common ancestors; branches represent lineages.

    • Key terminology: clade, monophyletic, paraphyletic, polyphyletic.

    • Students must be able to interpret a tree (identify sister groups, infer relative timing of divergence).

4. Population‑genetics fundamentals

4.1 Allele and genotype frequencies

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

  • p = frequency of allele A

  • q = frequency of allele a

  • p + q = 1

Genotype frequencies in a non‑evolving (equilibrium) population are:

\[

\begin{aligned}

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

\text{Aa} &= 2pq \\

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

\end{aligned}

\]

4.2 The Hardy–Weinberg equations (AO1)

Allele‑frequency equation:

\[

p + q = 1

\]

Genotype‑frequency equation (the classic HW expression):

\[

p^{2} + 2pq + q^{2} = 1

\]

5. Hardy–Weinberg equilibrium – assumptions and evolutionary forces

Assumption (equilibrium condition)What it preventsEvolutionary force that would act if the assumption is violated
Very large (effectively infinite) population sizeRandom sampling error (genetic drift)Genetic drift – especially strong in small populations.
No mutationCreation of new alleles or loss of existing onesMutation – introduces novel alleles or converts one allele to another.
No migration (gene flow)Introduction or removal of alleles from other populationsGene flow – homogenises allele frequencies between populations.
Random matingPreferential mating based on genotype or phenotypeNon‑random mating – assortative or disassortative mating changes genotype frequencies (but not allele frequencies directly).
No natural or artificial selectionDifferential reproductive success of genotypesSelection – directional, stabilising, disruptive, frequency‑dependent or sexual.

6. Calculating allele & genotype frequencies – worked example (AO2)

Problem: In a population of 200 beetles, 72 are homozygous dominant (AA), 96 are heterozygous (Aa) and 32 are homozygous recessive (aa). Determine the allele frequencies (p and q) and the expected genotype frequencies under Hardy–Weinberg equilibrium.

GenotypeNumber observed (N)Frequency observed (O)
AA7272 / 200 = 0.36
Aa9696 / 200 = 0.48
aa3232 / 200 = 0.16

Allele frequencies (using \(p = \frac{2N{AA}+N{Aa}}{2N}\) and \(q = 1-p\)):

\[

p = \frac{2(72) + 96}{2(200)} = \frac{240}{400} = 0.60,\qquad q = 1 - 0.60 = 0.40

\]

Expected genotype frequencies (Hardy–Weinberg):

\[

\begin{aligned}

\text{AA (}p^{2}\text{)} &= (0.60)^{2} = 0.36 \\

\text{Aa (}2pq\text{)} &= 2(0.60)(0.40) = 0.48 \\

\text{aa (}q^{2}\text{)} &= (0.40)^{2} = 0.16

\end{aligned}

\]

Observed frequencies match the expected values, indicating the population is in Hardy–Weinberg equilibrium provided the five assumptions are satisfied.

7. Fitness, selection coefficients and allele‑frequency change (AO2)

  • Absolute fitness (W) – average number of offspring produced by a genotype.

  • Relative fitness (w) – \(wi = \dfrac{Wi}{W{\max}}\) where \(W{\max}\) is the highest absolute fitness in the population.

  • Selection coefficient (s) – reduction in fitness relative to the most fit genotype: \(si = 1 - wi\).

Recursion formula for allele frequency change (derived from fitness values):

\[

p' = \frac{p^{2}w{AA} + pq\,w{Aa}}{\bar w},\qquad

q' = 1 - p'

\]

where \(\displaystyle \bar w = p^{2}w{AA} + 2pq\,w{Aa} + q^{2}w_{aa}\) is the mean fitness of the population.

Example (single generation):

GenotypeNumber (N)Absolute fitness \(W\)
AA401.00
Aa500.90
aa100.60

First calculate allele frequencies from the observed numbers:

\[

p = \frac{2(40)+50}{2(100)} = 0.65,\qquad q = 0.35

\]

Relative fitnesses:

\[

w{AA}=1.00,\; w{Aa}=0.90,\; w_{aa}=0.60

\]

Mean fitness:

\[

\bar w = (0.65)^{2}(1.00) + 2(0.65)(0.35)(0.90) + (0.35)^{2}(0.60) = 0.8625

\]

New allele frequency of A after one generation:

\[

p' = \frac{(0.65)^{2}(1.00) + (0.65)(0.35)(0.90)}{0.8625}=0.68

\]

Thus the advantageous AA genotype raises the frequency of allele A from 0.65 to 0.68 in one generation.

8. Other evolutionary forces – quantitative treatment (AO2)

ForceKey equation(s)Typical effect on allele frequencies
Genetic drift\(\displaystyle \Delta p \approx \pm\sqrt{\frac{p q}{2N_e}}\) (standard deviation of change per generation)Random fluctuations; strongest when effective population size \(N_e\) is small.
Mutation\(\displaystyle p' = p(1-\mu) + q\,\nu\) where \(\mu\) = A→a mutation rate, \(\nu\) = a→A mutation rateCreates new alleles or converts existing ones; at equilibrium mutation‑selection balance: \(q \approx \sqrt{\mu/s}\) for recessive deleterious alleles.
Migration (gene flow)\(\displaystyle p' = (1-m)p + m pm\) where \(m\) = proportion of migrants, \(pm\) = allele frequency in migrantsMoves allele frequencies toward those of the source population; can counteract drift or selection.
Non‑random matingAssortative: excess of homozygotes; Disassortative: excess of heterozygotes.
Change in genotype frequencies: \(f{AA}' = p^{2}+F p q\), \(f{Aa}' = 2pq(1-F)\) where \(F\) is the inbreeding coefficient.		Alters genotype frequencies (inbreeding coefficient \(F\)) but does not change allele frequencies directly.
Selection (already covered)Recursion formula in section 7.Predictable directional change in allele frequencies.

9. Testing for Hardy–Weinberg equilibrium – χ² test (AO2)

  1. Calculate observed genotype numbers (O) and total population size (N).

  2. Derive allele frequencies (p, q) from the observed data.

  3. Compute expected numbers (E) using \(E{AA}=p^{2}N\), \(E{Aa}=2pqN\), \(E_{aa}=q^{2}N\).

  4. Apply the chi‑square formula:

    \[

    \chi^{2}= \sum \frac{(O-E)^{2}}{E}

    \]

  5. Degrees of freedom = (number of genotype classes) – (number of alleles) = 3 – 2 = 1.

  6. Compare the calculated χ² with the critical value 3.84 (5 % significance).

    • χ² ≤ 3.84 → fail to reject the null hypothesis → population could be in HW equilibrium.

    • χ² > 3.84 → reject the null hypothesis → the population is evolving.

Sample AO2 activity (Cambridge style)

  1. Data: In a population of 500 flowers, 180 are red (RR), 260 are pink (Rr) and 60 are white (rr).

  2. Students must:

    • Calculate p and q from the observed genotypes.

    • Find expected numbers of each genotype.

    • Perform the χ² calculation.

    • Interpret the result (e.g., suggest possible directional selection favouring pink flowers if χ² > 3.84).

10. Linking selection to Hardy–Weinberg expectations (AO2)

  • Heterozygote advantage (balancing selection) – observed heterozygote frequency > 2pq. Example: sickle‑cell allele (HbS) in malaria‑endemic regions.

  • Directional selection – one homozygote is favoured; observed frequency of the favoured homozygote > p² (or q²) and the unfavoured homozygote is deficient.

  • Stabilising selection – intermediate phenotype favoured; often produces an excess of heterozygotes when the trait is additive.

  • Disruptive selection – extremes favoured; leads to a deficit of heterozygotes relative to 2pq.

  • Students should be able to explain how a χ² deviation from HW expectations provides evidence for any of the above selective regimes.

11. Summary (AO1)

  • Selection (natural or artificial) changes allele frequencies by altering the relative fitness of genotypes.

  • The Hardy–Weinberg principle gives a null model for a non‑evolving population; deviations indicate that at least one of the five assumptions is violated.

  • Allele frequencies (p, q) and genotype frequencies (p², 2pq, q²) can be calculated from observed data; the recursion formula shows how selection modifies p over generations.

  • Each of the five HW assumptions corresponds to a specific evolutionary force (drift, mutation, migration, non‑random mating, selection).

  • Quantitative tools – fitness calculations, selection coefficients, mutation‑selection balance, gene‑flow equations, and χ² testing – are essential AO2 skills for Cambridge examinations.