Evolution – Advanced Study Notes
Level: MIT‑style graduate course in Biology
Course: GNM Admission – Biology
Topic: Evolution
These notes are intended for students who already have a solid grounding in genetics, molecular biology, and basic evolutionary theory. They present a first‑principles view of evolution, interweaving rigorous mathematical reasoning with biological intuition, and highlight subtle edge cases that often appear only in high‑level research.
1. Evolution as a Population‑Genetics Problem
| Concept | First Principles | Key Equation |
|---|---|---|
| Evolution | Change in the distribution of genotypes/alleles in a population over time. | ( \Delta p = p' - p ) |
| Population | A statistical ensemble of individuals; allele frequencies are sample moments. | ( p = \frac{n_A}{2N} ) |
| Deterministic vs Stochastic | Deterministic forces (selection, mutation, migration) act continuously; stochastic forces (genetic drift) arise from finite sampling. | ( \text{Var}(p) = \frac{p(1-p)}{2N} ) |
Key Insight: Evolution is a probabilistic process governed by both deterministic selection gradients and random sampling noise.
2. Hardy–Weinberg Foundations
2.1. Equilibrium Conditions
Given a bi‑allelic locus ( A/a ), assuming random mating, no mutation, migration, selection, or drift, genotype frequencies satisfy:
[ \begin{aligned} f_{AA} &= p^2, \ f_{Aa} &= 2pq, \ f_{aa} &= q^2, \end{aligned} ] where ( p = \frac{n_A}{2N} ), ( q = 1-p ).
2.2. Deviations: The Causes
- Non‑random mating → Hardy–Weinberg disequilibrium (heterozygote excess/deficiency).
- Selection → changes in genotype frequencies.
- Mutation → influx of new alleles.
- Genetic drift → sampling variance in finite populations.
- Migration → gene flow between subpopulations.
3. Natural Selection
3.1. Fitness Landscape
Define absolute fitnesses ( w_{AA}, w_{Aa}, w_{aa} ). Relative fitnesses ( \tilde{w} ) are normalized by mean fitness ( \bar{w} ).
[ \bar{w} = p^2 w_{AA} + 2pq w_{Aa} + q^2 w_{aa} ]
3.2. Selection Coefficients
| Genotype | ( w ) | ( s ) | Interpretation |
|---|---|---|---|
| ( AA ) | ( 1 ) | – | baseline |
| ( Aa ) | ( 1 + hs ) | ( h ) | dominance coefficient |
| ( aa ) | ( 1 + s ) | ( s ) | selection coefficient (positive for advantage, negative for disadvantage) |
Allele Frequency Change Under Selection
[ \Delta p = \frac{p q [ (w_{AA} - w_{Aa})p + (w_{Aa} - w_{aa})q ] }{ \bar{w} } ]
For small ( s ) and ( h ), linear approximation:
[ \Delta p \approx spq [ h + (1-2h)p ] ]
3.3. Fisher’s Fundamental Theorem of Natural Selection
Statement: The rate of increase in mean fitness of a population equals the additive genetic variance in fitness.
[ \frac{d\bar{w}}{dt} = V_A ]
where ( V_A ) is the additive genetic variance. This theorem links selection to variance in a population.
3.4. Edge Cases
- Overdominance (( w_{Aa} > w_{AA}, w_{aa} )) → stable polymorphism.
- Underdominance (( w_{Aa} < w_{AA}, w_{aa} )) → disruptive selection, potential for speciation.
- Epistasis → interaction between loci alters fitness landscape.
4. Genetic Drift and the Coalescent
4.1. Wright–Fisher Model
Under random mating in a diploid population of size