[Leejohnston]

TITLE:
Population Genetics — How Evolution Becomes Mathematics

Evolution is often described as a story: organisms adapt, species change, and survival favours some traits over others.
But beneath the story is something far more precise.

Evolution is measurable.
Population genetics is the mathematics that makes it measurable.

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1. What population genetics actually tracks
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At its core, population genetics tracks allele frequencies.

Let:
p = frequency of allele A
q = frequency of allele a

Because these are the only two possibilities:

p + q = 1

This simple equation already reveals something deep:

Evolution does not create probabilities.
It redistributes them.

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2. The baseline: Hardy–Weinberg equilibrium
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If nothing acts on a population — no selection, no mutation, no migration, no drift — allele frequencies remain constant.

Genotype frequencies become:

AA = p²
Aa = 2pq
aa = q²

This state is not evolution.
It is mathematical equilibrium.

So the real question becomes:
What breaks this balance?

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3. Selection turns biology into motion
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Selection means different genotypes reproduce at different rates.

Let:
w₁₁ = fitness of AA
w₁₂ = fitness of Aa
w₂₂ = fitness of aa

The mean fitness of the population is:

w̄ = p²w₁₁ + 2pqw₁₂ + q²w₂₂

Evolution now becomes directional.

The change in allele frequency follows the direction of increasing fitness:

dp/dt ∝ ∇w̄

In words:
Allele frequencies move in the direction that increases average fitness.

This is natural selection written as mathematics.

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4. Mutation: slow but relentless
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Mutation introduces new alleles into the population.

Let:
μ = mutation rate from A to a
ν = mutation rate from a to A

Then allele frequency changes as:

Δp = −μp + νq

Mutation does not optimise.
It injects randomness — raw material for selection to shape.

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5. Genetic drift: randomness has weight
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In finite populations, chance alone can change allele frequencies.

The strength of drift depends on the effective population size:

Nₑ

The variance in allele frequency per generation is approximately:

Var(p) ≈ p(1 − p) / (2Nₑ)

This leads to an uncomfortable truth:

In small populations, randomness can overpower selection.

Evolution is not always optimisation.
Sometimes it is statistical accident.

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6. Fixation and loss are absorbing states
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Over time, alleles reach one of two endpoints:

p = 1 (fixation)
p = 0 (loss)

Once reached, evolution stops unless mutation or migration reintroduces variation.

Mathematically, these are absorbing boundaries.
Biologically, they are evolutionary dead ends.

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7. Why this matters beyond biology
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Population genetics equations appear far beyond biology:

• Epidemiology — pathogen strains
• Ecology — trait distributions
• Cultural evolution — ideas spreading
• Machine learning — genetic algorithms
• Risk and finance — probability flow

They describe a universal process:
How probabilities move through constrained systems under pressure and noise.

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8. The uncomfortable conclusion
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Evolution is not simply “survival of the fittest”.

It is:
• Gradient ascent on a fitness landscape
• With randomness
• Under constraints
• In finite systems

In plain terms:

Life evolves the same way complex systems everywhere evolve —
probabilistically, imperfectly, and mathematically.