Factors That Change Allele Frequencies
Why do cheetahs carry almost no genetic variation? Why do human populations remain enormously diverse? Why does sickle-cell allele frequency track malaria maps almost exactly? These are not separate mysteries. They are variations on one theme. The theme is this: forces push allele frequencies away from the static equilibrium. That equilibrium is described by The Hardy-Weinberg Principle. Hardy-Weinberg equilibrium is a null model. It tells us what happens when nothing happens. Real populations are never so quiet. Mutation introduces new alleles. Migration mixes gene pools. Selection filters alleles by their effect on survival and reproduction. Genetic drift randomly reshuffles frequencies from one generation to the next. This happens simply because populations are finite. This chapter is where population genetics stops being bookkeeping. It starts being evolutionary biology. Each of the four forces violates one Hardy-Weinberg assumption. Those assumptions are: no mutation, no migration, infinite population size, random mating with respect to the locus, and no selection. Each violation has a predictable mathematical signature. Learning to recognize those signatures is the practical skill this material builds. So is distinguishing them from each other in real data. The stakes are not abstract. Antibiotic resistance evolves because selection amplifies rare resistant alleles within days. Endangered species lose fitness because small populations drift toward fixation of harmful variants. Human population history can be read from these forces' signatures. This includes the routes of ancient migrations, bottlenecks, and admixture. It is read directly out of the allele-frequency fingerprints these forces leave behind. Conservation genetics, agricultural breeding, and epidemiology all run on the same four equations introduced here. This article works through each force's mechanism with explicit algebra. It then turns to the classical experiments and observational studies. These studies proved these forces act in real populations. Examples include Kettlewell's peppered moths, Endler's guppies, and the Grants' Galápagos finches. Also included are the founder-effect and bottleneck studies. These studies established drift as a real evolutionary agent, not just a mathematical curiosity.
- Hardy-Weinberg equilibrium requires no mutation, no migration, infinite population size, random mating, and no selection; violating any one changes allele frequencies.
- Mutation creates new alleles at rates around 10⁻⁵–10⁻⁶ per gene per generation — slow alone, but the ultimate source of all variation.
- Migration (gene flow) moves allele frequencies of a population toward the average of the populations it exchanges migrants with.
- Selection changes allele frequencies deterministically based on relative fitness; its speed depends on dominance, selection coefficient, and starting frequency.
- Genetic drift is random sampling error in small populations; it fixes or loses alleles regardless of their fitness, and is strongest when population size N is small.
- Real allele-frequency change is usually a combination of these forces — mutation-selection balance, migration-selection balance, and drift-selection tension are the standard analytical framings.
Allele frequencies change whenever real populations violate any Hardy-Weinberg assumption, and each violating force — mutation, migration, selection, or drift — leaves a distinct, quantifiable signature on the gene pool.
The core idea
The Hardy-Weinberg principle states that in an infinitely large, randomly mating population free of mutation, migration, and selection, allele and genotype frequencies remain constant across generations, with genotypes settling into the familiar p², 2pq, q² proportions. This is a null hypothesis, deliberately built from unrealistic assumptions, precisely so that departures from it are diagnostic. When a population's genotype frequencies or allele frequencies shift over time, the cause must be traceable to a violation of one or more of those five conditions. Four of those violations constitute the primary evolutionary forces treated here: mutation (heritable changes in DNA sequence that create new alleles), migration or gene flow (movement of alleles between populations via dispersing individuals or gametes), natural selection (differential survival and reproduction among genotypes owing to their phenotypic effects), and genetic drift (random, sampling-based fluctuation in allele frequency arising because real populations are finite). The fifth violation, non-random mating, changes genotype frequencies (notably inbreeding, which increases homozygosity) without necessarily changing allele frequencies itself — that topic is developed separately as a natural extension of this one. A crucial conceptual distinction runs through all four forces: mutation and drift are essentially random with respect to fitness, while selection is not. Mutation supplies raw material without foresight — it does not "aim" at adaptive alleles. Drift shuffles existing variation blindly. Only selection systematically enriches alleles that improve reproductive success, which is why selection, and only selection, produces cumulative adaptation. Migration is directional in the sense that it pushes allele frequencies toward a specific value (the migrant source frequency), but it is agnostic about fitness. Keeping these distinctions sharp — random versus non-random, direction versus magnitude — is the analytical backbone of population genetics.
Mechanism
Mutation. At the molecular level, mutation arises from replication errors escaping DNA polymerase proofreading and mismatch repair, from spontaneous chemical damage (deamination, depurination), and from mobile element insertion, among other sources. Typical eukaryotic mutation rates per locus per generation, μ, are on the order of 10⁻⁵ to 10�sup>-6. Consider a one-way mutation model in which allele A mutates to a at rate μ per generation, with no back-mutation. If pₙ is the frequency of A in generation n: ``` p_(n+1) = p_n(1 − μ) ``` Because (1 − μ) < 1, p declines geometrically toward zero — mutation alone drives an allele slowly toward loss, but the rate is glacial: with μ = 10⁻⁵, it takes roughly 70,000 generations to halve p. Add back-mutation (a → A at rate ν) and the two-way system reaches a stable equilibrium where forward and backward flux balance: ``` q̂ = μ / (μ + ν) ``` Mutation's real evolutionary importance is not its speed but its role as the ultimate source of all allelic variation — without it, every other force would eventually exhaust the genetic variance available to it, since selection and drift only sort and eliminate existing alleles.
Migration (gene flow). Consider a mainland-island model: an island population with allele frequency qᵢ receives migrants at rate m per generation from a mainland with frequency q_m. After migration, the island's new frequency is: ``` q'_i = (1 − m)q_i + m·q_m = q_i − m(q_i − q_m) ``` The change per generation is Δq = −m(qᵢ − q_m): migration moves the island frequency toward the mainland frequency, with the rate of approach set by m. Even small migration rates are evolutionarily powerful — as a rule of thumb, if Nm (number of migrants per generation) exceeds about 1, migration can prevent populations from diverging by drift alone, a benchmark used constantly in conservation genetics to judge whether isolated subpopulations remain a single management unit.
Natural selection. Selection acts through differences in relative fitness (w) among genotypes — the average per-capita contribution of offspring to the next generation. Define fitnesses for genotypes AA, Aa, aa as w₁₁, w₁₂, w₂₂. Given genotype frequencies p², 2pq, q² before selection, the mean population fitness is: ``` w̄ = p²w₁₁ + 2pq·w₁₂ + q²w₂₂ ``` and the change in allele frequency of a in one generation is: ``` Δq = pq[q(w₂₂ − w₁₂) + p(w₁₂ − w₁₁)] / w̄ ``` This single equation contains all the classical selection regimes as special cases:
| Regime | Fitness pattern | Long-run outcome |
|---|---|---|
| Directional (a recessive deleterious) | w₁₁ = w₁₂ = 1, w₂₂ = 1 − s | a slowly purged, slower as q→0 |
| Directional against dominant | w₁₁ = 1−s, w₁₂ = 1−s, w₂₂ = 1 | rapid removal, since heterozygotes exposed |
| Heterozygote advantage (overdominance) | w₁₂ > w₁₁, w₂₂ | stable polymorphism at intermediate q̂ |
| Heterozygote disadvantage (underdominance) | w₁₂ < w₁₁, w₂₂ | unstable equilibrium; whichever allele starts more common fixes |
For heterozygote advantage, the equilibrium allele frequency is set by the selection coefficients against each homozygote, s₁ against AA and s₂ against aa: ``` q̂ = s₁ / (s₁ + s₂) ``` This is exactly the algebra behind sickle-cell balancing selection, worked quantitatively below. Selection against a fully recessive lethal (s = 1) follows q_(n+1) = qₙ/(1 + qₙ), which shows the classic feature that removing rare recessive alleles is very slow — most copies hide, protected, in heterozygotes.
Genetic drift. Drift is not a force with direction but a consequence of finite sampling. Each generation's gametes are a finite sample drawn from the parental gene pool, exactly analogous to drawing marbles from an urn. With population size N (2N alleles at a diploid autosomal locus), the sampling variance in allele frequency from one generation to the next is: ``` Var(Δq) = pq / 2N ``` Smaller N means larger variance — smaller populations drift faster and further, purely as a statistical consequence, with no reference to which allele is "better." Two long-run consequences follow directly. First, drift eventually fixes one allele and loses the other at every locus, with no new mutation; the probability that a given allele eventually fixes equals its current frequency (a neutral allele at frequency p has exactly probability p of eventual fixation — this identity underlies the neutral theory of molecular evolution). Second, heterozygosity decays geometrically: ``` H_(t) = H₀ (1 − 1/2N)^t ``` so small, isolated populations lose variation measurably within tens of generations, while large populations retain it almost indefinitely. Drift is strongest in bottlenecks (a sudden crash in N) and founder events (a new population started by a small, non-representative sample), both of which are documented mechanistically and empirically below.
The evidence
Selection in real time: Kettlewell's peppered moths. The peppered moth (Biston betularia) exists in a pale, speckled typica form and a dark carbonaria form, controlled largely by a dominant allele. Before industrialization, carbonaria was rare; by the mid-20th century in polluted regions of England, it exceeded 90% of some populations. H. B. D. Kettlewell (1953–1956) tested the hypothesis that soot-blackened tree trunks made dark moths cryptic and pale moths conspicuous to bird predators, reversing their relative fitness. He released marked moths of both morphs in a polluted woodland near Birmingham and in an unpolluted woodland in Dorset, then recaptured survivors and directly observed bird predation. In the polluted site, recapture rates favored carbonaria; in the unpolluted site, typica fared better, and Kettlewell filmed birds preferentially picking conspicuous moths off matching or mismatching bark. Later work (Bruce Grant and colleagues, 1990s–2000s) documented that as Britain's Clean Air Act reduced pollution after 1956, carbonaria frequency declined again, tracking lichen recovery on tree bark — direct, reversible evidence that selection coefficients change with environment and produce measurable allele-frequency shifts within a human lifetime.
Selection quantified across generations: the Grants and Darwin's finches. Peter and Rosemary Grant's four-decade study (from 1973) of medium ground finches (Geospiza fortis) on the tiny Galápagos island of Daphne Major measured beak depth, a heritable, quantifiable trait, in every bird on the island across many generations. During the severe 1977 drought, small soft seeds were depleted and only large hard seeds remained; birds with deeper, stronger beaks survived disproportionately. Mean beak depth increased by roughly 4% in a single generation — one of the fastest documented cases of natural selection in a wild vertebrate population, later reversed in wet years favoring small-beaked birds exploiting small seeds. The Grants' work is exceptional because it combined pedigree-based heritability estimates with direct fitness measurement, letting them compute a realized selection differential and confirm it against the predicted response to selection (R = h²S), closing the loop between genotype, phenotype, and fitness.
Genetic drift demonstrated experimentally: Buri's Drosophila populations. Peter Buri (1956) established the definitive laboratory proof that drift alone, absent selection, changes and eventually fixes allele frequencies. He set up 107 replicate populations of Drosophila melanogaster, each maintained at a constant, small breeding size of 16 individuals (32 alleles at the locus), all starting at exactly q = 0.5 for two neutral eye-color alleles (bw and bw75, both homozygotes and heterozygotes visually distinguishable and presumed fitness-neutral). Over 19 generations, allele frequencies in the separate lines fanned out into a widening distribution — some lines drifted to q = 1 (fixation of bw75), others to q = 0 (loss), with the variance across replicate lines increasing generation by generation almost exactly as predicted by Var(Δq) = pq/2N. By generation 19, about 30% of lines had fixed one allele or the other. This is the textbook empirical confirmation that finite population size alone generates allele-frequency divergence, matching the sampling-variance formula quantitatively, not just qualitatively.
Founder effects and bottlenecks in nature: cheetahs and Northern elephant seals. Stephen O'Brien and colleagues (1980s) found that cheetahs (Acinonyx jubatus) show extraordinarily low genetic diversity — skin grafts between unrelated cheetahs are accepted almost as if between identical twins, and reciprocal MHC and allozyme surveys found near-zero polymorphism. This is attributed to one or more severe population bottlenecks, likely near the end of the Pleistocene, that reduced effective population size drastically and let drift fix alleles genome-wide. Northern elephant seals, hunted to a population of perhaps fewer than 100 individuals in the 1890s before legal protection, show similarly depressed variation at surveyed loci compared to the less-bottlenecked southern elephant seal — a founder/bottleneck signature confirmed by comparing pre- and post-bottleneck museum specimens where available, directly linking a known demographic event to a measured loss of heterozygosity.
Migration measured directly: Endler's guppies. John Endler's work (1970s–80s) on Trinidadian guppies (Poecilia reticulata) showed that male color-pattern allele frequencies differ predictably between pools with different predator regimes, but that gene flow (via fish moving between connected pools) continually reintroduces alleles selection is removing, producing a measurable selection-migration balance — a direct field demonstration that migration's homogenizing pull can be pitted quantitatively against selection's differentiating pull, exactly as the mainland-island equation predicts.
Worked examples
Example 1 — Selection against a recessive lethal, and how slow "purging" really is. Suppose a recessive allele a is lethal in homozygotes (w₂₂ = 0), fully neutral in heterozygotes and dominant homozygotes (w₁₁ = w₁₂ = 1), and starts at q₀ = 0.5 in a large population. Using q_(n+1) = qₙ/(1 + qₙ): ``` q1 = 0.5/1.5 = 0.333 q2 = 0.333/1.333 = 0.250 q3 = 0.250/1.250 = 0.200 q5 ≈ 0.143 q10 ≈ 0.0833 q20 ≈ 0.0455 ``` Notice the pattern: to go from q = 0.5 to q = 0.33 takes one generation, but to go from q = 0.1 to q = 0.05 takes roughly ten. Selection against a fully recessive allele decelerates sharply as q falls, because at low q nearly all copies of a are sheltered in heterozygotes, invisible to selection. This is why recessive genetic diseases are never eliminated by selection alone even under lethal selection — mutation-selection balance (μ against s = 1) predicts an equilibrium q̂ = √μ, not zero. For μ = 10⁻⁵, q̂ ≈ 0.0032, matching the observed persistence of many recessive disease alleles in human populations at low but stubbornly nonzero frequency.
Example 2 — Heterozygote advantage: solving for the sickle-cell equilibrium. In malarial West Africa, approximate relative fitnesses are:
| Genotype | Phenotype | Relative fitness |
|---|---|---|
| AA | normal, malaria-susceptible | w₁₁ = 1 − s₁ ≈ 0.85–0.90 (malaria mortality) |
| AS | sickle-cell trait, malaria-resistant | w₁₂ = 1.0 |
| SS | sickle-cell disease | w₂₂ = 1 − s₂ ≈ 0.10–0.25 (severe anemia mortality) |
Take representative published estimates s₁ ≈ 0.15 (selection against AA via malaria) and s₂ ≈ 0.80 (selection against SS via sickle-cell disease, historically near-lethal before modern treatment). The equilibrium frequency of the S allele is: ``` q̂ = s₁ / (s₁ + s₂) = 0.15 / (0.15 + 0.80) = 0.15/0.95 ≈ 0.158 ``` This predicts roughly 16% allele frequency at equilibrium — reasonably close to the 10–20% observed across many malarial West African populations (with the exact value depending on local malaria intensity, which sets s₁). Two features of this result matter pedagogically: first, the equilibrium is stable — perturb q away from 0.158 in either direction and Δq pushes it back, unlike the unstable equilibrium of heterozygote disadvantage. Second, the equilibrium exists only because selection acts oppositely on the two homozygotes simultaneously; remove malaria (as antimalarial campaigns and drainage of mosquito breeding sites have done in parts of the world), and s₁ collapses toward zero, sending q̂ toward zero over subsequent generations — a testable, and in some populations observed, prediction of the model.
Common misconceptions & traps
| Misconception | Correction |
|---|---|
| "Mutation drives most evolutionary change we observe." | Mutation supplies variation but changes frequencies far too slowly (μ ~10⁻⁵) to explain rapid shifts; selection or drift usually does the actual moving. |
| "Genetic drift only matters in very small populations, like zoos." | Drift's variance (pq/2N) is always present; it's negligible only when N is very large relative to the timescale considered — many wild populations are small enough for drift to matter over decades. |
| "A population in Hardy-Weinberg equilibrium is not evolving in any way." | Allele frequencies can be temporarily stable, but HW equilibrium concerns genotype proportions given fixed allele frequencies; the assumptions can still be silently violated in balanced, offsetting ways. |
| "Selection always eliminates deleterious recessive alleles eventually." | Selection against rare recessives becomes vanishingly slow (Example 1); mutation continually reintroduces them, producing a nonzero mutation-selection equilibrium, not elimination. |
| "Migration is weak compared to selection, so it rarely matters." | Even Nm ≈ 1 migrant per generation can homogenize populations that selection is trying to differentiate — migration-selection balance is a major real constraint in conservation and pest-resistance management. |
| "Genetic drift acts with a direction, gradually 'purifying' a population." | Drift is directionless sampling noise; it fixes alleles by chance regardless of whether they help or harm the organism — direction is the signature of selection, not drift. |
The scientist behind it
Peter Buri's 1956 experiment is the cleanest empirical demonstration that genetic drift, unaided by selection, changes allele frequencies according to the sampling-variance mathematics population geneticists (notably Sewall Wright, whose theoretical work on drift and small population size predates and motivated Buri's design) had derived decades earlier. Buri worked at the University of Chicago studying Drosophila melanogaster population genetics, choosing the brown eye-color locus precisely because its two alleles were visually scoreable and believed selectively neutral or nearly so under laboratory conditions, letting drift's signature emerge uncontaminated by differential survival.
Origins · 1 minPeter Buri and how this was discovered
Where this matters today
Antibiotic and antiviral resistance evolution is textbook directional selection in real time: a resistance allele present at very low frequency (often originating by mutation or pre-existing standing variation) sweeps rapidly through a bacterial or viral population once a drug imposes a large selection coefficient against susceptible genotypes — the same Δq equation used for peppered moths predicts clinical resistance timelines, and understanding it underlies stewardship strategies (rotating drugs, combination therapy) designed to slow the sweep. Conservation genetics leans on genetic drift and migration mathematics daily: small, fragmented populations of endangered species (cheetahs, Florida panthers, island foxes) lose heterozygosity and can fix deleterious recessive alleles purely through drift, motivating genetic rescue programs that deliberately introduce migrants to restore heterozygosity and counteract inbreeding depression — an applied, real-time use of the migration equation to reverse drift's damage. Agricultural pest and weed management applies mutation-selection and migration-selection balance to design refuge strategies for genetically modified crops: planting non-resistant refuge crops near Bt-toxin-expressing fields slows the spread of resistance alleles in pest populations by keeping migration high enough to dilute resistant alleles through mating with susceptible refuge-population insects, a direct field application of the mainland-island model. Human population genetics reconstructs migration routes, bottlenecks, and founder events from genomic diversity patterns: reduced heterozygosity in populations descended from small founding groups (documented in island populations and some Indigenous American populations) reflects the same founder-effect mathematics seen in elephant seals, while allele-frequency clines across geography (e.g., lactase persistence, sickle-cell trait) are read as signatures of selection layered on top of migration history — the analytical toolkit is identical whether applied to moths, finches, flies, or humans, which is precisely why this framework, together with The Hardy-Weinberg Principle and Testing Hardy-Weinberg Equilibrium, forms the shared quantitative language of the field before extending into Inbreeding and the Inbreeding Coefficient.
Ask the tutor: How would you calculate the minimum number of migrants per generation needed to prevent two subpopulations from diverging under drift?
Why doesn't natural selection completely eliminate harmful recessive alleles from a population?
Because selection can only act on phenotypes, and a rare recessive allele is mostly hidden in heterozygotes, whose fitness is normal. As shown by the recursion q_(n+1) = qₙ/(1+qₙ), the rate of removal decelerates sharply as q falls, since fewer and fewer homozygotes are formed each generation. Simultaneously, mutation continually recreates the allele at some low rate μ. These two opposing processes settle into a mutation-selection balance at q̂ ≈ √μ (for a fully recessive lethal), a small but stubbornly nonzero equilibrium frequency — which is exactly why recessive genetic diseases persist in human populations indefinitely rather than disappearing.
How can you tell, from real data, whether an allele-frequency change was caused by selection or by drift?
The key diagnostic is replication and comparison across loci or populations. Selection produces consistent, repeatable, environment-correlated changes — independent populations facing the same selective pressure (like polluted versus unpolluted woodlands for peppered moths) shift in the same direction. Drift produces divergent, uncorrelated changes across replicate populations, as Buri's 107 fly lines demonstrated by fanning out unpredictably from the same starting frequency. Genome-wide, selection at a specific locus stands out against a genome-wide neutral background shaped by drift and demography, which is the logic behind modern selection-scan statistics that compare a candidate locus to the rest of the genome.
Why is genetic drift stronger in small populations than large ones?
Drift is sampling error: each generation's gene pool is a finite random sample of gametes from the previous generation, exactly like drawing marbles from an urn. The variance of that sampling process is Var(Δq) = pq/2N, which is inversely proportional to population size N. In a population of N = 10, that variance is large, so allele frequencies can swing wildly and even fix or vanish within a handful of generations. In a population of N = 10 million, the same formula predicts a vanishingly small variance, so allele frequencies barely budge from drift alone across realistic timescales — which is why conservation biologists worry intensely about small, fragmented populations.
What is the difference between gene flow and migration in the everyday sense?
In population genetics, "migration" specifically means the movement of alleles between populations via dispersing individuals or gametes that then successfully breed — it is defined by successful gene flow, not mere movement. An animal that travels to a new region but never reproduces there contributes zero migration in this technical sense. The mainland-island model (q'_i = q_i − m(q_i − q_m)) treats m as the fraction of the next generation's gene pool on the island that is contributed by mainland migrants, making migration mathematically equivalent to a partial replacement of the local gene pool each generation.
Does heterozygote advantage explain all cases of stable genetic polymorphism?
No — overdominance is one mechanism producing a stable polymorphic equilibrium, and sickle-cell trait versus malaria is the best-documented human example, but it is not the only explanation for observed variation. Frequency-dependent selection (where rare alleles are favored simply because they are rare, as in some predator-prey and self-incompatibility systems), spatially varying selection combined with migration, and simple mutation-selection-drift balance for effectively neutral variants can all maintain polymorphism without any heterozygote fitness advantage. Distinguishing these mechanisms in real populations usually requires direct fitness measurements of all three genotypes, which is technically demanding and available for relatively few natural systems.
Can mutation alone explain rapid evolutionary changes, like antibiotic resistance spreading through a bacterial population in days?
Not by itself. Mutation rates (roughly 10⁻⁸ to 10⁻⁹ per base pair per bacterial generation, though effectively higher per gene across large populations) are far too low to explain an allele sweeping to high frequency within days through new mutation and gradual increase alone. What actually happens is that a resistance mutation, often already present at very low frequency in a large bacterial population (a case of pre-existing standing variation, as demonstrated for random mutation processes by the Luria-Delbrück fluctuation test), experiences an enormous, sudden selection coefficient once the antibiotic is applied, and selection — not mutation — does the fast work of driving it to high frequency.
If a population satisfies all Hardy-Weinberg assumptions perfectly, does that mean it cannot evolve at all?
Correct, and that is precisely the point of the model: an ideal HW population, by construction, has none of the four forces acting and therefore has constant allele frequencies indefinitely — it is evolutionarily static by definition. Real populations depart from this ideal to varying degrees, and evolutionary biologists use departures from HW genotype proportions, or year-over-year shifts in measured allele frequencies, as the empirical signal that one or more of mutation, migration, selection, or drift is actively operating, which is exactly why the HW model is taught first as a deliberate null hypothesis rather than as a description of real populations.