Modeling Selection at a Single Locus
Why does a lethal recessive allele never disappear from a population? It kills homozygotes in infancy, generation after generation. Yet it persists forever. The answer cannot be found by staring at a single family. Nor can it be found in a single death. It requires tracking an allele's fate across an entire population. This must be done over many generations. That requires arithmetic, not intuition. The one-locus, two-allele selection model is the mathematical engine for this task. It lets evolutionary biologists convert a biological premise into a prediction. The prediction is precise and testable. The premise: genotype A survives or reproduces better than genotype B. The prediction concerns how allele frequencies shift from one generation to the next. This model matters because it is the bridge between two ideas. Natural Selection and the Concept of Fitness is a qualitative idea. It concerns differential survival and reproduction. The Hardy-Weinberg Equilibrium as a Null Model is the frequency bookkeeping system. It tells us what happens when nothing is pushing allele frequencies around. Selection is what happens when Hardy-Weinberg's "no selection" assumption is violated on purpose. This lets us watch what changes. The big picture: a single parameter, relative fitness, multiplies genotype frequencies each generation. The resulting recursion equation predicts how fast an allele rises or falls. It also predicts whether the allele goes to fixation. Finally, it predicts whether the allele can persist forever at an intermediate frequency. This single framework explains very different phenomena. These include the spread of insecticide resistance. They include the persistence of sickle-cell allele in malarial regions. They also include the slow, stubborn presence of lethal recessive genetic diseases in every human population. This article builds the model term by term. It derives the standard recursion equation. It walks through the classic empirical demonstrations of selection acting in real populations. It also works two numerical examples to full quantitative resolution. One involves asymmetric fitness costs. The other involves a clinically important disease allele.
- Selection at one locus is modeled by assigning relative fitness (w) to each genotype and recomputing genotype and allele frequencies each generation.
- The core recursion, Δq = pq[q(w22−w12) + p(w12−w11)] / w̄, predicts the direction and speed of allele-frequency change.
- Selection against a recessive allele slows dramatically as the allele becomes rare, because rare recessive alleles hide in heterozygotes.
- Heterozygote advantage (overdominance) produces a stable polymorphism at an equilibrium frequency, rather than fixation of either allele.
- Kettlewell's peppered moth experiments and the sickle-cell/malaria correlation are the classic empirical anchors for this model.
- The same recursion, with different fitness values plugged in, generates directional, balancing, and purifying selection — the outcomes covered in later chapters.
Selection changes allele frequency by exactly the amount that genotypes differ in relative fitness, weighted by how common each genotype currently is — nothing more, nothing less.
The core idea
The one-locus, two-allele selection model asks a narrow but powerful question: given a locus with alleles A₁ and A₂, at frequencies p and q (p + q = 1), and given that the three genotypes A₁A₁, A₁A₂, A₂A₂ have different relative fitnesses w₁₁, w₁₂, w₂₂, how does q change from one generation to the next? Relative fitness is not survival probability or litter size in absolute terms; it is each genotype's contribution to the next generation rescaled so that the best-performing genotype gets w = 1. This rescaling is what lets fitness differences of any biological origin — viability, mating success, fecundity, longevity — be combined into a single multiplicative factor. The model assumes random mating (so genotype frequencies start out at Hardy–Weinberg proportions p², 2pq, q² before selection acts), discrete non-overlapping generations, and an infinite population (so genetic drift is ignored). Selection then acts as a differential-survival filter: each genotype's Hardy–Weinberg frequency is multiplied by its relative fitness, and the population is renormalized to sum to 1. The quantity that falls out of this renormalization, mean population fitness w̄ = p²w₁₁ + 2pq·w₁₂ + q²w₂₂, is the single number that summarizes how well-adapted the population is, on average, at this locus, this generation. Three qualitatively different outcomes emerge depending only on the ordering of w₁₁, w₁₂, w₂₂, and this is the conceptual payoff of the whole exercise: the same equation, with different fitness values plugged in, generates directional replacement of one allele by another, permanent balanced polymorphism, or the elimination of extreme phenotypes — the three regimes formally developed as Directional Selection and Stabilizing and Disruptive Selection in later treatments. Understanding this one model well is therefore the single most efficient investment a student can make in evolutionary genetics.
Mechanism
Step 1 — Start from Hardy–Weinberg proportions. Before selection, random mating guarantees genotype frequencies
| Genotype | A₁A₁ | A₁A₂ | A₂A₂ |
|---|---|---|---|
| Frequency | p² | 2pq | q² |
| Relative fitness | w₁₁ | w₁₂ | w₂₂ |
Step 2 — Apply fitness as a survival/reproduction filter. Each genotype's frequency is multiplied by its relative fitness. This produces frequencies after selection but before renormalization: p²w₁₁, 2pq·w₁₂, q²w₂₂. These no longer sum to 1 because fitness has removed some fraction of individuals differentially.
Step 3 — Renormalize by mean fitness. Divide each term by their sum, w̄ = p²w₁₁ + 2pq·w₁₂ + q²w₂₂, so the surviving/reproducing population again sums to 1:
Step 4 — Extract the new allele frequency. The frequency of A₂ in the next generation, q₁, equals the frequency of A₂A₂ individuals plus half the A₁A₂ individuals:
Step 5 — Convert to a change equation. Subtracting q from both sides and simplifying algebraically yields the standard result every population geneticist memorizes:
This equation is the entire model compressed into one line. Read it carefully: Δq is zero whenever p = 0 or q = 1 (fixation — no further change possible), and it is also zero whenever the bracketed term is zero, which happens either when all three fitnesses are equal (no selection) or, remarkably, at a specific intermediate q when w₁₂ exceeds both homozygote fitnesses (heterozygote advantage) or falls below both (heterozygote disadvantage). The sign of Δq — whether A₂ rises or falls — depends entirely on the sign of that bracket, not on the absolute magnitude of any single fitness.
Special case: selection against a recessive lethal. Set w₁₁ = w₁₂ = 1 and w₂₂ = 1 − s, where s is the selection coefficient (0 ≤ s ≤ 1, with s = 1 for a full recessive lethal). The recursion simplifies to
The crucial biological insight is in the q² term: because A₂ must be homozygous to be exposed to selection at all, and homozygotes become rare as q shrinks, the rate of elimination collapses as the allele becomes rare. A fully recessive lethal falls quickly from high frequency but is essentially immune to further removal once rare — heterozygotes shield it from selection's view. This is why recessive genetic diseases are never actually purged from populations by selection alone; mutation constantly resupplies the allele, and selection removes it only in the vanishingly rare homozygous state.
Special case: heterozygote advantage (overdominance). Set w₁₁ = 1 − s, w₁₂ = 1, w₂₂ = 1 − t, with s, t > 0 (both homozygotes worse than the heterozygote). Setting Δq = 0 and solving the bracket gives a stable internal equilibrium:
Unlike directional selection, this equilibrium is an attractor: populations above q̂ decline toward it, populations below q̂ rise toward it, so both alleles are permanently maintained — a balanced polymorphism. This is precisely the regime that explains the sickle-cell allele's persistence, worked numerically below.
The evidence
Kettlewell's peppered moth experiments (1953–1956). The rise of the dark ("melanic") form of the peppered moth Biston betularia in industrial England during the nineteenth and twentieth centuries had long been attributed to selection, but it was H. B. D. Kettlewell who supplied direct fitness data. Kettlewell released marked light and dark moths onto tree trunks in polluted woodland near Birmingham and unpolluted woodland in Dorset, then recaptured survivors and, separately, filmed birds preying on moths resting on trunks. In the sooty, lichen-free Birmingham woods, dark moths were recaptured at roughly twice the rate of light moths; in unpolluted Dorset woodland, the pattern reversed, with light moths surviving better. Direct observation confirmed birds (mainly great tits and other insectivorous species) preferentially took whichever morph contrasted more against the bark. This supplied real relative-fitness values (not merely a plausible story) that could be plugged directly into the recursion equation above, and the predicted allele-frequency shift matched the historical rise and fall of melanism as industrial soot accumulated and later declined following clean-air legislation. Later methodological critiques (Majerus and others) refined release protocols and predator behavior assumptions, but repeated, better-controlled trials in the 1990s–2000s reaffirmed differential bird predation as the dominant selective force, making this one of the best-documented cases of selection at a single, largely one-locus-behaving trait acting in the wild.
The sickle-cell allele and malaria (Allison, 1954 onward). Anthony Allison demonstrated that carriers of one sickle-cell allele (genotype HbA/HbS, heterozygotes) suffered substantially lower mortality from Plasmodium falciparum malaria than HbA/HbA homozygotes, while HbS/HbS homozygotes suffered severe, often fatal, sickle-cell anemia. Allison's field surveys across East Africa showed the sickle-cell allele frequency tracked malarial endemicity almost precisely: high in holoendemic malarial regions, low where malaria was absent, exactly as predicted by an overdominance model. Later case-control hospital studies in Kenya and Nigeria quantified this directly: heterozygotes showed roughly 60–90% protection against severe and cerebral malaria relative to homozygous normal individuals in several studies. This is a textbook empirical demonstration of heterozygote advantage generating a stable polymorphism — not a hypothetical illustration, but a locus where both homozygote fitnesses are independently measurable (death from malaria vs. death from anemia) and match the mathematical prediction of a stable intermediate q̂.
Endler's guppy predation experiments (1980s). John Endler transplanted Trinidadian guppies (Poecilia reticulata) between pools with differing predation intensity and tracked coloration-associated allele frequencies over multiple generations in both field and greenhouse mesocosms, directly measuring selection differentials predicted by the same class of model, confirming that predator-imposed selection coefficients of the magnitude used in textbook problems (s in the range of 0.1–0.5 per generation) are biologically realistic, not merely convenient algebra. Together these three bodies of evidence anchor the model at both ends: Kettlewell and Endler show selection coefficients estimated from direct survival/predation data reproduce observed allele-frequency change; Allison shows a genuine stable polymorphism, maintained exactly where the overdominance mathematics predicts an internal equilibrium.
Worked examples
Example 1 — Asymmetric selection against a recessive allele in an insect pest. A crop pest population has an insecticide-susceptibility locus. Susceptible homozygotes (ss) die from spraying at a high rate; heterozygotes (Rs) survive nearly as well as resistant homozygotes (RR) because resistance is largely, but not fully, recessive-masked. Suppose:
- w(RR) = 1
- w(Rs) = 0.95
- w(ss) = 0.20
- Current frequency of the susceptible allele s: q = 0.7 (R is rare — spraying has just begun)
Mean fitness: w̄ = p²(1) + 2pq(0.95) + q²(0.20), with p = 0.3, q = 0.7.
Apply the general recursion with w₁₁ = 1 (RR), w₁₂ = 0.95 (Rs), w₂₂ = 0.20 (ss):
Bracket: q(0.20 − 0.95) + p(0.95 − 1) = 0.7(−0.75) + 0.3(−0.05) = −0.525 − 0.015 = −0.540
So the susceptible allele frequency drops from q = 0.7 to q₁ ≈ 0.507 in a single generation — a large jump, because at this frequency the ss homozygote is still common (q² = 0.49) and exposed to strong selection. Repeating this recursion generation after generation shows the classic pattern: rapid initial decline in q, then dramatic slowing as q² shrinks and the s allele hides increasingly in resistant-appearing heterozygotes — precisely why insecticide-resistance management programs find early spraying highly effective but later "mopping up" of residual susceptibility alleles nearly impossible without additional tools (e.g., rotating chemical classes, since selection cannot see the recessive allele once homozygotes become rare).
Example 2 — Heterozygote advantage: predicting the equilibrium sickle-cell frequency. Suppose field and hospital mortality data from a holoendemic malarial region give the following relative fitnesses (normal allele A, sickle allele S):
- w(AA) = 1 − s, with s = 0.15 (moderate malaria mortality among non-carriers)
- w(AS) = 1 (heterozygotes protected from severe malaria, negligible sickling pathology)
- w(SS) = 1 − t, with t = 0.90 (severe sickle-cell anemia, high childhood mortality)
The equilibrium frequency of the sickle allele is:
This predicts roughly 14.3% allele frequency for HbS at equilibrium — translating to a heterozygote (carrier) frequency of 2p̂q̂ = 2(0.857)(0.143) ≈ 24.5% of the population, a figure squarely in the range reported for several high-malaria-burden regions of West and Central Africa (many populations there show sickle-cell trait carrier frequencies between roughly 10% and 30%, consistent with locally varying s and t). The model further predicts that if malaria were eliminated from this region (s → 0), the equilibrium collapses to q̂ = 0, and the sickle allele should decline toward loss over many generations, purely through the excess mortality of SS homozygotes with no compensating heterozygote advantage — a prediction consistent with the lower HbS frequencies observed in African-descended populations in non-malarial regions after multiple generations, modulated further by Genetic Drift and admixture history in smaller populations.
Common misconceptions & traps
| Misconception | Correction |
|---|---|
| "Selection acts on genotype frequencies directly, the same way each generation." | Selection acts on relative fitness differences; genotype frequencies are reset to Hardy–Weinberg proportions by random mating each generation before selection acts again. |
| "A recessive lethal allele should be selected out of a population within a few generations." | Because rare recessive alleles are hidden in heterozygotes, elimination slows sharply as q shrinks (Δq ∝ q²); complete removal by selection alone would take, in principle, infinite time. |
| "Heterozygote advantage means the heterozygote is always fittest in absolute terms." | Overdominance is a relative-fitness statement within one locus; it doesn't guarantee the heterozygote is optimal for the whole organism, only that it outperforms both homozygotes at this locus in this environment. |
| "Dominant alleles are automatically more common." | Dominance is about phenotypic expression in heterozygotes, not frequency; a dominant allele can be rare, and a recessive allele can be common (see The Hardy-Weinberg Equilibrium as a Null Model). |
| "Fitness values are fixed biological constants for a species." | Relative fitness is environment- and context-dependent (Kettlewell's moths reversed fitness ranking between polluted and unpolluted woodland); it is not an intrinsic property of a genotype alone. |
The scientist behind it
The mathematical formalism of one-locus selection was built jointly by the three founders of population genetics — R. A. Fisher, J. B. S. Haldane, and Sewall Wright — each deriving overlapping versions of the recursion between the 1920s and 1930s, unifying Natural Selection and the Concept of Fitness with Mendelian inheritance for the first time. J. B. S. Haldane in particular computed selection coefficients from real field data (notably estimating s for industrial melanism in moths decades before Kettlewell's direct experiments) and coined much of the standard notation still used today, including explicit generation-by-generation recursions for dominant, recessive, and intermediate selection.
Origins · 1 minJohn Burdon Sanderson Haldane and how this was discovered
Haldane was a physiologist-turned-geneticist whose 1924–1932 series "A Mathematical Theory of Natural Selection" derived selection recursions for one- and two-locus systems, estimated real-world selection coefficients from field data decades before molecular confirmation was possible, and helped found the Modern Synthesis linking Mendelian genetics to Darwinian selection.
Where this matters today
Antibiotic and pesticide resistance management. Public-health and agricultural resistance-management programs explicitly model resistance alleles with the recursion above, predicting how quickly resistance spreads under a given drug/pesticide regime and — crucially — how slowly it can be reversed once common, guiding rotation and refuge strategies designed to keep resistance-allele frequencies below dangerous thresholds.
Human disease genetics and carrier-frequency prediction. Genetic counseling programs use mutation-selection balance (an extension of the recessive-lethal case above, adding recurrent mutation as an input) to predict and explain stable carrier frequencies for diseases such as cystic fibrosis, Tay-Sachs, and sickle-cell disease, and to explain why prenatal screening cannot simply "select disease alleles away."
Conservation genetics. Managers modeling endangered populations use one-locus selection models combined with The Hardy-Weinberg Equilibrium as a Null Model and drift to judge whether an observed allele-frequency shift in a small population reflects adaptive selection, inbreeding depression, or pure demographic chance — a distinction that changes management recommendations entirely.
Cancer cell population dynamics. Tumor evolution is now routinely modeled with exactly this recursion, treating tumor subclones as "genotypes" with fitness differences driven by growth rate and drug sensitivity, predicting how quickly resistant subclones sweep through a tumor under chemotherapy — directly foreshadowing the directional-selection framework developed next.
Ask the tutor: How would this recursion change if the population also experiences recurrent mutation from A₁ to A₂, and what equilibrium does mutation-selection balance predict for a recessive lethal?
Why do we use "relative" fitness instead of absolute survival or offspring numbers?
Absolute fitness values differ enormously across species and environments and depend on arbitrary units (offspring per year, survival percentage, etc.), making comparisons unwieldy. Rescaling so the best genotype equals 1 strips away those units and isolates what actually matters for allele-frequency change: how genotypes perform relative to one another in the same environment. Only relative differences appear in the Δq equation; multiplying every genotype's fitness by the same constant leaves Δq unchanged. This is why population geneticists always report selection coefficients (s = 1 − w) rather than raw survival or fecundity numbers when discussing evolutionary dynamics.
Does this model assume discrete, non-overlapping generations, and does that matter biologically?
Yes — the standard recursion assumes generations don't overlap, matching organisms like annual insects or annual plants cleanly. For organisms with overlapping generations (most vertebrates, perennial plants), continuous-time differential equation versions of the same logic are used instead, but they yield qualitatively identical conclusions: direction of change depends on fitness differences, rate depends on genotype frequencies and how strongly they differ. The discrete model is taught first because its algebra is transparent and its predictions are directly testable across annual generations in real short-lived study organisms like Drosophila or agricultural pests.
Can this one-locus model be used for polygenic traits like human height?
Not directly. The model assumes selection acts differentially on three discrete genotype classes at a single locus with clear allele-specific fitness values. Traits like height are influenced by many loci of small effect plus environmental variance, so selection's effect on any single contributing allele is diluted and difficult to isolate experimentally. Quantitative genetics uses a different but related toolkit (breeder's equation, heritability) for such traits. The one-locus model remains essential, however, precisely because it isolates the cleanest possible case — a single Mendelian switch — making its logic the necessary foundation before tackling multi-locus complexity.
How is a selection coefficient (s) actually measured in a real population?
Researchers compare survival or reproductive success across known genotypes, often using mark-recapture (as Kettlewell did), controlled common-garden or transplant experiments (as Endler did with guppies), or case-control mortality studies (as Allison did with malaria and sickle-cell genotype). The measured survival/fecundity ratio between genotypes, rescaled so the best genotype equals 1, directly gives w for each genotype, and s = 1 − w for the disadvantaged genotype. Modern studies increasingly combine genomic time-series data (allele frequencies sampled across known generations or even across archaeological/ancient-DNA time points) to estimate s statistically from the trajectory of frequency change itself.
Does heterozygote advantage always lead to a stable polymorphism, or can it be lost?
Given the assumptions of the model — infinite population, constant fitness values, random mating — yes, the internal equilibrium q̂ = s/(s+t) is stable: perturbations in either direction are pushed back toward it. In real finite populations, however, genetic drift can carry an allele to loss or fixation even under heterozygote advantage, especially when the population is small or the fitness advantage is weak, illustrating why real-world overdominant polymorphisms (like sickle-cell) require both a strong, spatially consistent selective pressure (malaria) and reasonably large population sizes to remain stable over evolutionary time.
What happens to Δq if all three genotypes have identical fitness?
If w11 = w12 = w22, the bracket term in the Δq equation becomes zero for every value of p and q, so Δq = 0 at every generation regardless of starting allele frequency. This recovers exactly the Hardy–Weinberg null expectation: with no selection (and no other evolutionary force), allele frequencies remain constant generation after generation. This is precisely why the selection model is best understood as a deliberate, parameterized departure from the Hardy–Weinberg baseline rather than a wholly separate framework.
Why does industrial melanism make a good textbook example if later studies criticized Kettlewell's methods?
Later scrutiny (particularly by Michael Majerus) identified genuine methodological weaknesses in Kettlewell's original protocols, such as unnaturally high moth densities and non-natural resting positions on tree trunks. Rather than overturning the conclusion, subsequent, more rigorously controlled experiments — including Majerus's own multi-year replication published posthumously in 2012 — reproduced the core finding: differential bird predation by morph and habitat, matching the direction and rough magnitude of the original result. This makes peppered moths a valuable teaching case not despite the controversy but because of it, illustrating how science self-corrects methodology while confirming the underlying causal claim.