9.1 Frequencies at one locus #
9.1.1 Allele and genotype frequencies #
For N diploids, p = (2 n_AA + n_Aa) / (2N) and q = 1 − p. Homozygotes contribute two copies.
Random mating repackages genotypes and does not, by itself, change allele frequencies. Selection, mutation, migration, and drift change allele frequencies. Inbreeding changes genotype frequencies only. A sentence about “the frequency” should say which one it means.
9.1.2 Hardy–Weinberg equilibrium #
Hardy and Weinberg (1908): with random union of gametes, genotypes become p², 2pq, and q² in one generation and stay there. Allele frequency comes back as p² + pq = p. The five assumptions are no selection, no mutation that moves frequencies, no migration, random mating, and a population large enough that sampling error is negligible. The simple formula is also autosomal, diploid, and sexual, with the same p in both sexes. X-linked genes need a different expression because males are hemizygous.
Hardy-Weinberg
One round of random mating restores the genotype ratios from any starting genotype frequencies, if p is unchanged and the other assumptions hold. Genotype frequencies are rebuilt; allele frequencies have the memory. A chi-square test that estimates p from three classes has one degree of freedom left. It detects a departure and does not name which assumption failed. Hardy–Weinberg is a null model, as 9:3:3:1 is.
9.1.3 Carrier frequency from q² #
For a rare autosomal recessive near equilibrium, incidence ≈ q² and carrier frequency ≈ 2q. Constructed numbers, not a country rate: incidence 1/10,000 gives q = 1/100 and 2pq ≈ 1/50. Carriers outnumber affected people by about 2/q, here about 200 to 1.
Almost every copy of a rare recessive allele is hidden in a heterozygote. That is why selection on affected people is slow, and why a family history can be empty where carriers are common. The figure is for one locus in one population. Consanguinity pushes homozygotes above q², so treating incidence as q² then overestimates q. The affected class must also be one allele, not a mixture of similar recessives.
9.2 Forces that change frequencies #
9.2.1 Selection #
Fitness w is relative reproductive success and s = 1 − w. Against a complete recessive, the fraction of a alleles sitting in homozygotes is only q. At q = 0.01, about 1% of copies are exposed. For a recessive lethal, q′ = q / (1 + q), so 0.01 moves only to about 0.0099. Balance with mutation is q̂ ≈ √(μ / s): even a lethal recessive lingers near the square root of the mutation rate.
A deleterious dominant is exposed in heterozygotes and is removed efficiently. One that prevents reproduction persists only as new mutation, with incidence on the order of 2μ, and equilibrium frequency about μ / s. Achondroplasia is toward this end. Huntington disease is dominant but often late, so s is smaller and the allele need not be vanishingly rare. Selection has no target in this arithmetic. Dominance and rarity set the speed.
9.2.2 Mutation, migration, and non-random mating #
Mutation from A to a at rate μ shifts q by about μp per generation if back-mutation is ignored. That is a small push beside strong selection, which is why the balances exist. Rates of about 10^−6 to 10^−5 per locus per generation are the Chapter 7 scale.
If migrants replace a fraction m of an island and carry frequency p_m, then Δp = m (p_m − p). One generation moves the island a fraction m of the way to the source. Continued migration homogenises, and a little gene flow can swamp weak local selection. The geography of that tension is Evolutionary Biology.
Inbreeding raises identity by descent and does not change allele frequency. With inbreeding coefficient F,
f(aa) = q²(1 − F) + qF, and f(Aa) = 2pq(1 − F).
The extra homozygotes expose recessives: inbreeding depression. For non-inbred parents, F is 1/2 after selfing, 1/4 for a child of full siblings, 1/8 for a child of an uncle–niece pair, and 1/16 for a child of first cousins. Assortative mating, pairing by phenotype, is not inbreeding. It homozygoses the loci under assortment, not the whole genome.
9.2.3 Drift, founders, and bottlenecks #
Drift is sampling error. In the simplest diploid model the variance of one generation’s change in p is p(1 − p)/(2N), largest near p = 1/2 and large when N is small. A neutral allele’s fixation probability is its current frequency, so a new mutant, at frequency 1/(2N), usually dies.
Effective size N_e is the ideal size that would drift as fast as the real population. Unequal sex ratio, uneven family size, and crashes make N_e smaller than the census. A bottleneck, or a founder event at the start of a new population, takes a sample. Later growth does not restore missing alleles. Commonness after a founder event is not evidence of advantage. Chapter 12 is that point in human history.
| Force | Allele frequency | Genotypes | Strongest when |
|---|---|---|---|
| Selection | Changes directionally | Change | The allele is expressed |
| Mutation | Slow change | Slight change | Selection is weak |
| Migration | Moves toward the source | Change | m or the difference is large |
| Inbreeding | Unchanged | More homozygotes | Relatives mate |
| Drift | Random walk | Less heterozygosity | N_e is small |
9.3 Keeping an allele in the population #
9.3.1 Sickle cell and malaria #
If the heterozygote is the fittest genotype, both alleles are retained. With fitnesses 1−s, 1, and 1−t for AA, Aa, and aa, q̂ = s / (s + t). Sickle cell is the case. HbS is the beta-6 glutamic acid to valine allele. HbA/HbA is most susceptible to severe falciparum malaria. HbA/HbS has a relative resistance and does not have sickle-cell disease in ordinary conditions. HbS/HbS has the disease, a large cost where care does not remove it.
Where severe falciparum malaria has long been a cause of death, s holds q̂ far above a mutation-only frequency. Where that pressure has been absent, HbS is simply deleterious and stays rare except by migration or founder sampling. The qualitative geography is the historically malarial regions of tropical Africa, the Mediterranean, the Middle East, and parts of the Indian subcontinent. No country incidence is stated here. Heterozygotes are not immune, and the allele is not advantageous in every environment. The usual cellular account is impaired parasite growth or preferential removal of parasitised cells under low oxygen.
9.4 Key terms, summary, and self-check #
Key terms. Allele frequency; Hardy–Weinberg; carrier frequency; fitness; mutation–selection balance; migration; inbreeding coefficient; drift; founder effect; bottleneck; heterozygote advantage.
Summary. Random mating yields p², 2pq, and q² if selection, mutation, migration, and drift are absent. Carrier frequency for a rare recessive is about 2√(incidence). Selection hardly sees a rare recessive; it sees a dominant at once. Migration shifts p toward the source, inbreeding raises homozygosity at constant p, and drift dominates when N_e is small. HbS is held by heterozygote advantage where falciparum malaria makes HbA/HbS fittest, and not where that advantage is absent.
Self-check.
1. In 50 AA, 20 Aa, and 30 aa, what is q, and why is this not Hardy–Weinberg?
2. Incidence 1/40,000 under Hardy–Weinberg implies what carrier frequency?
3. Why does a recessive lethal move only from 0.01 to about 0.0099 in one generation?
4. What does F not change, and what is F for a child of first cousins?
5. Why is the equilibrium frequency of HbS different where severe falciparum malaria is absent?
