Appearance
3.3 — Population Genetics: the Arithmetic of Change
About 1 in 25 people of northern European descent carries a cystic fibrosis allele. About 1 in 27 Ashkenazi Jews carries a Tay–Sachs allele. Among the Old Order Amish of Pennsylvania, a form of dwarfism that is vanishingly rare worldwide occurs in about 1 in 200 births. On the Atlantic island of Tristan da Cunha, asthma affects roughly half the population.
None of those numbers are about the environment. They are about arithmetic, and this page is the arithmetic.
Counting alleles instead of people
Evolution is a change in allele frequencies in a population over time. So the first thing to do is count alleles rather than individuals.
Take 100 people, of whom 36 are AA, 48 are Aa and 16 are aa. Each person carries two alleles, so there are 200 in total.
Count the A alleles: the 36 AA people contribute 72, the 48 Aa people contribute 48. That is 120 out of 200, so the frequency of A is 0.6. Count the a alleles: 16 aa people give 32, plus 48 from the heterozygotes. That is 80 out of 200, so the frequency of a is 0.4.
By convention p is the frequency of one allele and q of the other, and since these are the only two possibilities, p + q = 1.
Hardy–Weinberg: the null model
In 1908, G. H. Hardy — an English mathematician who thought the point too obvious to be worth publishing until a geneticist pressed him — and Wilhelm Weinberg independently derived what happens to genotype frequencies when nothing is acting on a population.
The derivation is a Punnett square with probabilities in it. An egg carries A with probability p and a with probability q. Same for a sperm. Fertilisation combines them at random, so:
- AA arises when both gametes carry A: p \times p = p^2
- aa arises when both carry a: q \times q = q^2
- Aa arises two ways — A egg with a sperm, or a egg with A sperm: pq + qp = 2pq
p^2 + 2pq + q^2 = 1
Read aloud: the proportion of one homozygote, plus the heterozygotes, plus the other homozygote, must add to the whole population.
And the frequencies do not change from one generation to the next, provided five conditions hold: no mutation, no selection, no migration, random mating, and an infinitely large population.
That is the point of the model. No real population meets those conditions, so Hardy–Weinberg is not a description of reality. It is the null hypothesis — what you would see if nothing were happening. When a real population deviates from it, something is happening, and the deviation tells you what.
The clinical use, which is immediate
Cystic fibrosis affects about 1 in 2,500 births in a northern European population. That is the frequency of the affected homozygote, so q^2 = 1/2500.
q = \sqrt{1/2500} = 1/50 = 0.02
So p = 0.98, and the carrier frequency is:
2pq = 2 \times 0.98 \times 0.02 = 0.0392 \approx \tfrac{1}{25}
One person in 25 is a carrier, from a disease that affects one in 2,500. This is the calculation that gives the counselling numbers, and it makes a general point worth internalising: for any rare recessive condition, carriers vastly outnumber affected people. The rarer the disease, the more extreme the ratio. At q = 0.001 — one affected person per million — carriers are about 1 in 500, five hundred times more numerous than cases.
Which is why you cannot eliminate a recessive disease by preventing affected people from reproducing. Almost all copies of the allele are in healthy carriers, invisible without testing. The eugenic programmes of the early twentieth century were founded on not understanding this equation, among their many other and much greater faults.
The five forces that break the equilibrium
1. Mutation introduces new alleles. On its own it changes frequencies extremely slowly — mutation rates are around 10⁻⁸ per base per generation — but it is the ultimate source of all variation, so nothing else can happen without it.
2. Selection changes frequencies according to fitness (Chapter 3.2). Fast when selection is strong.
Selection against a recessive allele gets slower and slower as the allele becomes rare, for the same reason as above: the allele hides in heterozygotes where it is invisible to selection. This is why deleterious recessive alleles persist indefinitely at low frequency rather than being cleared.
3. Migration (gene flow) moves alleles between populations and makes them more similar. Human populations have exchanged genes continuously throughout history, which is why there are no discrete human races in a genetic sense — variation is a set of overlapping gradients, and about 85 percent of all human genetic variation is found within any single local population, with only about 15 percent distinguishing continental groups. Chapter 3.5 returns to this.
4. Non-random mating changes genotype frequencies without changing allele frequencies. Consanguinity is the case that matters medically: cousin marriage increases homozygosity, which is why it raises recessive disease risk (Chapter 2.6) while leaving the alleles themselves as common as they were.
5. Genetic drift — random change from sampling. This is the one that surprises people, and it deserves the rest of the page.
Genetic drift, and why small populations are different
Selection is not the only thing that changes allele frequencies. Chance does too, because which individuals reproduce is partly luck — being in the wrong place when a rock falls has nothing to do with your genes.
Flip a fair coin ten times and you will often get 7 heads. Flip it ten thousand times and you will get very close to half. Sampling error is large in small samples and negligible in large ones, and a generation of reproduction is a sampling event.
So in a small population, allele frequencies wander at random from generation to generation. Given enough generations, an allele will wander all the way to 0 or to 1 — it is lost or fixed — and once fixed, no variation remains at that site.
Drift removes variation, and it does not care whether an allele is good or bad. In a small enough population, drift overwhelms selection: a mildly beneficial allele can be lost by chance and a mildly harmful one can become universal.
Founder effects
When a small group leaves a large population and starts a new one, they carry only a sample of the original variation — and whatever they happened to carry becomes the baseline for everyone descended from them.
The Amish. Around 200 founders arrived in Pennsylvania in the eighteenth century, and the community has grown largely without marrying outside. Ellis–van Creveld syndrome — short stature, extra fingers, heart defects — is extremely rare worldwide but occurs in about 1 in 200 Amish births in Lancaster County. It has been traced to a single couple who arrived in 1744.
Ashkenazi Jews. Historical bottlenecks and centuries of relative endogamy left elevated frequencies of several conditions: Tay–Sachs (carrier rate about 1 in 27 before screening), Gaucher disease, Canavan disease, familial dysautonomia, and specific BRCA1 and BRCA2 founder mutations. Community-based carrier screening reduced Tay–Sachs births by over 90 percent, and it is one of the most effective genetic screening programmes ever implemented.
Afrikaners. Variegate porphyria in South Africa traces to one Dutch couple who married in 1688.
Finland. Around forty conditions are enriched in the Finnish population and rare elsewhere, from a founder population followed by long isolation. This is now scientifically valuable — the Finnish genetic isolate is heavily used in research precisely because rare variants are common enough there to study.
And on Tristan da Cunha, settled by about 15 people in the early nineteenth century, several of whom happened to have asthma, roughly half the population is asthmatic.
None of these communities is doing anything wrong. The pattern is a mathematical consequence of small founder numbers, and understanding it is what makes targeted screening possible.
Bottlenecks
A population crash has the same effect: the survivors are a small sample, and their variation is what remains.
Cheetahs show almost no genetic variation, apparently from a severe bottleneck around 10,000 years ago. Skin grafts between unrelated cheetahs are accepted without rejection, which is essentially what happens between identical twins — an extraordinary demonstration of how little variation is left. It leaves the species dangerously vulnerable to any new pathogen.
Northern elephant seals were hunted to about twenty animals in the 1890s. They have recovered to over 100,000 but carry almost no genetic diversity.
Humans went through a bottleneck too. Human genetic diversity is strikingly low for a species of our range — two unrelated humans differ at about 0.1 percent of their DNA, less than two chimpanzees from neighbouring troops. The population that left Africa was small, perhaps a few thousand breeding individuals, and everyone outside Africa descends from it. This is why African populations retain more genetic diversity than all non-African populations combined, and why African genomes are disproportionately important for medical research.
Balancing selection: why bad alleles stay common
Some harmful alleles do not get eliminated because the heterozygote is fitter than either homozygote. This is heterozygote advantage, and it is the answer to the sickle cell puzzle from Chapter 2.8.
- HbA/HbA — normal, susceptible to falciparum malaria.
- HbA/HbS — carrier, mild or no symptoms, roughly tenfold protection against severe malaria.
- HbS/HbS — sickle cell disease, historically often fatal in childhood.
In a malarial region, the carrier is the fittest genotype. Selection therefore maintains the allele at an intermediate frequency, because pushing it up produces more affected homozygotes and pushing it down produces more malaria deaths. The equilibrium sits where those two costs balance, which in high-malaria regions lands the allele frequency around 10 to 20 percent.
And the prediction the model makes has been tested. Populations that have left malarial regions should see the allele decline, since the benefit is gone and the cost is not. African-American populations, several centuries removed from endemic malaria, do have measurably lower sickle allele frequency than West African source populations. The model is not just a story; it makes checkable predictions.
The same logic appears elsewhere. Thalassaemia and G6PD deficiency are both maintained by malaria protection. The cystic fibrosis allele may have been maintained by some resistance to cholera or typhoid, though the evidence is much weaker and this should be labelled a hypothesis rather than a finding.
Where this shows up in real medicine
Screening programmes are designed with these numbers. Carrier screening is offered where allele frequency is high enough to make it worthwhile: Tay–Sachs in Ashkenazi communities, thalassaemia in Mediterranean and South Asian populations, sickle cell in African and Indian populations.
Drug responses vary by population for the same reason. The enzymes that metabolise drugs (Chapter 1.5) come in variant forms whose frequencies differ between populations, so the same dose produces different blood levels. Around 15 to 20 percent of East Asians carry a variant that makes them poor metabolisers of some drugs, against a few percent of Europeans. Carbamazepine causes a severe and sometimes fatal skin reaction in people carrying a particular immune-system variant that is common in Han Chinese and Thai populations and rare in Europeans, and testing for it before prescribing is now standard in those populations — one of the clearest cases where population genetics changes a prescription. Chapter 22.13.
And a warning about how this is used. Population-level allele frequencies say something about groups and very little about individuals. Ancestry is a rough proxy for genetic risk and is now being replaced by direct testing wherever possible, because using ancestry as a shortcut both misses people who do not fit the assumption and reinforces categories that are not biologically sharp.
What the next page fixes
Everything so far describes change within a population. It does not explain how one population becomes two species that can no longer interbreed. Chapter 3.4 covers speciation — what a species actually is, how populations split, how fast it happens, and how the whole tree of life is reconstructed from the results.