Fifty generations of perfect selection halves a recessive gene, and that is the best case

By the 42Math team · 25 June 20266 min read

A gene carried by one person in twenty-five, causing a condition in one in two-and-a-half thousand. Apply the strongest selection pressure that can exist for 1,250 years, and it is still carried by one person in fifty.

Take a recessive allele at a frequency of 2%. About one person in twenty-five carries a single copy and is entirely unaffected; roughly one in 2,500 inherits two copies and has the condition.

Now apply selection of a severity no real population has ever experienced: every individual with two copies fails to reproduce, every generation, without exception. Not reduced fertility — zero. Run it for fifty generations, which for humans is something over a thousand years.

Set the Hardy–Weinberg panel to p = 0.98 with fitness w(aa) = 0 and fifty generations. The allele frequency at the end is 0.010.

Halved. That is the whole return on a millennium of the strongest selection pressure the mathematics allows.

Where the copies are hiding

The reason is a two-line argument about accounting.

Selection can only act on what it can see, and what it can see is the double-copy individuals. With the allele at q = 0.02, the panel gives the carrier frequency 2pq as 0.039, about 3.9% of the population walking around with exactly one copy. The affected fraction is q², which is 0.0004, or 0.04%; the panel rounds that row to three decimals and prints 0.000, which is its own comment on the scale of the thing selection is being asked to grip.

Count the alleles rather than the people. Carriers hold one copy each, affected individuals two. So of every 100 copies of the allele in the population, 98 are sitting in carriers where selection cannot reach them, and 2 are in affected individuals where it can. Removing every affected individual removes 2% of the copies. And 2% is exactly q. The smaller q gets, the smaller the fraction of it that is exposed.

That gives the rate directly. Under complete selection against the recessive, the frequency after t generations is

q_t = q₀ / (1 + t·q₀)

which is not exponential decay. It is a hyperbola, and hyperbolas have long tails. Starting from 0.02, the time to halve is 1/q₀ = 50 generations. Halving again takes another 100. The one after that takes 200. Each halving costs double the previous one, so the allele approaches zero and never usefully arrives. Compare it to the exponential decay familiar from radioactive decay, where every half-life is the same length: here the half-life doubles each time, because the hiding place gets proportionally larger as the allele gets rarer.

Set p = 0.5, so that q starts at 0.500, keep the same lethal fitness and the same fifty generations, and the panel finishes at p = 0.981: q has fallen from 0.500 to 0.019, a twenty-six-fold reduction. The rare allele managed two-fold over an identical run. It is precisely the rare alleles, the ones associated with serious disease, that are almost perfectly protected.

What this settled

The equilibrium result has an unusual origin. In 1908 the geneticist Reginald Punnett was arguing with the statistician George Udny Yule, who had claimed that a dominant trait would inexorably spread through a population until three-quarters of it was affected. Punnett could not construct the algebraic rebuttal, so he asked his cricket partner, the Cambridge pure mathematician G. H. Hardy.

Hardy's reply to Science runs barely two pages and opens by apologising for the triviality of the point. He shows that with random mating the genotype proportions p², 2pq and q² reproduce themselves unchanged in every subsequent generation: nothing spreads, nothing dwindles, the frequencies simply stay. Wilhelm Weinberg had published the same result in German months earlier, which is why the law now carries both names.

Hardy considered it beneath his interests — he wrote later that he had never done anything useful — and it became the founding equation of population genetics. It is also the reason the numbers above are not an opinion. Once you know that allele frequencies are stable by default, the effect of selection is calculable, and the calculation says that removing the affected removes a share of the gene pool equal to the frequency of the allele itself.

That result had a target when it was fresh. Compulsory sterilisation programmes in the first half of the twentieth century were justified on the explicit promise of eliminating heritable conditions within a few generations. The arithmetic on this page was already published, already simple, and says the promise was arithmetically impossible: the overwhelming majority of copies of any rare recessive allele are in unaffected carriers, invisible to any selection scheme that acts on who is affected. The programmes were monstrous on grounds that have nothing to do with mathematics. They were also, on their own stated terms, never going to work.

The modern reading

The same arithmetic runs forwards, and it is why carrier screening exists as a separate concept from diagnosis.

If you know the affected frequency you can recover the carrier frequency, because q = √(q²) and carriers are 2pq. One in 10,000 affected means q = 0.01 and carriers around 1 in 50. One in a million affected still means q = 0.001 and roughly 1 in 500 people carrying it. Every one of us carries a handful of alleles that would be serious in a double dose; they are invisible because the odds of meeting a matching partner are low. Consanguinity raises the risk not by creating anything but by raising exactly that probability of a match.

And the persistence is not always a defect to be regretted. Some heterozygotes are fitter than either homozygote — the sickle-cell allele's protection against malaria is the standard case — and the Overdominance preset shows what that does. It starts at p = 0.200 with the heterozygote fittest, w(AA) = 0.90 against w(aa) = 0.80, and after 120 generations it settles at p = 0.667. That is not somewhere the simulation wandered: the balance point is (1 − w_aa) ÷ [(1 − w_AA) + (1 − w_aa)] = 0.20 ÷ 0.30, exactly two thirds. Selection is holding the allele in place rather than removing it, and the condition is the price of the protection.

One warning about all of the above, and the panel makes it in its own words: this model never rolls dice. It holds p = 0.500 for fifty generations exactly, and runs the identical curve every time, because there is no randomness in it. Real populations are finite, so chance alone shifts frequencies each generation — genetic drift — and in a small population drift can lose an allele that selection could never touch. Infinite population size is the assumption doing the most work here, and it is the first one reality breaks.

Published 25 June 2026 · corrections welcome via the corrections page.