Chance and Selection

Chapter 2 The force with a direction

Natural selection

When carriers of one gene variant leave more offspring, that variant becomes more common, at a speed set by how it is inherited.

11 min read · 5 interactive figures, 1 question

Sixty beetles in which orange carriers leave more offspring: beetles with two orange copies have fitness 1 + s, mixed beetles 1 + hs. The dashed line is the same selection in an infinite population; the orange line is this one, with chance included. Set s to zero and the climb stops; make it negative and orange is pushed out. Switch orange to recessive and watch how long it idles while rare.
On this page
  1. Fitness, and the one equation
  2. The peppered moth: up, then down
  3. Lactase persistence: how strong must selection have been?
  4. Sickle cell: selection that keeps a harmful allele common

Natural selection is a consistent difference in survival or reproduction between individuals that carry different gene variants. If carriers of allele A leave more offspring, on average, than carriers of a, then A makes up a larger share of the next generation’s genes, and the change repeats every generation the difference persists. In the language of the previous chapter, selection is the force that moves the allele frequency p in a direction. Everything else about it, how fast, how far, and whether it can be stopped, comes out of one short equation.

Selection needs three things that Darwin identified and that are still the working definition: individuals differ, some of the differences are inherited, and the differences affect how many offspring they leave. Selection does not create variants. The black form of the peppered moth below existed as a mutation before soot made it useful; selection only changed how common it was.

Fitness, and the one equation

Population geneticists summarize “how many offspring” with relative fitness, w. Give the three genotypes fitnesses

wAA = 1 + s,    wAa = 1 + hs,    waa = 1(2.1)
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Here s, the selection coefficient (it means the same thing on every page of this site, and in the population lab), is the advantage of AA over aa (s = 0.1 means 10 % more surviving offspring; a negative s means A is selected against), and h, the dominance coefficient, says where the heterozygote falls: h = 1 if A is dominant (Aa does as well as AA), h = 0 if A is recessive (Aa does as badly as aa), h = ½ if each copy counts equally (“additive”). Start a generation in Hardy–Weinberg proportions, let each genotype survive in proportion to its fitness, and count the A copies among the survivors[1]:

p′ = p²(1 + s) + pq(1 + hs)w̄,     w̄ = p²(1 + s) + 2pq(1 + hs) + q²(2.2)
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The denominator, w̄, is the population’s mean fitness; dividing by it turns survivors back into frequencies. A worked example: p = 0.3, s = 0.1, h = ½. Then w̄ = 0.09 × 1.1 + 0.42 × 1.05 + 0.49 = 1.03, and p′ = (0.099 + 0.2205) / 1.03 = 0.310. One generation of a 10 % advantage moved p by one percentage point. That is typical: selection is relentless rather than fast.

When s is small the change per generation is close to

Δp ≈ s p q [h + (1 − 2h) p](2.3)
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The factor pq is the key to the whole chapter. Selection can only act on variation, and pq is small when either allele is rare. A new favourable mutation, at p ≈ 0, barely moves; neither does an allele that has almost won. The bracket says how dominance shifts that slowness to one end or the other.

Question 1 Prediction before the next figure.#
Figure 1 Three alleles with the same advantage s: dominant (solid), additive (long dashes) and recessive (short dashes). The first panel is the frequency over time from p₀; the second is the change per generation, Δp, at each frequency (side by side on a wide screen, stacked on a phone). Drag the p₀ marker along the second panel’s axis, or use the slider. Notice that the recessive curve is flat near zero (rare recessives are hidden in heterozygotes) and the dominant curve is flat near one (the last a alleles hide in heterozygotes, which look like AA). Set s below zero and all three fall instead: the same equation describes selection against A.#

Read the Δp panel from left to right. When A is rare, almost every copy of it sits in a heterozygote, since AA individuals are rarer still (p² against 2pq). If A is recessive, heterozygotes look like aa and get no advantage, so selection sees nothing: Δp is essentially zero. If A is dominant, every heterozygote benefits and Δp takes off at once. Near fixation the roles swap. The last a alleles also hide in heterozygotes, which a dominant A makes indistinguishable from AA, so selection cannot find them. With s = 0.1, the dominant allele needs about 1,100 generations to climb from 50 % to 99 %; the recessive one needs 59.

That asymmetry has a sobering practical consequence. Selection against a rare recessive disease allele, even complete selection in which no affected person has children, barely changes its frequency, because nearly all copies are carried by healthy heterozygotes. At q = 0.01, carriers (2pq ≈ 0.02) outnumber affected people (q² = 0.0001) by about 200 to 1. Set the figure to a recessive allele near zero and watch how little happens.

The peppered moth: up, then down

The peppered moth, Biston betularia, rests by day on tree trunks and branches, where its speckled grey and white wings match lichen-covered bark. In nineteenth-century Britain a black form, carbonaria, became common around industrial cities, where soot had killed the lichens and blackened the trees. The melanic form is inherited through a single dominant allele, C[2]. In 1924 J. B. S. Haldane used its spread to make what Cook and colleagues call the earliest measurement of natural selection on a Mendelian gene in the wild: to rise as fast as it did, melanic moths must have had a large advantage[2]. documented

The mutation itself has now been found. In 2016 Arjen van’t Hof and colleagues showed that the carbonaria allele is a large transposable element, a stretch of “jumping” DNA, inserted into the first intron of a gene called cortex; from the way the surrounding DNA has been broken up by recombination since, they dated the insertion to around 1819[3]. One mutation, in one moth, at the start of the industrial revolution. documented

Then the air was cleaned up. After clean-air legislation the melanic form declined across Britain[4], and that decline is a second, independent test of the selection story. The case attracted heavy criticism in the 1990s: H. B. D. Kettlewell’s classic 1950s experiments had placed moths on tree trunks at high densities, which critics argued was unnatural, and one popular book went so far as to suggest fraud, a charge Cook and colleagues call “quite groundless”[2]. The Cambridge geneticist Michael Majerus responded with the largest predation experiment ever run on the species: between 2001 and 2007 he released 4,864 moths, one per sleeve on randomly chosen branches, at natural densities, in a garden near Cambridge, and watched birds eat them. He died in 2009 before publishing; Cook, Grant, Saccheri and Mallet analysed his data in 2012. Melanic moths survived each day at about 91 % of the rate of typical ones (a daily selection coefficient of 0.091 against melanics, 95 % interval 0.028 to 0.157; in this site’s convention s = −0.091 per day), which the authors estimate amounts to selection of 0.1 to 0.2 against melanics per generation[2]. Saccheri and colleagues, fitting the retreat of melanism across north-west England and north Wales, put the cost of being melanic at about s ≈ 0.2[4]. established

Majerus also recorded the share of melanic moths in light traps at nearby Madingley each year, and those seven numbers are enough to test the equation yourself. Carbonaria is dominant, so the melanic share is 1 − q², where q is the frequency of the typical allele; one generation per year.

Figure 2 The share of melanic moths in Majerus’s light traps at Madingley, 2001–2007 (points), against the selection model of equation (2.2) with melanics at fitness 1 + s, so a decline needs s below zero. Adjust s and the starting frequency, or press the button to fit both by least squares. Notice how weak selection (s = −0.05) cannot produce the decline, and what the fit implies for 2008 onward.#

The best fit gives s ≈ −0.28: melanics leaving about 72 % as many offspring as typical moths each year. That is at the strong end of the published estimates, and it should be. The seven frequencies were read off a graph by Cook and colleagues, the yearly samples were small, and moths migrate into Madingley from places where melanics are rarer or commoner. The point of the exercise is not the second decimal place. It is that no small advantage, the kind you might hope to miss in a field study, can explain a fall from 12 % to 1 % in six years: the equation demands strong selection, and the direct predation experiment found it.

Lactase persistence: how strong must selection have been?

Most mammals stop making lactase, the enzyme that digests milk sugar, after weaning. In several populations with a long history of herding cattle, a variant that keeps the gene switched on into adulthood has become common. The frequency of the European lactase persistence allele has been tracked in ancient DNA: it is absent from samples more than about 5,000 years old in Central Europe and reaches 70 % or more in many European populations today[1]. The haplotype carrying it is unusually long for its frequency, a signature of a recent, rapid rise; Bersaglieri and colleagues concluded that strong selection acted on it within the past 5,000 to 10,000 years[5]. The authors add that the signal is among the strongest yet seen for any gene in the genome. Non-persistence is an autosomal recessive condition[6], so persistence is dominant: one copy is enough. established

Equation (2.2) can turn that history into a number. If the allele started at 0.5 % and reached 70 % in about 5,000 years, at 28 years per human generation, how large must s have been?

Figure 3 A dominant allele starting at 0.5 %, with s = 0.02 to begin with: at that strength it takes 429 generations, about 12,000 years, to reach the dashed 70 % line, far too slow. Raise s until the curve crosses the line near 180 generations (about 5,000 years at 28 years per generation). Notice how much the answer depends on the starting frequency, which nobody knows well.#

The answer is a few percent per generation: s ≈ 0.05 takes 175 generations, s ≈ 0.04 takes 217. Lactase persisters had to leave something like 4 to 5 % more surviving children than non-persisters, every generation, for thousands of years. For a dietary enzyme that is a large effect, in line with the strength of the genetic signal. Why milk digestion mattered so much (calories, clean fluid, calcium, survival in famines) is still debated. active

Sickle cell: selection that keeps a harmful allele common

Selection does not always drive one allele to fixation. In 1954 Anthony Allison, working in East Africa, showed that people carrying one copy of the sickle-cell haemoglobin allele, HbS, had fewer and lighter infections with Plasmodium falciparum, the deadliest malaria parasite, than people with normal haemoglobin[7]. Two copies cause sickle-cell disease, which before modern treatment was usually fatal in childhood. So the carrier, the heterozygote, has the highest fitness of the three genotypes: heterozygote advantage, or overdominance. established

Write the fitnesses as 1 − c for normal homozygotes (c, the cost of malaria), 1 for carriers and 1 − t for HbS homozygotes (t, the cost of the disease). Many textbooks call c “s”; here s keeps its one meaning from equation (2.1), and c and t are costs. Equation (2.2), with general fitnesses, then has a stable equilibrium in the middle[1]:

q̂ = cc + t(2.4)
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Figure 4 Set the cost of malaria to normal homozygotes (c) and the cost of sickle-cell disease (t). The first panel follows the HbS allele from four different starting frequencies; all converge on the same value. The second panel (beside it on a wide screen, below it on a phone) is the population’s mean fitness, which peaks at that frequency. Notice what happens when c goes to zero, as with malaria control. The default fitnesses are illustrative, not measurements.#

With c = 0.15 and t = 0.85 the equilibrium is q̂ = 0.15, which is the right order of magnitude. Mapping hundreds of surveys, Piel and colleagues predicted HbS frequencies above 9 % across a broad belt from southern Ghana to northern Zambia, with a maximum of about 18 % in northern Angola, and found that the global distribution of the allele matches the historical distribution of malaria, strongly in Africa[8]. established About 2 % of births at that frequency are HbS homozygotes; the price of protecting the carriers is paid by them. This is balancing selection, and it is why “selection favours the fittest” is misleading at the level of genes: here no allele is fittest, a combination is.

Directional selection (the moths, lactase) and balancing selection (sickle cell) are both deterministic stories: given the fitnesses, the curve is fixed. Real populations are finite, and in a finite population even an allele with no effect on fitness at all changes frequency every generation. That second force, which has no direction, is the subject of the next chapter.