Chance and Selection

Chapter 6 The tug-of-war

Selection versus drift

A beneficial mutation starts as one copy in a lottery; whether its advantage or its luck decides its fate depends on the population size times the selection coefficient.

8 min read · 4 interactive figures, 2 questions

Ten beetles; orange-orange beetles have fitness 1 + s and mixed ones 1 + s/2, with s = 0.1, so Ns = 1. Press New population a few times: orange often loses anyway. Raise N to 100 (Ns = 10) and it nearly always wins. The advantage is the same; Ns is not.
On this page
  1. Most beneficial mutations are lost
  2. Kimura’s formula
  3. The borderland, |Ns| ≈ 1
  4. Slightly harmful mutations, and why size matters
  5. The whole argument in one paragraph

Natural selection and genetic drift act at the same time on every variant in every finite population. Selection pushes a frequency in a direction, by about ½ s × pq per generation for an additive allele (s as in equation (2.1): AA has fitness 1 + s, Aa 1 + s/2); drift jostles it, with no direction, by about √(pq/2N). Which one decides a variant’s fate depends on the product of the population size and the selection coefficient, Ns. When |Ns| is much larger than 1, selection wins and the deterministic curves of Chapter 2 are a good guide. When |Ns| is much smaller than 1, the variant behaves as if it were neutral and its fate is a coin toss weighted by its frequency, as in Chapter 3. The same advantage can be decisive in a large population and irrelevant in a small one.

Most beneficial mutations are lost

Question 1 Prediction before the next figure.#
Figure 1 Two hundred populations of N individuals, each starting with a single copy of a beneficial mutation (AA has fitness 1 + s, carriers 1 + hs; the default s = 0.02 with h = ½ gives carriers a 1 % edge). Run them. Nearly all the tubes stay blue: the mutation is lost within a few generations. The sentence compares the number that fix with Kimura’s formula. Then make A recessive: with the same s it fixes about half as often (Kimura gives 0.011 against 0.020), because a lone copy sits in a heterozygote that selection cannot see; dominant, Kimura gives 0.0325.#

The reason is the lottery of Figure 1 in the drift chapter. A single new copy is carried by one heterozygous individual. With a fixed population size, that individual leaves on average about one copy in the next generation, give or take: and in roughly a third of cases (the Poisson chance of zero offspring copies, e−1 ≈ 0.37) it leaves none. A 1 % advantage for its carrier barely changes those odds. Only if the mutation survives long enough to reach a few hundred copies does its advantage start to dominate, and the chance of surviving that long is small. In 1927 J. B. S. Haldane treated the early spread as a branching process and found that, for one new copy whose carriers have a small advantage hs (hs ≪ 1) and in a population large enough that Nhs ≫ 1, the probability of eventual success is approximately twice the carriers’ advantage[1]:

u ≈ 2hs    (= s for an additive allele)(6.1)
#

So adaptation is wasteful. A favourable mutation usually has to arise many times before one copy of it survives the lottery, which is one reason why large populations, which produce more mutations, can adapt faster. established

Kimura’s formula

Haldane’s result covers a new mutation in a large population. Motoo Kimura found the general answer in 1962, using a diffusion approximation that treats the allele frequency as a continuous random walk with a drift term from selection and a variance term from sampling[2]. For an additive allele (h = ½: fitnesses 1, 1 + s/2 and 1 + s for aa, Aa and AA, the convention of the whole site), in a population of effective size N, an allele now at frequency p eventually fixes with probability

u(p) = 1 − e−2N s p1 − e−2N s(6.2)
#

Two limits check it. As s → 0, both exponentials expand to first order and u(p) → p, the neutral result of equation (3.2). For a new mutation, p = 1/2N, and when 2Ns is large the denominator is close to 1 and the numerator is close to 1 − e−s ≈ s = 2hs, which is Haldane’s answer. In between, the formula interpolates smoothly, and the only combination that matters is Ns. A worked value: N = 100, s = 0.02, p = 1/200 gives u = (1 − e−0.02)/(1 − e−4) = 0.0202, four times the neutral chance of 0.005. For other dominance Kimura’s paper gives the answer as a ratio of two integrals with no closed form; Figure 1 and the population lab compute it numerically for whatever h you choose.

A note on notation, because textbooks differ. Here, as on every page of this site, s is the fitness difference between AA and aa, and the heterozygote gets hs (equation 2.1). Kimura’s paper, and many texts, instead give each copy of A an advantage s, so that AA has 1 + 2s: an additive allele with s = 0.02 here has s = 0.01 there, their 4Ns is our 2Ns, and their Haldane result “2s” is our 2hs. The biology is the same; only the bookkeeping differs, which is why a number for Ns means nothing until the convention is stated.

Figure 2 The chance that one new mutant copy takes over, divided by the neutral chance 1/2N, against Ns (an additive allele, s as everywhere on this site); the vertical axis is logarithmic, the horizontal one is not. Black: Kimura’s formula. Dashed: Haldane’s 2hs, which is s here. Drag the orange point along the curve (or use the slider) and press Test it to run 4,000 Wright–Fisher populations from a single copy; the square and bar show the observed fraction with a 95 % interval. Notice that there is no flat stretch: across the shaded band (|Ns| below 1) the ratio rises about sevenfold, from about 1/3 to about 2.3 (at the default N = 50). Outside the band, harmful mutants are purged and beneficial ones run along Haldane’s line.#

The engine behind this site also computes the exact Wright–Fisher answer, solving the full Markov chain for populations of up to 400 individuals, and checks the diffusion formula against it: for N = 50 to 200 the two agree within 3 % when N|s| is up to about 1, and within 6 % up to N|s| = 2 (the worst case is a harmful allele in the smallest population). The simulated points in Figure 2 come from exact binomial sampling, not from the formula.

The borderland, |Ns| ≈ 1

Read Figure 2 from the middle outward. At Ns = 0 the ratio is 1: neutral. At Ns = 1, a beneficial mutant is about 2.3 times as likely to fix as a neutral one; at Ns = −1, about three times less likely. By Ns = 10 the beneficial mutant is 18 times favoured, close to Haldane’s line at 20, and by Ns = −4 a deleterious one is about 350 times less likely to fix than a neutral one. So the transition from “drift decides” to “selection decides” is not a cliff but it is fast, and it is centred on |Ns| of order one[3].

The heuristic behind the boundary is worth having. Drift’s characteristic time is about 2N generations, the time for heterozygosity to fall to 1/e of its value. Over that time selection, pushing at about ½ s pq per generation for an additive allele, moves the frequency by about Ns pq, while drift has moved it by about pq: selection wins when Ns ≫ 1. The same advantage that is decisive in a population of a million (Ns = 10,000 for s = 0.01) is invisible in a population of 50 with s = 0.001.

Question 2 Prediction before the next figure.#
Figure 3 The tug-of-war. Populations start with A at 10 %, with AA at fitness 1 + s and carriers 1 + s/2. Run them with N = 10, then with N = 500 (the dashed line is the same selection in an infinite population). The sentence gives Ns and Kimura’s prediction for the fraction that fix.#

Slightly harmful mutations, and why size matters

The boundary applies to harmful mutations too, and that has consequences for whole genomes. A mutation that lowers the fitness of its homozygotes by 0.001 (s = −0.001, carriers half that) is purged efficiently in a population with Ne = 10,000 (Ns = −10: it fixes with a probability around 4 × 10−8 of the neutral rate), but behaves almost neutrally in a population with Ne = 100 (Ns = −0.1). In 1973 Tomoko Ohta proposed that a large share of molecular evolution consists of such slightly deleterious mutations, fixed by drift in species with small effective sizes and purged in species with large ones[4]. Her nearly neutral theory predicts that small populations accumulate a load of slightly harmful variants, which is one of the concerns about species, like the northern elephant seal, that have been through severe bottlenecks. How much of genome evolution is nearly neutral, and how the distribution of fitness effects of new mutations varies between species, remain active research questions. active

The whole argument in one paragraph

Evolution at a single gene is a change in how common its variants are. With no force acting, frequencies stay put and random mating sorts them into Hardy–Weinberg proportions in one generation. Selection moves them in a direction, at a rate set by s and slowed by rarity and by dominance. Drift moves them at random, at a rate set by 1/2N, and in the end fixes one variant with probability equal to its frequency. Founders and bottlenecks are drift at its strongest; gene flow pulls populations back together, and a few migrants per generation are enough. Selection and drift act together everywhere, and the product Ns decides which one is in charge. Every one of these statements can be checked against real data, and on this site each figure does.