Chance and drift

Every result so far has quietly assumed a population large enough that a frequency behaves like a probability, and no real population is.

The previous lessons built a mechanism: mutation supplies alleles at a measured rate, and selection changes their frequencies at a rate the algebra predicts. Both arguments treated allele frequencies as exact. In a population of finite size the gametes that make the next generation are a sample, and a sample deviates from the proportions it was drawn from. This lesson works out how much, and finds that the answer reorganises the whole subject: most of what happens at the molecular level is not selection at all.

Sampling is a force

Take a population of $N$ diploid adults, so $2 N$ gene copies, with allele $A$ at frequency $p$ . The next generation is formed by drawing $2 N$ copies from a gamete pool in which $A$ is at frequency $p$ . The number of $A$ copies drawn is binomial, so the new frequency $p^{'}$ has expectation $p$ and variance

operatorname {V a r} (p^{'}) = \frac{p (1 - p)}{2 N}

The expectation being $p$ is what makes drift undirected: it is as likely to go up as down. The variance being nonzero is what makes it a force: the frequency will not stay put, and it will not return. There is no restoring term anywhere. A frequency that wanders to 0 or 1 stops, because a population with no copies of $A$ cannot produce one by sampling, and those two states are therefore absorbing.

Repeating the sampling compounds it. The heterozygosity $H = 2 p q$ , which is the standard measure of how much variation a population holds, decays as

H_{t} = H_{0} {(1 - \frac{1}{2 N})}^{t}

so a population loses a fraction $1 / 2 N$ of its variation every generation whatever the alleles are doing. That is the first substantive claim: finite populations run down. Mutation puts variation in at rate $μ$ and drift takes it out at rate $1 / 2 N$ , and the standing level of variation is where those balance.

Buri's flies

The theory was thirty years old before anyone tested it properly, and the test is a good one because both the prediction and the measurement are exact.

In 1956 Peter Buri set up 107 independent populations of Drosophila melanogaster, each founded with 8 males and 8 females, all heterozygous at the bw locus so that the allele $b w^{75}$ started at exactly $p = 0.5$ in every line. Each generation he picked 8 males and 8 females at random from the offspring and used them as the next generation's parents. The genotypes are distinguishable by eye colour, so he could score every fly. He ran it for 19 generations.

Example. With $N = 16$ and $p_{0} = 0.5$ , what variance in $p$ across the 107 lines does theory predict after 19 generations, and what fraction of lines should have gone to fixation?

The variance accumulates as $operatorname {V a r} (p_{t}) = p_{0} q_{0} [1 - (1 - 1 / 2 N)^{t}]$ . With $p_{0} q_{0} = 0.25$ and $2 N = 32$ ,

operatorname {V a r} (p_{19}) = 0.25 [1 - (0.96875)^{19}] = 0.25 \times 0.4530 = 0.1132

a standard deviation of 0.337, on a quantity that can only run from 0 to 1. After nineteen generations the lines should be scattered right across the range, with a substantial number already at 0 or 1: the heterozygosity remaining is $(0.96875)^{19} = 0.547$ , so nearly half the original variation is gone. Buri's observed result was that 30 lines had fixed for $b w^{75}$ and 28 had lost it, 58 out of 107, and the rest were spread across every intermediate frequency. The qualitative prediction is confirmed emphatically: identical populations under identical conditions with no selection whatever ended up in completely different places.

Now you. Buri's observed variance was larger than the prediction above, and matching it requires putting $N$ at about 11.5 rather than 16. He counted his flies, so the census number is not in doubt. What is going on, and what is the quantity that actually belongs in the formula?

Answer

The formula does not want the number of adults. It wants the number of adults in an idealised population that would drift at the observed rate, which is the effective population size $N_{e}$ . The two differ whenever the real population departs from the idealisation, and it almost always does.

Three departures matter here. Offspring number varies between parents: in the ideal case it is Poisson, and in real flies a few females contribute far more eggs than others, which concentrates the next generation's ancestry and raises the sampling variance. The sexes may contribute unequally, with $N_{e} = 4 N_{m} N_{f} / (N_{m} + N_{f})$ , though Buri's 8 and 8 makes this term neutral. And the parents were themselves drawn from a larger pool of offspring, adding a round of sampling the formula does not count.

The lesson generalises well beyond flies. $N_{e}$ is typically a fraction of the census size, often a tenth or less, so drift is stronger than a headcount suggests. The largest single effect is a bottleneck, because $N_{e}$ over a period is the harmonic mean of the sizes, not the arithmetic one. A population sitting at 1,000 for four generations and dropping to 10 for one has an arithmetic mean of 802 and a harmonic mean of $5 / (4 / 1000 + 1 / 10) = 48$ . One bad generation costs almost everything.

The consequences are visible in real species. Northern elephant seals were hunted to perhaps twenty individuals by 1892 and now number over 200,000, and when 24 protein loci were surveyed in 1974 every one was monomorphic: the census recovered and the variation did not. Cheetahs are similar. Estimates of the long-term human $N_{e}$ from genetic diversity come out near 10,000 to 20,000 despite a census in the billions, because the harmonic mean reaches back through every bottleneck our ancestors passed.

What happens to a new mutation

Drift matters most at the moment when a new allele is rarest, which is the moment it appears. A new mutation exists as one copy out of $2 N$ , so its frequency is $1 / 2 N$ and its fate is almost entirely a matter of luck.

For a strictly neutral allele the answer is immediate and requires no algebra. Every one of the $2 N$ copies at a locus is equally likely to be the ancestor of all copies in the distant future, and exactly one of them will be, so the probability that a given new copy is the winner is $1 / 2 N$ . In a population with $N_{e} = 10^{4}$ that is $5 \times 10^{- 5}$ : a neutral mutation is lost 99.995 per cent of the time.

Example. A beneficial mutation with a selective advantage of 1 per cent appears as a single copy. What is the probability that it ever reaches fixation?

Haldane worked this out in 1927 using a branching process. Ask what fraction of lineages founded by one copy eventually die out. If the number of surviving offspring copies is Poisson with mean $1 + s$ , the extinction probability $x$ satisfies $x = e^{- (1 + s) (1 - x)}$ , and for small $s$ the solution is $x \approx 1 - 2 s$ . So the fixation probability is about $2 s$ , here $0.02$ .

A mutation that is genuinely and permanently 1 per cent better than everything around it is lost, at random, 98 times out of 100. This is the most under-appreciated number in the subject. The previous lesson showed that such an allele sweeps in 924 generations once it is common; this lesson shows it almost never gets the chance. Adaptation is therefore not the story of a good mutation arising, it is the story of the same good mutation arising fifty times before one of them survives its first few generations, which is why the mutation supply arithmetic of the previous lesson mattered.

Now you. Why does the fixation probability depend on $s$ but not on the population size, when the neutral probability $1 / 2 N$ depends on nothing else?

Answer

Because the danger is concentrated entirely in the first handful of generations, when the allele is present in a few copies and the rest of the population is irrelevant to it. A lineage starting from one copy either grows past the point where chance can kill it or does not, and whether it does depends on its own growth rate, $1 + s$ , not on how many other individuals are in the population. Once it has a few hundred copies its trajectory is essentially deterministic and the previous lesson's recursion takes over.

Population size returns through a different door. What decides whether an allele behaves as beneficial or as effectively neutral is the comparison between $s$ and $1 / (2 N_{e})$ : when $| s |$ is much smaller than $1 / (2 N_{e})$ , drift dominates and selection cannot see the allele at all. In a population of $N_{e} = 10^{4}$ that threshold is $5 \times 10^{- 5}$ , so a mutation with an advantage of one part in a hundred thousand is invisible to selection in humans and clearly visible to it in a bacterial population of $10^{9}$ . The same mutation is beneficial in one species and neutral in another, purely because of population size, which is Tomoko Ohta's nearly neutral theory in one sentence.

Kimura's argument

In 1968 Motoo Kimura drew a conclusion from these pieces that provoked twenty years of argument. Take a genome with $n$ neutral sites mutating at rate $μ$ per site. Each generation the population of $N$ individuals produces $2 N μ$ new neutral mutations at a given site, and each has probability $1 / 2 N$ of eventual fixation. The rate at which neutral substitutions accumulate in the lineage is therefore

k = 2 N μ \times \frac{1}{2 N} = μ

The population size cancels completely. Neutral substitutions accumulate at the mutation rate, in a mouse and in an elephant, in a population of a thousand and a population of a billion. That is a molecular clock, and it falls out of the neutral assumption with no further hypotheses.

Kimura's claim was that this, and not selection, accounts for most molecular change: the variation seen between species at the sequence level is largely the accumulated debris of mutations that were never worth anything. The evidence he pointed to was the constancy of protein evolution rates that Emile Zuckerkandl and Linus Pauling had noticed in 1962, and the fact that the observed rate of amino acid substitution, extrapolated across a whole genome, implied a substitutional load that no population could pay if every change were driven by selection.

The clock's most persuasive feature is that different proteins run at different but characteristic speeds, and the speeds line up with how much of the molecule matters. Fibrinopeptides, which are cut out of fibrinogen and discarded, change at roughly 8 substitutions per site per billion years. Haemoglobin runs at about 1. Cytochrome c, which must dock precisely with two large protein complexes, runs at about 0.3. Histone H4, which wraps DNA and whose surface is almost entirely functional, differs at only 2 of its 102 residues between a cow and a pea. Under a selectionist reading these differences are hard to interpret; under a neutral one they are the fraction of each protein that is free to change.

The clock, and where it disagrees with itself

A clock is only useful if it can be checked against something independent, and here the honest answer includes a discrepancy the field has not resolved.

Example. Take the human pedigree mutation rate of $1.2 \times 10^{- 8}$ per base per generation, a generation time of 25 years, and a human-chimpanzee split at 6.5 million years ago from the fossil record. What sequence divergence does the neutral clock predict?

Two lineages each accumulate substitutions since the split, so the divergence per site is $2 μ t$ with $t$ in generations. Here $t = 6.5 \times 10^{6} / 25 = 2.6 \times 10^{5}$ generations, giving

2 \times 1.2 \times 10^{- 8} \times 2.6 \times 10^{5} = 6.2 \times 10^{- 3}

or 0.62 per cent. The measured single-base divergence between the human and chimpanzee genomes is about 1.2 per cent, roughly twice the prediction.

Now you. Something in that calculation is wrong. List the candidates and say which way each would push, and what the disagreement does to the standing of the neutral theory.

Answer

There are four candidates and no consensus. The split could be older: taking the 1.2 per cent divergence at face value with the pedigree rate puts it at 12.5 million years, which some fossil interpretations can accommodate and most cannot. The generation time could be longer in the ancestral lineage, which multiplies the years per generation and stretches the date the same way. The pedigree rate could understate the long-term rate, if for instance the phylogenetic rate calibrated over tens of millions of years genuinely differs from the rate measured in trios today, which is the "hominoid slowdown" hypothesis. Or the older phylogenetic calibrations, which gave $2.5 \times 10^{- 8}$ and fit the 6.5-million-year date exactly, were themselves circular, having been calibrated on assumed divergence dates.

What the disagreement does not do is threaten the neutral theory. The prediction $k = μ$ is a statement that the substitution rate equals the mutation rate, and both quantities are being measured to within a factor of two of each other across seven million years, which for a parameter-free prediction in biology is a good result. What it threatens is the practice of reading absolute dates off molecular data without an independent calibration. The clock is real, it is noisy, and its variance across lineages is larger than a strict Poisson process allows, which is why it is described as overdispersed and why any date derived from it should carry a factor-of-two error bar rather than three significant figures.

What drift settles and what it leaves

Two of the subject's recurring confusions dissolve here. Change is not evidence of adaptation: two isolated populations of the same species will diverge at neutral sites simply by sampling, at a rate set by the mutation rate, and most sequence differences between species mean nothing about their circumstances. And absence of change is not evidence of stasis in the population, since a locus at fixation is invisible to every method until a mutation arrives.

Everything in this half of the course, though, has described one population changing. The Hardy-Weinberg result, the selection recursion, the mutation supply and the sampling variance all treat the population as a single interbreeding pool that persists through time. That describes a lineage getting different. It does not describe a lineage becoming two, and without splitting there is one species on earth, however well adapted. What it takes to divide a pool is the subject of the next lesson.