A filter cannot produce what is not poured into it, and the previous lesson's algebra has no term anywhere in it that makes a new allele.
That is the gap Darwin admitted in the Origin when he wrote that the laws governing inheritance are quite unknown, and it is the last hole in the mechanism. This lesson fills it with three processes that can be measured rather than assumed: mutation, which makes new alleles from copying error; duplication, which makes new genes from whole copies; and recombination, which makes new combinations from old alleles. It also settles the question that decides whether the process is Darwinian at all, which is whether a mutation appears because it is needed.
Counting mutations directly
Until sequencing was cheap, mutation rates were inferred indirectly, from the frequency of a visible disease or from mutant colonies on a plate. Both estimates are contaminated by selection. The clean measurement is to sequence a mother, a father and their child, and count the bases in the child that are in neither parent.
In 2012 Kári Stefánsson's group at deCODE did this for 78 Icelandic families and found about 63 new single-base mutations per child. Dividing by the number of bases the method could reliably call in both copies of the genome, roughly per haploid set counted twice,
per base per generation. The study also found something that indirect methods could never have shown: about 87 per cent of the new mutations came from the father, and the paternal number rose by roughly two mutations for every year of the father's age. That asymmetry has a mechanical cause. An egg is produced after about 23 rounds of cell division and then waits; sperm are produced continuously, through hundreds of divisions by middle age, and each division is a chance to copy a base wrongly.
Example. Of the 63 new mutations in a child, how many land in protein-coding sequence, and what does the answer say about how much of a genome selection can be watching?
Protein-coding exons are about 1.5 per cent of the human genome, so the expected number is , call it one. Roughly a quarter of random changes in coding sequence are synonymous and change no amino acid, so about 0.7 of a mutation per child alters a protein. That is the entire raw material, per person, on which selection for a better protein can possibly act. The other 62 mutations fall in sequence where most changes have no measurable consequence, which is a fact the neutral theory of a later lesson is built on. Note also what the number rules out: an organism cannot be carrying a large hidden reserve of new coding variants each generation, so evolution has to work with very small per-generation increments, exactly as the previous lesson's selection coefficients require.
Now you. Escherichia coli has a genome of bases and a per-base mutation rate measured by mutation-accumulation experiments at per generation. How often does a cell acquire a mutation, and how many times is a given base mutated in a single overnight culture of cells?
Answer
Per genome per generation the rate is , so about one cell in a thousand acquires a new mutation each division. That sounds negligible until the second calculation. In a culture of cells, the expected number of cells carrying a mutation at any one nominated base is , and for one specific substitution at that base it is 0.35.
In other words, a tube of broth left overnight on a bench contains a cell mutated at essentially every position in the genome. The bacterium's low per-base rate and its enormous population size cancel almost exactly, and the practical consequence is that for a bacterial infection the relevant question is never whether a resistance mutation will arise. It is already there before the drug is given.
Do mutations arise in response to need?
That last remark conceals the deepest question in the subject. If bacteria become resistant when exposed to a drug, two accounts fit the observation. Either resistant cells are produced at random beforehand and the drug merely selects them, which is Darwinian, or the drug induces the change in the cells that meet it, which is Lamarckian. Until 1943 nobody had a way to decide, and serious microbiologists held the second view.
Salvador Luria found the test while watching a slot machine at a faculty dance in Bloomington, and worked out the statistics with Max Delbrück. The design is elegant because it does not require you to see a mutation at all, only to count survivors.
Grow many small independent cultures of E. coli from tiny inocula. Also grow one large culture and divide it at the end into samples of the same size. Then plate everything on agar covered with bacteriophage T1, which kills every sensitive cell, and count the resistant colonies.
Example. What does each hypothesis predict for the distribution of colony counts across the independent cultures?
Under the induced hypothesis, resistance is conferred at the moment of contact with the phage, with some small probability per cell. Each plate is then a large number of independent trials with a small success probability, so counts follow a Poisson distribution and the variance equals the mean. That prediction holds whatever the probability is, so it needs no fitted parameter.
Under the mutation hypothesis, resistance arises at random during the growth of the culture, before any phage is present. A mutation in the last division before plating contributes one resistant cell. A mutation twenty divisions earlier contributes a clone of about a million. So the count depends on when the first mutation happened, which varies wildly from culture to culture, and the distribution has a long tail of jackpots and a variance far exceeding its mean. The two hypotheses differ not in the average but in the scatter, which is why the experiment is called a fluctuation test.
Now you. In their experiment 23, twenty independent cultures gave a mean of 11.35 resistant colonies with a variance of about 694, while ten samples drawn from one bulk culture gave a mean of 16.7 with a variance of about 15. Which hypothesis survives, and why is the second set of numbers essential to the argument?
Answer
The independent cultures give a variance-to-mean ratio of , sixty times what Poisson allows, with several cultures at zero and at least one over a hundred. The induced hypothesis is dead. The mutation hypothesis is exactly what a jackpot distribution looks like.
The bulk-culture samples are the control, and without them the result proves nothing. If resistant cells are simply hard to count, or clump, or the plating is erratic, every set of counts would be overdispersed and the first result would be an artefact of technique. Samples drawn from one culture share their entire mutational history, so the only remaining variation is sampling and plating error, and they give a ratio of , essentially Poisson. Technique is clean; the excess variance in the first set is biology.
The cleanest demonstration came later. In 1952 Joshua and Esther Lederberg pressed a velvet pad onto a plate grown without phage and transferred the colony pattern to selective plates, showing that resistant colonies appear at the same positions on replicas, so the resistant cells could be traced back to a plate that had never met the selective agent at all.
Mutation on its own is a feeble force
Mutation supplies alleles, but it is a very weak director of frequencies, and it is worth seeing how weak. If mutates to at rate per generation and nothing else acts, then increases by per generation, so .
With , reaching a frequency of one half takes generations, and after a million generations the allele is still at 1 per cent. Against selection coefficients of the order of 0.01, which the previous lesson showed sweep an allele in under a thousand generations, mutation pressure is negligible as a force. Its role is entirely as a source: it decides what is available, not what happens next. The one place the rate does matter directly is in balancing selection against a deleterious recessive, where the equilibrium frequency is set by the rate, which is why rare recessive diseases persist at the frequencies they do.
Duplication, and where a genuinely new gene comes from
Point mutation modifies an existing gene. It does not obviously explain how a genome comes to have more genes than it had, and the objection that selection can only tinker with what exists has real force against point mutation alone.
The answer, argued by Susumu Ohno in 1970, is duplication. Unequal crossing over, retrotransposition and whole-genome duplication all produce a second copy of a gene, and a second copy is free in a way the first is not: while the original continues doing the job, the spare can accumulate changes that would otherwise be lethal. Most spares simply decay into pseudogenes, and the genome is full of those. Occasionally one acquires a function the original did not have.
The cases are specific enough to check. Old World primates see in three colours because an ancestral long-wavelength opsin gene duplicated on the X chromosome and the two copies diverged by a handful of amino acid substitutions, shifting one peak from 560 to 530 nanometres; New World monkeys mostly retain the single gene. Antarctic notothenioid fish survive at temperatures below the freezing point of their blood using an antifreeze glycoprotein whose gene is a modified copy of a trypsinogen, still carrying recognisable fragments of the digestive enzyme's sequence at both ends. The vertebrate globins, myoglobin and the alpha and beta chains of haemoglobin, are one ancestral gene copied and recopied, which is why the beta cluster on human chromosome 11 has five working genes and a pseudogene lying in the order they are used through development.
Rates can be estimated. Michael Lynch and John Conery's 2000 survey put gene duplication at roughly 0.01 per gene per million years, which for a genome of 20,000 genes is about 200 duplications per million years, most of them doomed.
Recombination shuffles what mutation makes
The third source manufactures nothing new at the level of the allele but a great deal at the level of the combination. A human produces gametes by choosing one of each of 23 chromosome pairs independently, which alone gives distinct combinations, and crossing over then breaks and rejoins the chromosomes at one to three points each, so the number of distinguishable gametes is effectively unbounded.
This matters for a reason the previous lessons set up. Selection acting on a population without recombination has to wait for two beneficial mutations to occur in the same lineage, one after the other. With recombination they can arise in different individuals and be brought together, which is Fisher and Hermann Muller's argument for why sex exists at all. Recombination is also what makes the multiple-factor model of the fifth lesson generate transgressive offspring: the child assembling more adding alleles than either parent has is a recombination product.
The arithmetic of supply
Put the rate and the population size together and the picture changes character.
Example. There are about people alive. Each carries roughly 63 mutations absent from their parents. How many times over has every base in the human genome been mutated afresh in the living population?
The total is new mutations, spread over sites, which is new mutations per site. Every single position in the human genome exists in a mutated form in someone alive today, roughly a hundred and sixty times over, and every possible single-base variant compatible with reaching birth is currently present somewhere in the species. The limiting resource in human evolution is emphatically not the supply of point mutations.
Now you. Tuberculosis is treated with three or four drugs at once, never one. Resistance to rifampicin arises by point mutation at about per cell per generation and to isoniazid at about . An untreated lung cavity can hold bacilli. Explain the treatment regime in numbers.
Answer
With bacilli, the expected number already resistant to rifampicin is , and to isoniazid . Single-drug therapy therefore does not fail through some new adaptation; it fails because the resistant cells are present on day one and the drug clears their competitors for them. This was observed directly in the streptomycin trials of the late 1940s, where monotherapy produced resistant relapse within months.
Resistance to both requires both mutations in one cell, and since the mechanisms are independent the joint probability is , so a population of would be needed, which is a million times more bacilli than a patient has. Adding a third drug removes any doubt. The whole logic of combination therapy is a direct application of mutation-supply arithmetic, and it is worth noticing that it works only because mutations are independent and pre-existing. If the Lamarckian account had been right, the drugs would induce resistance in whatever cells they met and no combination would help.
What is still missing
The mechanism is now complete in outline. Mutation and duplication supply new alleles at a measured rate, recombination assembles them into new combinations, Mendelian heredity conserves them, and selection changes their frequencies at a rate the algebra predicts and the peppered moth confirms.
It is complete, and it is also wrong about one thing, which the next lesson takes up. Everything so far has assumed a population large enough that frequencies behave like probabilities. Real populations are finite, gametes are drawn as a sample rather than in exact proportion, and a beneficial mutation, when it first appears, is a single copy in a single individual whose fate is mostly decided by whether that individual happens to get run over.