An engineer reviewing the Origin of Species in 1867 found a hole in it that Darwin could not close, and closing it required knowing something about heredity that nobody in Britain knew.
The previous lesson set out the four premises of the argument and named its weakest joint: Darwin needed offspring to resemble their parents and had no account of why they do. This lesson is about what goes wrong when you assume the obvious account, and about the seven years of pea breeding that gave the right one.
Jenkin's objection
Fleeming Jenkin was a telegraph engineer, a colleague of Kelvin's, and a good enough critic that Darwin said he had given him more trouble than any other reviewer. His review appeared in the North British Review in June 1867 and it makes two arguments.
The first is about the limits of selection. Breeders, Jenkin observed, get rapid change at first and then hit a wall: a racehorse line improves and then stops improving, and no amount of further selection carries it past. If domestic selection has a boundary, why should natural selection not have one too?
The second is the swamping argument, and it is the serious one. Suppose a single individual is born with some markedly favourable variation. It must mate with an ordinary member of the population. If inheritance blends, meaning that the offspring is intermediate between its parents, then its young carry half the novelty, their young a quarter, and the variation is diluted out of existence long before selection has had time to act on it. Jenkin illustrated this with an offensive parable about a shipwrecked European on an island, which is the part everybody remembers and the least important part of the argument.
Darwin's defence was that he did not rely on single sports but on the small continuous variation present everywhere in a population. That is a real answer to the parable, but it does not touch the underlying arithmetic, and the arithmetic is worth doing.
Example. Under strict blending, an offspring's value for some measured character is the average of its two parents' values. If parents pair at random, what happens to the variance of the character in each generation?
Let the character have variance in the parental generation, and let the two parents of any offspring be drawn independently. The offspring's value is , so its variance is
The variance halves every generation, whatever it started at, and it does so with no selection acting at all. After 5 generations is down to 3.1 per cent of its original value, after 10 generations to 0.098 per cent, and after 20 to one part in a million. A blending population becomes uniform, and a uniform population cannot be selected on, because every individual is the same.
Now you. To keep the variance steady under blending, how much new variation must be supplied each generation, and what does that requirement imply?
Answer
Exactly half of the standing variance must be created anew every generation, since half is destroyed. R. A. Fisher pressed this in 1930: the required rate of fresh variation is so enormous that it would have to be visible, with something like half the population carrying a newly arisen difference in every character in every generation, which nobody observes. Under particulate inheritance the requirement collapses to almost nothing, because the variance is not destroyed in the first place and mutation has only to replace what selection and chance remove. The blending model is not merely inconvenient for Darwin. It is quantitatively incompatible with populations that are observably variable.
Why peas, and why counting
The answer was already in print. Gregor Mendel, an Augustinian friar at St Thomas's Abbey in Brno, had studied physics and mathematics at Vienna under Christian Doppler between 1851 and 1853, and returned to a monastery with an experimental garden. Between 1856 and 1863 he grew something like 28,000 pea plants, and he read the resulting paper to the Brno Natural History Society on 8 February and 8 March 1865, publishing it in the society's proceedings in 1866.
Almost everything about the design is better than what botanists were doing. The garden pea has many varieties that breed true, so the starting material is clean. Its flower is closed and normally self-pollinates, so a plant left alone is a controlled experiment and a cross has to be made deliberately with forceps. Above all, Mendel chose characters that come in two sharply distinct states with nothing in between: the seed is round or wrinkled, the cotyledons yellow or green, the stem tall or short. There is no judgment involved in scoring them, so a character can be counted rather than described.
Counting is the innovation. Earlier hybridisers, including some very good ones, recorded which forms appeared. Mendel recorded how many, in numbers large enough to have a standard error worth quoting, and he treated one character at a time rather than trying to describe the whole plant at once.
Segregation, and a ratio of three to one
Crossing a true-breeding round-seeded plant with a true-breeding wrinkled-seeded one gives an F1 generation that is entirely round. The wrinkled character has not been blended away to something intermediate, and Mendel's crucial observation is that it has not been destroyed either: self-pollinating the F1 gives an F2 in which wrinkled seeds reappear, unchanged, at about a quarter.
The model that explains this is that each plant carries two copies of a factor for the character, one inherited from each parent; that the copies do not mix; that one form of the factor (round) masks the other (wrinkled) when both are present; and that the two copies separate when gametes are made, so each gamete carries one at random. A cross of two F1 plants, each carrying one of each, then produces the four equally likely combinations, three of which contain at least one round factor.
Example. Mendel counted 5,474 round and 1,850 wrinkled seeds in the F2. Test this against the 3:1 prediction.
The total is , so the expected counts are round and wrinkled. The chi-square statistic is
On one degree of freedom that has a probability of about 0.61, meaning a random sample would deviate from 3:1 by more than this in about three cases out of five. The observed ratio is 2.96 to 1. Pooling all seven characters gives 14,949 dominant to 5,010 recessive, a ratio of 2.984, on nearly 20,000 seeds.
Now you. Why is it essential to the argument that the wrinkled seeds in the F2 are indistinguishable from the original wrinkled parent, rather than merely wrinkled-ish?
Answer
Because that is the whole difference between particulate and blending inheritance. Under blending, a character that vanishes in the F1 is gone: there is nothing left to reappear. Mendel's wrinkled seeds come back at full strength after a generation of complete concealment, which shows that the factor passed through the F1 plant unaltered by the round factor it shared a cell with. Heredity is therefore the transmission of discrete objects, not the mixing of fluids, and the variance-halving arithmetic of the previous section simply does not apply. Everything Darwin needed follows from this one observation.
Two characters at once
Mendel then crossed plants differing in two characters at once, round yellow against wrinkled green, and asked whether the two behave independently. If they do, the F2 should show the product of two independent 3:1 ratios, which is 9:3:3:1.
Example. Mendel's dihybrid F2 gave 315 round yellow, 108 round green, 101 wrinkled yellow and 32 wrinkled green. Does that fit 9:3:3:1?
The total is 556, so the expected counts are , twice, and . Then
On three degrees of freedom that has a probability of about 0.93. The fit is excellent, and two of the four F2 classes, wrinkled yellow and round green, are combinations that did not exist in either grandparent. Independent assortment does not merely preserve variation; it manufactures new combinations of it, which is a second thing Darwin needed and could not supply.
Now you. Independent assortment is not a general law. What breaks it, and does the breach damage the argument above?
Answer
Genes sitting close together on the same chromosome are inherited together and do not assort independently. William Bateson and Reginald Punnett found ratios badly departing from 9:3:3:1 in sweet peas around 1905 and could not explain them; Thomas Hunt Morgan's group explained them from 1911 as linkage, with the frequency of recombination between two loci measuring the distance between them. Mendel's seven characters actually map to only four of the pea's seven chromosome pairs, so some of his pairs were on the same chromosome and happened to be far enough apart to assort nearly freely. The breach does not damage the argument. Linkage reduces the rate at which new combinations are produced without abolishing it, because crossing over reshuffles even linked loci, and the essential point, that factors are discrete and are not diluted, is untouched.
The data are too good
There is an awkwardness about Mendel's numbers that an honest account has to include. In 1936 R. A. Fisher, who admired the work and reconstructed the whole experimental programme, showed that the agreement between Mendel's observed counts and his expected ratios is closer than sampling would ordinarily produce. Combining the chi-squares across all the experiments Fisher obtained a total of about 41.6 on 84 degrees of freedom, where the expected value of a chi-square is its degrees of freedom. Deviations that small arise by chance far less than one time in ten thousand.
Fisher also identified a specific technical problem. To distinguish a true-breeding round F2 plant from a segregating one, Mendel raised its offspring and looked for any wrinkled seed. With ten seeds scored, a segregating plant has a probability of producing no wrinkled seed at all and being misclassified. That shifts the expected ratio of segregating to constant plants from 2:1 down to about 1.70:1. Mendel reported 372 segregating to 193 constant, a ratio of 1.93, which sits close to the naive expectation and away from the correct one.
What to conclude is disputed and the honest answer is that we do not know. Proposals include unconscious bias in scoring ambiguous seeds, a gardening assistant who knew what was wanted, stopping data collection when the ratio looked right, and Fisher having mismodelled the number of seeds actually scored. Nobody has suggested the conclusions are wrong: the ratios have been reproduced thousands of times since, in peas and in everything else. The episode is a good illustration of the difference between a result being true and a data set being trustworthy, and of the fact that the two can be separated by experiment.
Rediscovery, and a thirty-year war
Mendel's paper was distributed to about 130 institutions and cited a handful of times before 1900, when three botanists, Hugo de Vries, Carl Correns and Erich von Tschermak, published segregation ratios independently and found the paper in the literature. William Bateson read it on a train to London, abandoned his lecture notes, and became its advocate in Britain.
The expected outcome would have been the immediate union of Mendel with Darwin. What happened instead was twenty-five years of hostility. The Mendelians, led by Bateson and by de Vries, studied characters with two sharply distinct states and concluded that evolution proceeds by discrete jumps: de Vries's mutation theory, based on abrupt new forms appearing in the evening primrose, held that new species arise in a single step and that selection merely weeds out the failures. The biometricians, led by Karl Pearson and W. F. R. Weldon, studied characters like height, weight and beak size, which vary continuously with no distinct classes at all, had built statistics in order to measure the resemblance between relatives, and pointed out that these characters are the ones selection actually works on in nature.
Each camp was right about its own data and wrong about the other's, and the dispute was bitter enough to be personal. It looked like a real contradiction: if heredity comes in discrete units producing sharp ratios, where does smooth continuous variation come from, and how can selection move a population by small degrees if the underlying units come only in whole numbers?
The resolution is arithmetical rather than experimental, and it needs a shift of attention from the family to the population. That shift, made in a single page of Science in 1908, is the subject of the next lesson.