G. H. HARDY
From Science, vol. 28, 1908, pp. 49-50.
| This short paper has more of the air of a kindly old professor gently
reprimanding an irrepressible student inclined to go off half-cocked than
that of a major contribution to genetic thought. Hardy had noted the tendency
of non-mathematically inclined biologists to make assumptions and draw
conclusions based upon erroneous interpretations of the statistics in Mendel's
work, and wrote this letter to the editor of Science to correct these errors.
The consequences of the paper have been quite far-reaching, however, for
it gave rise to the field of population genetics, which forms one of the
primary bases for the contribution of genetics to evolutionary thought.
Hardy, as a mathematician, did not differentiate between the individual
and the genes that individual carries, so he based his calculations of
frequency on the numbers of homozygotes and heterozygotes in the population.
Because of the redistribution of genes between individuals, his first generation,
which was made up entirely of "pure" individuals, differs in proportions
from his second generation, which includes heterozygotes. Geneticists soon
recognized that the constancy and stability Hardy observed after his second
generation existed equally in the transition from first to second, if one
compares the total number of "A" and "a" genes in the population,
rather than the numbers of different kinds of individuals. A direct consequence
of this awareness is the "gene-pool" concept, which is concerned primarily
with the total number of genes and their proportions in a population, and
not with the appearance of the individuals carrying those genes. From the
viewpoint that the number of genes in a gene pool tends to remain stable
and unchanging comes the concept of evolution defined as any situation
which tends to change the proportional distribution of genes in a gene
pool. Hardy perceived several of the factors that could affect the proportional
distribution, and pointed them out in his concluding paragraph. He missed
one of the primary forces, however, in that the fruit of Darwin's thought,
natural selection, is omitted.
The concept of stability of gene proportions in a population has come to be known as the "Hardy-Weinberg Law," as a consequence of another of those dramatic coincidences that were pointed out earlier, for Weinberg (Über den Nachweis des Verebung beim Menschen, 1908) pointed out the same facts at much the same time as did Hardy. This law still forms the core about which the field of population genetics revolves today. |
| To the Editor of Science: I am reluctant to intrude in a discussion
concerning matters of which I have no expert knowledge, and I should have
expected the very simple point which I wish to make to have been familiar
to biologists. However, some remarks of Mr. Udny Yule, to which Mr. R.
C. Punnett has called my attention, suggest that it may still be worth
making.
In the Proceedings of the Royal Society of Medicble (Vol. I., p. 165) Mr. Yule is reported to have suggested, as a criticism of the Mendelian position, that if brachydactyly is dominant "in the course of time one would expect, in the absence of counteracting factors, to get three brachydactylous persons to one normal." It is not difficult to prove, however, that such an expectation would be quite groundless. Suppose that Aa is a pair of Mendelian characters, A being dominant, and that in any given generation the numbers of pure dominants (AA), heterozygotes (Aa), and pure recessives (aa) are as p:2q:r. Finally, suppose that the numbers are fairly large, so that the mating may be regarded as random, that the sexes are evenly distributed among the three varieties, and that all are equally fertile. A little mathematics of the multiplication-table type is enough to show that in the next generation the numbers will be as (p + q)2:2(p + q)(q + r):(q + r)2, or as p1 :2ql :r1, say. The interesting question is-in what circumstances will this distribution be the same as that in the generation before? It is easy to see that the condition for this is q2 = pr. And since q12 = p1rl, whatever the values of p, q and r may be, the distribution will in any case continue unchanged after the second generation. Suppose, to take a definite instance, that A is brachydactyly, and that we start from a population of pure brachydactylous and pure normal persons, say in the ratio of 1:10,000. Then p = 1, q = 0, r = 10,000 and p1 = 1, q1 = 10,000, r1 = 100,000,000. If brachydactyly is dominant, the proportion of brachydactylous persons in the second generation is 20,001:100,020,001, or practically 2: 10,000, twice that in the first generation; and this proportion will afterwards have no tendency whatever to increase. If, on the other hand, brachydactyly were recessive, the proportion in the second generation would be 1:100,020,001, or practically 1:100,000,000, and this proportion would afterwards have no tendency to decrease. In a word, there is not the slightest foundation for the idea that a dominant character should show a tendency to spread over a whole population, or that a recessive should tend to die out. I ought perhaps to add a few words on the effect of the small deviations from the theoretical proportions which will, of course, occur in every generation. Such a distribution as p1:2q1:r1, which satisfies the condition q12 = p1r1, we may call a stable distribution. In actual fact we shall obtain in the second generation not p1:2q1:r1, but a slightly different distribution p1':2q1':r1', which is not "stable." This should, according to theory, give us in the third generation a "stable" distribution p2:2q2 :r2, also differing slightly from p1:2q1:r1; and so on. The sense in which the distribution p1:2q1:r1 is "stable" is this, that if we allow for the effect of casual deviations in any subsequent generation, we should, according to theory, obtain at the next generation a new "stable" distribution differing but slightly from the original distribution. I have, of course, considered only the very simplest hypotheses possible. Hypotheses other than that of purely random mating will give different results, and, of course, if, as appears to be the case sometimes, the character is not independent of that of sex, or has an influence on fertility, the whole question may be greatly complicated. But such complications seem to be irrelevant to the simple issue raised bv Mr. Yule's remarks. 'P.S. I understand from Mr. Punnett that he has submitted the substance of what I have said above to Mr. Yule, and that the latter would accept it as a satisfactory answer to the difficulty that he raised. The "stability'' of the particular ratio 1: 2: 1 is recognized by Professor Karl Pearson [Phil. Trans. Roy. Soc. (A), vol. 203, p. 60]. |