(Letter to Science, 1908) 17kB, PDF

JULY 10, 1908 SCIENCE N. S. Vol. XXVIII:49-50

Available from: Electronic Scholarly Publishing http://www.esp.org

DISCUSSION AND CORRESPONDENCE

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 Medicine (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 p1:2q1:r1, say.

The interesting question is – in what circum-stances 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 q = p1r1, 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 propor-tions

which will, of course, occur in every genera-tion. Such a distribution as p1:2q1:r1, which satisfies the condition q = 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 p:2q:r, which is not "stable."

This should, according to theory, give us in the third generation a "stable" distribution p2:2q2:r2, also differing 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 effects 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 that [sic] 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 by Mr. Yule’s remarks.

G. H. Hardy

Trinity College, Cambridge,

April 5, 1908

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).

Reprinted from Hardy, G. H. 1908. Mendelian proportions in a mixed population, Science, N. S. Vol. XVIII:49-50. (letter to the editor)

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