Mendelian proportions in a mixed population. 1908.

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.O. 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 thatAa 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 asp1: 2q1: rl, say. The interesting question isin 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 = pir1, whatever the values ofp, 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 = 10,0,00 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