Am. Nat. 73: 185-188 (1939)


STUDIES of hybridization in natural populations have shown (Riley, 1938; Delisle, 1937; Goodwin, 1937; Anderson and Hubricht, 1938; Anderson and Turrill, 1938) that the characters of the parental species tend to stay together in such populations. While this had been predicted (Anderson, 1936) from a consideration of the linkage of multiple factor characters with each other, the phenomenon is of such practical and theoretical importance as to deserve rigid mathematical exposition. The following paper attempts to estimate the hindrance to free recombination imposed upon hybrids by gene linkage alone. It does not consider the effect of such other restraints as pleiotropy, selective fertilization, zygotic elimination and gametic elimination.

It has been immediately apparent to those who have considered the question that linkage greatly reduces the chances of recombination. Jones (1920), for instance, has computed the chances of recombining favorable genes in maize. "Two factors in each chromosome so spaced as to have 10 per cent. breaks in the linkage with each other would necessitate 2020 individuals in the segregating generation to have an even chance of securing the one plant desired. This number of corn plants would require an area roughly 3,700,000,000,000 times the area of the United States."

The restraint of linkage, however, is not confined to frequencies. It also imposes severe restrictions upon the kinds of gene combinations which are possible with any frequency. When all the loci of a germ-plasm are considered this restriction is even greater than that imposed upon frequencies and runs into figures of astronomical magnitude. Some notion of this restriction may be gained by considering recombination in a single cross-over segment of the germ-plasm. Let us take the simple case of a short chromosome in which there is regularly a single cross-over. Let us further suppose that in the two species, or races, which are to be crossed, there are ten pairs of gene differences within this chromosome. This seems a conservative number for a length of germ-plasm which might well be fifty units long genetically and made up of two hundred or more genes.

In the gametes of the first generation hybrid, as a result of four-strand crossing-over, one half of the gametes will have one crossed-over section in this chromosome and the other half will have none. The number of cross-overs per chromosome will be increased the same way in each generation; double cross-overs will not be possible until the F2 generation forms its gametes, triple cross-overs until the F3, etc. In each generation one half the gametes will acquire an extra cross-over, one half will continue the previous number. The number of cross-overs per gamete and the proportions of each kind of gamete can therefore be obtained from expanding (1/2 + 1/2)n where "n" equals the number of hybrid generations. For the ten gene pairs under consideration complete recombination can not be attained until gametes are produced in which all nine breaks between the original sets of ten differing gene pairs have occurred. To obtain such a gamete will require a minimum of nine hybrid generations, and even then these gametes may be expected only once in 29 (= 512). It will require twice as many hybrid generations before gametes of this degree of recombination will be in the majority. The above figures have all been obtained on the hypothesis of random crossing-over along the chromosomes. Actually, of course, cross-overs would tend to recur in particular areas and thus greatly reduce the possibility of complete recombination.

A more precise estimate of the hindrance to recombination can be obtained by considering the ratio of the possible gene combinations in the F1 to random combination. With three pairs of differing loci, abc/ABC, there can be a cross-over between the "a" locus and the "b" locus and between the "b" and the "c." Each of these will permit two recombinations, viz., aBC, Abc; and abC, ABc. The total number of recombinations will therefore be equal to twice the number of gene abutments or 2(n-1) where "n" equals the number of differing gene pairs. With the two original combinations the total number of kinds of gametes will be 2n. Since the total number of possible combinations, were it not for the restrictions imposed by linkage, is given by 2, the ratio we are seeking will be 2n/2n. For three pairs of gene differences this becomes 3/4; for four pairs 1/2; for ten pairs 10/512, or less than 2 per cent.

Since the same principle will be operating in every cross-over region (tempered only by the occurrence of multiple crossing-over) the total hindrance in the entire germ-plasm will be enormous. An estimate can be obtained by considering the not impossible case of an organism which regularly has a single chiasma in each chromosome. For such an organism the ratio of the possible kinds of gametes to the total number of recombinations will be (2n/2n)N where "n" equals the numbers of differing loci per chromosome and "N" is the number of pairs of chromosomes. For even such a slight difference as four genes per chromosome and with only six pairs of chromosomes this ratio becomes 1/64. For ten gene differences per chromosome and with ten pairs of chromosomes it becomes (10/512)10 or roughly less than one in 100,000,000,000,000,000.

It should be emphasized that this restriction is independent of the size of the F2 and constitutes an absolute upper limit to gene recombination in that generation. The ratio (10/512)10, inconceivably small though it may be, represents the fraction of the total combinations which could be achieved in a population of infinite size. In any actual F2 the additional restrictions of combination frequencies will reduce the actual gene combinations to a fraction of this infinitesimal fraction.

It may therefore be predicted that linkage alone will greatly hinder recombination in species crosses, or in any cross where there is a considerable number of genes involved. Even with small numbers of gene differences in each cross-over segment, the possible recombinations among the gametes of the F1 will be only a fraction of the total imaginable combinations. With any considerable number of gene differences the possible combinations will be only an infinitesimal fraction of the total combinations, and numerous hybrid generations will be required before there can be anything like complete recombination of those gene differences which have survived. These theoretical considerations suggest that the conditions in the hybrid populations which have been studied are general phenomena.