Chromatid segregation of tetraploids and hexaploids.

N an earlier paper (GEIRINGER 1948) the author has investigated the matheI matical genetics of autopolyploids if one locus is considered and chromosome segregation is assumed. A paper on the linkage theory of autopolyploids (m loci), under the same assumption, is in press (GEIRINGER 1949). However, while chromosome segregation might be assumed as an approximation theory, the study of polyploids should actually be based on the consideration of chromatid segregation. In the case of one locus this theory can be worked out and is presented in this paper as far as random mating is concerned. This study, for m = 1, as well as the approximation theory for general m, may serve as a necessary preparation for a general linkage theory of polyploids under chromatid segregation. A basic paper by J. B. S. HALDANE (1930) is mainly concerned with the chromosome segregation theory of polyploids; “random chromatid segregation” of tetraploids is briefly discussed too and a limit formula (our formula (24)) follows. In a paper of a more recent date R. A. FISHER (1947) deals with the linkage theory of polyploids under chromatid segregation. The paper uses important earlier investigations by the same author (1941, 1944) as well as a paper by FISHER and MATHER (1943) and papers by MATHER (1935, 1936). As in previous papers it is our aim to consider a heredity problem like the one in question as a probability problem. From certain basic probability distributions which must be known other genetical distributions are derived by means of probability calculus. The fate of these distributions, from generation to generation, is the main subject of the mathematical investigation. If we assume distinct, non overlapping generations, numbered 0, 1, . . we may denote by w ( ~ ) ( x ; y) the distribution of genotypes in the nth generation. Here x denotes the genetic material the individual has received from its mother and y designates the paternal heritage. We assume that there is no genotypic difference between an individual of type (x; y) and one of type (y; x); that is: (x; y )= (y ; x). Accordingly wcn)(x; y)=w(”)(y; x); and^,^, w ( ~ ) ( x ; y) = 1. We may assume that a t the beginning, for n =0, the distributions of genotypes are the same for males and females; if they are not the same, that is, if w(O)(x; y) is the genotype for females and W(Q(x; y) the one for males, they will be the same after one generation of random breeding. In the formation of a new individual a parent transmits to the offspring one set of genes, the other set coming from the other parent. The kinds of gametes which an organism may produce correspond to the possible combinations of the genetic material it has inherited. This segregation takes place according to a * . n,