Some general formulations of linkage effects in inbreeding.

effect of a given system of consanguineous mating on the degree of homoTrEgosity of resulting individuals is conveniently described by means of the coefficient of inbreeding (WRIGHT 1921 ) . The coefficient of inbreeding, F, of an individual may be defined to be the probability that the two genes possessed by that individual at a locus are identical by descent (MAL~COT 1948; KEMPTHORNE 1957). The probability that with respect to a given set of n loci the two gametes of an individual are occupied throughout by pairs of genes identical by descent is equal to F" provided that the loci in question are not linked. A more general formulation of this probability, allowing for arbitrary linkages, is needed in certain problems of population genetics. Such problems include the interpretation of covariances between relatives (COCKERHAM 1956). The present article gives several relations which may be useful for formulating and applying the required generalization of the coefficient of inbreeding. Gametic frequencies and recombination values: We shall have to employ recombination values as well as frequencies of modes of gamete formation. Therefore any general connection between those two categories of probabilities would be helpful. It has been stated by several writers (see for exampIe GEIRINGER 1944, OWEN 1950) that the gametic frequencies cannot be expressed entirely as linear functions of recombination values, if the number of loci involved is greater than three. This obstacle, however, can be surmounted by introducing new classes of recombination values. Consider four linked loci, i, i, k, I , arranged in this order on the chromosome. A diploid individual produces 24 = 16 different types of gametes with respect to those loci. The probability that the individual transmits to its progeny a certain gamete will be denoted by y followed by a four-digit subscript, where the digits refer to the corresponding loci in the sequence given above. The figures 0 and 1 will be used to distinguish the two cases that a locus carries the maternal or the paternal gene, respectively. Thus yoloo is the probability that a transmitted gamete contains the maternal genes at loci i, k, I, and the paternal gene at locus i. Recombination values will be denoted by p with subscripts specifying the loci involved, i.e. p i j , etc. Since the gametic frequencies must add to unity, and owing to obvious sym-