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Average recombination frequencies.

ODELS in population genetics and quantitative inheritance often include, as unknown parameters, the recombination fractions between pairs of interacting loci. For example, GRIFFING (1960) demonstrated that the expected response to mass selection includes terms which are functions of the recombination frequencies between all pairs of relevant loci. Since these linkage parameters are unknown even in the most intensively studied quantitative traits, GRIFFING used DARLINGTON’S (1939) concept of the recombination index to estimate the average degree of recombination between randomly chosen pairs of loci. GRIFFING (1960), HANSON ( 1 962), and SVED (1 964) used combinatorial methods to obtain expressions for the average recombination frequency, and it is the purpose of this note to demonstrate that their results can be derived more simply using the following argument. Consider a chromosome with map length 1 (morgans), and suppose that the number of loci per map unit is large and constant over the length of the chromosome. Let two loci map at positions x1 and zz such that Izl zgl = z. The recombination fraction ( y ) will be some function of z, say g ( z ) ; and the average recombination frequency is found by summing g ( z ) over all pairs of relevant loci. The average recombination fraction is a function of the number of loci and the recombination frequencies among pairs of these loci. In general we know none of these parameters and have to make assumptions about their values. One way of looking at the problem is to ask what is the expected recombination value between a pair of loci which have been randomly chosen from all of the possible sites on the chromosome. It is assumed above that these sites are a sample from a uniform distribution with range ( O , l ) , and the average recombination frequency is given by the integral