Supplementary materials for An Investigation of the Power for Separating Closely Linked QTL in Experimental Populations Genotypic Distributions of Three and Four loci under Self and Random Mating for the Derivation of Correlations between Pairwise QTL

For any pair of QTL, they can be located in neighboring or non-neighboring marker intervals. For neighboring case, distribution of three anking marker loci is needed to obtain the correlation between the two QTL. For non-neighboring case, distributions of four markers is required to obtain the correlation. In general, there are 2 m diierent gametic genotypes and 2 2m;1 + 2 m =2 zygotic genotypes for m loci in the populations from two i n bred lines. For example, there are 4, 8 and 16 gametic genotypes and 10, 36 and 136 zygotic genotypes for m = 2, 3 and 4. As diierent populations will undergo various number of meiosis cycle, the distributions of gametic and zygotic genotypes will vary. F or sellng, Haldane and Waddington (1931) formulated the transition equations of the ten genotypic frequencies for m = 2. Kao and Zeng (2009) obtained the transition equations of the 36 genotypic frequencies for m = 3. The procedures of obtaining transition equations for m = 4 a r e given below. Let 1 and 0 represent the capital and small-letter alleles, respectively, from P 1 and P 2 , so that the conngurations of the 16 gametes can be r 2 and r 3 are the recombination rates between the rst and second genes, between the second and third genes, and the third and four genes, respectively. The random uniication of these 16 gametes frequencies will produce the 136 diierent zygotes in a population. Because of symmetry, the complementary pairs have the same frequencies, there are 72 diierent g e n o t ypic frequencies. Let 1