464 FRED H. HULL 



Complete dominance 



^ = 1 , V = I at equilibrium 



k = 2 , V = 3/ 4 at equilibrium 



For the more general case where Pi and Pj are (not inbred) individuals in 

 a crossbreeding population, equations paralleling (2), (3), and (4) are second, 

 fourth, and second degree, respectively. The simplification obtained with 

 homozygous parents is reduction of the three functions to first, second, and 

 first degree, respectively, by removing dominance effects (allelic interactions) 

 from parent phenotypes Pj and Pj. Mendel found the simplification obtained 

 with homozygous parents to be of considerable value in his early studies of 

 monogenic inheritance. 



The Mendelian model (2), (3), (4) may be complicated with innumerable 

 kinds of interactions (epistasis) by simple, compound or complex transforma- 

 tions (log, anti-log, exponential, etc.) of (2), (3), and (4). It is not intended 

 to imply, however, that interactions of alleles must precede interactions of 

 non-alleles in living organisms. 



The estimate of bp for any tester parent line is independent of gene fre- 

 quencies of the other parent lines with respect to dominance interactions. If 

 obtained estimates of bp for the same tester with samples of weak and strong 

 lines respectively should differ significantly, the necessary interpretation 

 would seem to require some kind of interaction other than between alleles, 

 or that the lines were not strictly homozygous. 



Interpretation of bp, b^, etc., by the Mendelian model presented here will 

 not be biased by linkage of two loci if frequencies of ab, aB, Ab, and AB do 

 not deviate significantly from expectation from random recombination of 

 gene frequencies of the two loci with respect to all of the parent lines. Free 

 assortment of the two loci is then effectively simulated. But any union of two 

 unlike gametes must contain some cases of repulsion linkage close enough to 

 retain the aB and Ab combinations in high frequency through several gen- 

 erations. A sample of lines all derived directly by selfing from one heterozy- 

 gous parent plant may well contain many cases of repulsion linkage to simu- 

 late overdominance. This effect would not be counterbalanced by high fre- 

 quencies of coupling linkage of other pairs of loci. Lines within each of the 

 25 samples reported here are in most or all cases no more closely related than 

 plants within one or more varieties. 



Variations of d and k from locus to locus would contribute to total vari- 

 ance, but would not seem to impair seriously the validity of the estimates of 

 regression coefficients, nor of Pc when bp — 0. 



When all loci are aa or all loci are A A, Pi = Pj = Fi = F2 = x. With 

 this restriction (3) becomes a quadratic with roots equal to the phenotypes 

 at the two limits. The difference is n{2d), the genetic range, the denominator 

 of 62 = —k/nld. Values of k, calculated thus, for the nine cases where parent 



