24 GENETIC STOCKS AND BREEDING METHODS 



(or AA) in the ratio c 2 :2c(\ - c) : (1 - c) 2 . These in turn, when crossed to the inbred 

 strain, yield incrosses, crosses, and backcrosses with respect to the a-locus in the ratio 

 c 2 :{\ - c) 2 :2c{\ - c). Finally, the backcrosses, with probability r m , in the first genera- 

 tion of C m , each produce two kinds of offspring, ARIAr and AR/ar, or aR/ar and aRjAr, 

 in the ratio 1/2:1/2. These yield three kinds of matings in the second generation o 

 G m , incrosses, backcrosses, and intercrosses, in the proportions 1/4:1/2:1/4. 



The incrosses in the second generation of C m each produce one kind of rr progeny, 

 with relative frequency 1/4, and thus yield only incrosses in the first generation of the 

 next cycle, C m+1 . The backcrosses each produce two kinds of rr progeny, A A and Aa, 

 or aa and Aa, in the ratio (l/2)f : (1/2)(1 - e) and so yield incrosses and backcrosses 

 in the same ratio in the first generation of C m + x . The intercrosses produce three kinds 

 of rr progeny, AA, Aa, and aa, or aa, Aa, and AA, in the ratio (l/4)c 2 : (l/2)c(l - c) : 

 (1/4)(1 - c) 2 , and so yield incrosses, crosses, and backcrosses in the proportions 

 (l/4)c 2 :(l/4)(l - r) 2 : (l/2)t(l - c) in the first generation of C m + 1 . In summary, 

 the backcrosses in the first generation of C m yield incrosses, crosses, and backcrosses 

 in the first generation of C m + 1 in the proportions (1/4)(1 +c) 2 :(l/4)(l - c) 2 : 



+ (l/4)f 2 = 0/4)0 + <> 2 , 



0/4)(l -c) 2 = (1/4)(1 -c)\ 



+ o/2Mi -c) = (1/2)0 -<- 2 )- 



The probabilities of the various mating types in C m + x may then be represented as 

 functions of the probabilities in C m by three linear equations: 



A» + i = P m + c 2 q m + (1/4)(1 +c) z r m , 



7. + i = (1 ~c) 2 q m + (1/4)(1 -c) 2 r m , 



2c{\ -c)q m + (1/2)(1 -c 2 )r m . 



Again the incrosses are an absorbing barrier; once matings of like homozygotes are 

 reached, they hold their type in succeeding generations. At the same time fractions 

 of the matings of other types yield more incrosses which are thenceforward fixed, the 

 rate depending upon c. By means of these equations, the probability of the desired 

 type of mating (incrosses) may be computed for any number of cycles and for selected 

 values of c. The results for 6 cycles, or 12 generations, are outlined in figure 6, for the 

 case q = 1 , that is when breeding starts with a cross AR/AR x arjar, or aRjaR x ArjAr. 

 When c = 1/2 and q = 1, the probabilities p m of incrosses, the matings of the 

 desired type, are the successive values 



0, 1/4, 19/32, 203/256, 1835/2048,... 



form =0, 1,2, 3,4 



The probability h m of heterozygotes is 



h m = r„ . 



