.34 GENETIC STOCKS AND BREEDING METHODS 



and its nonmutant allele R in another. A preparatory cross between the strains will 

 produce double heterozygotes, ARjar (or aR\Ar). When these are mated with the 

 mutant-bearing strain ar/ar (or ArjAr) the initial matings are backcrosses of the third 

 kind, i.e., /„ = 1 . The probability p of incrosses increases more rapidly with this 

 starting point (figure 9). 



The probability of heterozygosity is less informative than the probability of in- 

 crosses in this system of breeding. When it is desired to compute h n , however, it must 

 be observed that two probabilities are required, one for rr (or dd) and one for Rr (or 

 Dd) animals. They are: 



P(Aa\rr) = h' n = s n + v n , 



P(Aa\Rr) = h" n = r n + t n + v n . 



In general, the probability h' n of heterozygosity at the a-locus among the homo- 

 zygous mutants will be less than the probability h" n of heterozygosity at the a-locus among 

 the heterozygous mutants for all loci linked with the mutant locus. By forcing 

 heterozygosity upon the locus of interest, one also forces heterozygosity upon loci linked 

 with it. The probabilities, h' n and hi, for the first twelve generations, starting with 

 either a cross (q = 1) or a backcross (/ = 1), are shown in figure 10. 



If c is near zero, and if q = 1, the probability is approximately 2/3 that in a 

 given line the heterozygotes Rr will remain heterozygous Aa at the closely linked 

 a-locus. This is so because the probability t n of backcrosses of the third kind approaches 

 2/3 as n increases, while s n and v n each approach zero. The probability p n of incrosses 

 equals 1 — t n ; as n increases /> n approaches 1/3. 



If t — 1 and c is near zero, there is practically no chance of getting incrosses, 

 since backcrosses of the third kind with probability t n yield only backcrosses of the 

 third kind; 



to = h = t 2 = ■ ■ ■ = t n = 1 

 approximately. 



The determinant formed from coefficients of q n , . . ., v n in the equations for 

 q n+1 , . . . ., v n + 1 is expressible as an equation of the 5th order: 



(i 5 - (3 - 2c - 2k) ^ - c/u 3 + {(5 - 7c - 2*)(1 - 2k) + 4(1 - 2c)k 2 }/u 2 



- (1 - 2c)(\ - 2k)(\ - Ak)fi - 2(1 - 2c){\ - 2k) 2 = 

 where ju — 2X. 



When c = 1/2, the characteristic root is, as expected, the same as for brother-sister 



inbreeding, i.e., 



X x = (1/4)(1 + y/b). 



The characteristic root rises to one as c decreases to zero. The values of Xj for 

 selected values off are: 



c = 0.5 0.4 0.3 0.2 0.1 



X x = 0.8090 0.8135 0.8270 0.8514 0.8984 1. 

 It has not been possible to solve the equation 



|G - X,I| = (8) 



