SYSTEMS OF MATING 



37 



1 — 4£ + 2k 2 :2k 2 in the next generation. Finally, the intercrosses of the second 

 kind produce four kinds of progeny and all five kinds of matings in the ratio (1/2) 

 (1 - 2£) 2 :(1/2)(1 - 2k) 2 Ak(\ - 2k) :2k 2 :2k 2 in the next generation. 

 These results may be shown as a set of five linear equations : 



Ai+i =P»+ (l/4)r„ + 2k 2 v n + (1/2)(1 - 2k) 2 w n , 



<7n + i = 2k 2 v n + (1/2)(1 -2k) 2 w n , 



r n+ i = (l/2)r„ + 4*(1 - 2k)v n + 4A(1 - 2k)w n , 



v n + 1 = (l/2)y B + (1/4)(L- 2k)r n + (!_« + 2A a )i/ B + 2^ n5 



(l/2)?n + 



:i/2)Ar B + 



2* 2 * n + 



2£ 2 w r 



Fig. 11. Probability of incrosses with heterozygosis forced by intercrossing. 



1.0 



PROBABILITY 



4 6 8 10 

 GENERATION, n 



The probability of incrosses p n for selected values of c = 0, 1/10, 1/2 for the system of 

 brother-sister inbreeding with heterozygosis forced by intercrossing, starting with q = 1. 



The probabilities p n of incross matings, the desired type, for selected values of c 

 are shown in figure 11, assuming that q = 1 . This is the case if the mutation of interest 

 arose in a noninbred stock and the initial mating between carriers of the mutation r is 

 AA x aa. This is the least favorable starting point with respect to the value o£p n ; other 

 starting points will produce higher values of p n in fewer generations. The nonlinked 

 loci become fixed at the usual rate for brother-sister inbreeding. Linked loci reach 

 fixation more slowly at rates dependent upon c. When c is near zero, about 2/3 of the 

 matings will remain as intercrosses of the first kind and 1/3 become incrosses of the 

 desired type (v n = 2/3 and^ n = 1/3 when n is infinite). 



The probability of heterozygosity decreases depending upon c, as seen in figure 12, 

 except when c = 0. In general, 



h n = (l/2)r n + v n + w n . 



The determinant formed from the generation matrix by omitting the p n + 1 row and 

 the p n column yields the general equation : 



// 4 - (3 - 4/ + 2l 2 )/n 3 + (1 - U + 4l 2 - 4/ 3 )// 2 



+ (3 - 10/ + 14/ 2 - 12/ 3 + 8/ 4 )/* - 2(1 - 5/ + 10/ 2 - 10/ 3 + 4/ 4 ) = 



