SYSTEMS OF MATING 



29 



incrosses to begin the next cycle. The CROSS-crosses, probability q m , produce back- 

 CROSS-backcrosses, and these produce intercrosses of three kinds, incrosses, back- 

 crosses, and intercrosses, in the ratio c 2 :2c(\ —c):(\ — c) 2 . These in turn yield 

 crosses of three kinds to begin the next cycle, incrosses, crosses, and backcrosses, in the 

 ratio c 2 (2 — r) 2 :(l — c) 4 :2r(l — r) 2 (2 — c). The CROSS-backcrosses, probability r m , 

 produce BACKCROss-incrosses and BACKCROSS-backcrosses in the ratio 1/2:1/2. The 

 BACKCROSS-incrosses yield iNTERCROSS-incrosses and thence CROSS-incrosses to begin the 

 next cycle. The BACKCRoss-backcrosses yield intercrosses of three kinds, incrosses, 

 crosses, and intercrosses, in the ratio {\j2)c 2 :c{\ — c) : (1 /2) ( 1 — c) 2 ; and these in turn 

 yield crosses to begin a new cycle with incrosses, crosses, and backcrosses, in the ratio 

 (1/2V 2 (2 - ^ 2 :(1/2H1 - r) 4 :r(l-r) 2 (2 - c). 



Fig. 8 



. Probability of incrosses for the cross-backcross-intercross system. 



I. Or 



PROBABILITY 

 Pm 



CYCLE m 



The probability of incrosses p m for the cross-backcross-intercross system beginning with 

 q = 1," for values of c = 1/10, 2/10, 3/10, 4/10, 5/10. The ratio of the probabilities of 

 heterozygosity in successive generations is constant, h m + 1 /h m = (1 — c) 2 , when m > 1. 



The probabilities of mating types in C m + 1 may thus be shown as functions of the 

 probabilities of mating types in C m : 



Pm + l = Pm + 



q n +\ = 



c 2 (2 - c) 2 q m + {(1/2) + (l/2)c 2 (2 - c) 2 }r m , 

 (1 -cYq m + (1/2) (1 -c)*r n , 



2r( l _ C ) 2 {2 - c)q m + 



r(l - c) 2 {2 - c)r n 



Again the familiar result emerges. The incrosses, probability p m , increase at the expense 

 of the backcrosses, q m , and the intercrosses, r m . The rate of increase of/? m depends upon 

 c, the probability of crossing over between the a-locus and the r-locus. 



When the initial mating is a cross, that is, q = 1 , the desired types of matings 

 increase as shown in figure 8, for selected values off. 



When c = 1/2 and q — 1 , the successive values of p m are 



0, 9/16, 57/64, 249/256,..., 



