46" GENETIC STOCKS AND BREEDING METHODS 



The larger root, (1/4)(1 + V5) = 0.8090, is the desired ratio. In more complicated 

 cases it is convenient to derive this from the characteristic equation of the P matrix. 



X x = (1/4)[1 + V5] = 0.8090 as the larger root. 



This 2 x IP matrix compares with a 6 x 6 matrix of mating types assuming- 

 multiple alleles in both cases or with the 4 x 4 matrix of mating types with respect to 

 pairs of alleles as in Green and Doolittle's analysis. The P matrix is usually of lower 

 order than the mating-type matrix, but the root (largest) that gives the limiting ratio, 

 PjP', is always exactly the same. 



The case of a population of limited size (N m males, N f females in each generation) 

 in which every mating is of the type Dd x Dd in the locus of interest, will now be 

 considered. The objective is the ratio, PjP' for linked genes, A,a. For the sake of 

 simplicity, it will be assumed that these are related symmetrically to D and d in the 

 total population. Let c and c s be the amounts of recombination in oogenesis and 

 spermatogenesis respectively; figure 15 shows the relation between a sperm carrying 

 D, (D s ) and an ovum carrying d, (d ) . Gamete D s may derive its A allele from either 

 of the uniting gametes of the preceding generation but with probability (1 — c s ) from 

 the D gamete and probability c s from the d gamete. These are then the path coefficients 

 in these cases. Similarly the A allele of gamete d is related to the d gamete of the 

 preceding generation by a path coefficient with the value 1 — c and to the D gamete 

 by one with value c . The correlation between the A alleles of D s and d is designated 

 r s0(JV) in which the subscript N indicates nonidcntical D alleles. The correlation between 

 uniting gametes (F) is of this sort. The correlation between A alleles of sperm and egg 

 that carry identical D alleles (r soU) ) and the corresponding correlations between two 

 spermatozoa (r ss(N) , r ss0) ) and between two eggs {r 00(N) , r ooU) ) are also of concern. 

 Because of the postulated symmetry of the system of mating, it makes no difference 

 whether a parent came from a union of D sperm with d ovum or the reverse. Thus the 

 correlation between A alleles both associated with D or both with d is the average of 

 r ss(/)> r so(n, and r 00(I) with weights of 1/4, 2/4, and 1/4 respectively. 



r, = (l/4)|/ ss(7) + 2r s0(/) + r 00(I) \ 

 Similarly 



r N ~ ( * /4) |/ss(JV) + ^ r so(N) 4" r oo(.N)\' 



On tracing the connecting paths in figure 15 it is found that 



'sow = F = [(1 - c s ){\ - c ) + c s c o y N + 0,(1 - c ) + (1 - c s K]r;. 

 If d is replaced by D , coefficients c and (1 — c ) must be exchanged; and, if D s is 

 replaced by d s , c s and (1 — c s ) must be exchanged. Union of d s with D gives the same 

 result as above. 



r so(n = [(* - «t)0 - c o) + WoVi + 0,0 - O + (1 - c s )c o y N . 



