38 



MUTATION AND PLANT BREEDING 



below). In other words, the equilibrium is possible only when the 

 fitness of the heterozygote, w 2 , is greater or smaller than those of both 

 homozygotes. One way of representing the former situation is as 

 follows: 



Wi = 1-/. | 



( w-2—Wi = t, \ 

 w 2 = 1, ; w z — Wi = —s + I 



{ W2—W3 — s, ) 

 w 3 = 1—5, j 



Substitution of these values in the general expression for equi- 

 librium yields 



t—F s s—Ft 



q=- — , p 



(l-F)(s + t) (I -F)(s + ty 



where F must be smaller than t/s and s/t, one of them being a frac- 

 tion. To facilitate the appreciation of this type of selectional balance, 

 a numerical illustration is provided in Table 3. Even without the 

 details of mathematics, it is quite obvious that this equilibrium is 

 stable. Since Aa has the greatest fitness, neither gene A nor gene a can 

 be eliminated from the population. Furthermore, if heq(A) or freq(fl) 

 is too low, the greater fitness of Aa tends to raise it to a certain level 

 and vice verso. The greater fitness of Aa and the loss of A A and aa are 

 balanced in such a way that the gene frequency remains unchanged. 



It was mentioned previously that any set of three numbers pro- 

 portional to w 1 :w 2 :w3 will yield the same result. Since 80:100:70 = 

 100:125:87.5, the latter three numbers have been used in the lower 

 portion of Table 3. The results are the same, of course. 



On the other hand, if the heterozygote fitness value is lower than 

 that of either homozygote, the equilibrium is unstable. This, again, 

 should be obvious. The elimination of a certain number of Aa from 

 the population implies equal loss of the number of A and a genes. 

 This inflicts proportionally a greater loss of the gene whose frequency 

 is already lower than that required for equilibrium, and hence its 

 frequency will decrease further. The frequency of the more common 

 gene will, in turn, increase. Consequently, selection against heterozy- 

 gotes will lead to near elimination of one of the alleles (to be kept at 

 a very low level by new mutations). Unstable equilibria are not 

 expected to exist in nature. 



