Li: MUTATION, SELECTION, AND POPULATION FITNESS 41 



Probably in most cases, if not in all, natural selection favors not the 

 extremely large or the extremely small, but some intermediate value. 

 This value may then be called the optimum size with respect to fitness 

 and be denoted by x . In any realistic situation in which the metrical 

 trait is controlled by a number of loci, the optimum size x is prob- 

 ably close to but does not necessarily coincide with the mean size 



x of the population. 



One way of representing the fitness that decreases with large as 

 well as with small sizes is to let 



w = 1— c(x— x ) 2 



where c is a constant. That is, the selection coefficient against an 

 organism of size x is proportional to the square of the deviation of x 

 from optimum value. Suppose that the size of the three genotypes 

 (AA, Aa, aa) are %i = 11, x 2 = 10, x 3 = 1, respectively, and that the 

 optimum size is x = 8. Then the fitness of the three genotypes will be 

 (taking c = 0.01) 



Wi = 1 — c ( 1 1 -8) 2 = 0.91 



Wz = 1 _ c (i0-8) 2 = 0.96 



iv 3 = 1 _ C ( 1 -8)2 = 0.51 



Such a set of fitness values will lead to a stable equilibrium, as 

 shown in section 4. If the optimum size is x = 4, the three fitness 

 values would be 0.51, 0.64, 0.91, leading to fixation of the small size. 



For a metrical trait determined by a number of loci, the situation 

 is very complicated and selection for an optimum size does not always 

 lead to stable equilibrium. Wright (14) found no stable equilibrium 

 when there is no dominance or complete dominance with respect to 

 size at all loci. 



Kojima (6, 7) recently showed that stable equilibrium points do 

 exist when there is partial dominance or overdominance in size at all 

 loci, and proposed a method of locating such stable points. In nature 

 there must be a large number of metrical traits stabilized at certain 

 optimum levels. It is unfortunate that the mathematics involved in 

 this type of investigation should be so intricate that the most impor- 

 tant type of selection in nature has been also the least understood. 

 Kojima's finding must be hailed as a significant landmark in selection 

 genetics. 



In all of the previous examples there is only one equilibrium 



