290 AN INTRODUCTION TO MODERN GENETICS 



which can be solved to give 



kn= u„- Uq-\- log, ( -^ 



If we consider the population to have originally consisted entirely of 

 A A, Uq = 1 and kn = u„ -{- log^ «„ — i . If this equation is plotted, it 

 will be found that selection is very slow for, or of course against, rare 

 recessives. 



Haldane has worked out similar equations for many different types of 

 selection and different types of inheritance and breeding. In nearly all 

 these investigations the method is in principle the same as that shown 



Fig. 127. The Rate of Natural Selection. — The table gives the number of genera- 

 tions required for a given change in proportions of the dominant, it being supposed 

 that the dominants have 1,000 offspring for every 999 of the recessives (k = 0,001). 

 Notice that the change is very slow when recessives are rare. Exactly the same 

 figures are obtained if selection is against the dominant, but the table must then 

 be read from the right towards the left; e.g. it will take 309,780 generations to 

 decrease the dominants from 99,999 to 99 per cent. 



Proportion of dominants: 0,001-1% 1-50 °o 50-99% 99-99,999% 

 Autosomal gene - 6,920 4,819 11,664 309,780 



Sex-linked - - 6,916 4,668 5,593 10,106 



(in homogametic sex) 



6,928 5,164 11,070 20,693 



(in heterogametic sex) 



above; a finite difference equation is derived relating the value of w in 

 two successive generations, and then some method is found of solving 

 these equations, usually with the help of approximations which only 

 hold when k is small. The only other example we have space to mention 

 here concerns the effect of mutation. 



Natural selection is usually balanced by natural mutation; in fact, 

 perhaps the most common function of natural selection is to eliminate 

 harmful genes which otherwise would quickly spread throughout a 

 species. If a recessive gene a has a selective disadvantage ky and the 

 mutation rates A-^ a = p and a -^ ^ = ^, we have in the w + ith 

 generation 



^ A genes ^ (u„^ + u„)(i - p) + (u„ -{- i - k)q 

 "" + ^ " a genes («„ + i - k){i - q) -^ p{u^ + wj 



hi 



