SOME QUANTITATIVE ASPECTS OF EVOLUTION 



way. If rate A-^a = u and a— >A = v, the frequency of a at equilibrium = 



q = — ; — . For example, if the rates are equal, then q = -, — —7 = ;:, or 



equilibrium will occur when the frequencies of A and a are equal. If 



2 

 u = 2v, then a is being formed twice as rapidly as A, and q = = 



2 



■^, or equilibrium will be reached when two-thirds of the genes are a. If 



u = 4v, then 80 per cent of the genes will be a at equilibrium. 



The sickle cell trait is also apropos here. The Hardy-Weinberg formula 

 gave an expectation of 0.64 ss : 0.32 Ss : 0.04 SS in the African tribes stud- 

 ied. Random breeding should give normals and heterozygotes in a ratio 

 of 2 : 1, or 0.67 to 0.33, with SS being subject to 100 per cent negative 

 selection. This should cause a drop of 7 per cent in the frequency of S 

 (and a similar rise in 5) in one generation. Thus S should be rapidly elim- 

 inated, yet the population appears to be in equilibrium. The reason is 

 that selective elimination of ss individuals by malaria gives a selective 

 advantage to the genotype Ss, which confers resistance to malaria, in spite 

 of the severe selection against homozygotes for S. 



In nature, neither mutation nor selection will ordinarily occur alone, 

 and so the two will act simultaneously, perhaps in the same direction, 

 perhaps in opposite directions, to upset the Hardy-Weinberg equilibrium. 

 Most frequently, selection will work against mutation, as the majority of 

 possible mutations are deleterious. This will result in very slow change, 

 if any. But if a particular mutation is favored by selection, and if its muta- 

 tion rate is appreciable, the combined action of mutation and selection 

 might well cause a rather rapid change. 



Haldane's calculations showing the extremely slow rate at which small 

 selection pressures could establish a new mutant or fix one which is al- 

 ready well established have been referred to above. This type of calcu- 

 lation is more than an exercise in statistics, because of the importance of 

 mutations of quantitative genes ( see Chapter 13 ) . And very minor muta- 

 tions are unlikely to be subject to selection pressures of a much greater 

 order than that used in the calculations. But the differentiation of species 

 must involve the accumulation of a great many such differences, partly 

 simultaneously, partly in sequence, if the neo-Darwinian theory be cor- 

 rect. Dobzhansky * has pointed out that ". . . The number of generations 

 . . . needed for the change may, however, be so tremendous that the effi- 

 ciency of selection alone as an evolutionary agent may be open to doubt, 

 and this even if time on a geological scale is provided." It is partly for this 

 reason that Goldschmidt believed that the neo-Darwinian theory places 

 too great a burden upon natural selection, and hence that the work of 

 selection must be shortened by some other process, namely, systemic 

 mutation. 



* Dobzhansky, Th., "Genetics and the Origm of Species," 1st Ed., Columbia Univer- 

 sity Press, 1937. 



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