SOME QUANTITATIVE ASPECTS OF EVOLUTION 



But there is an additional factor here, to be discussed below: there is 

 severe selection against genotype SS (sickle cell anemia). 



The Hardy-Weinberg Law, then, operates to maintain the status quo. 

 It is a conservative factor in evolution. In order to apply the formula to 

 evolutionary problems, it is necessary to take into account the factors 

 which might upset the equilibrium and cause a change in the relative 

 frequencies of the alleles. The principal calculable factors are mutation 

 and selection. As the mathematics of mutation and selection is rather com- 

 plicated, the details will not be presented here. References to the papers 

 of Fisher, Haldane, and Wright may be found in Dobzhansky's book. 



Selection Pressure and Rates of Evolution. Calculations of the effect 

 of selection are most easily made where selection favors a gene with com- 

 plete dominance. If 1000 AA or Aa survive for every 999 aa that survive, 

 the dominant form may be said to be favored by a positive selection pres- 

 sure of 0.001 ( or conversely, aa may be said to be subject to a negative 

 selection pressure of the same magnitude). Thus, selection will modify 

 the Hardy-Weinberg equilibrium by this small but calculable factor in 

 each generation. Haldane has calculated the results of such a selection 

 rate upon populations with varying initial proportions of the favored dom- 

 inant gene. He found that the rate of increase in the proportion of the 

 gene, in large populations, would be extremely slow when the initial pro- 

 portion of the favored dominant is either very low or very high. But, when 

 the initial proportion of the gene is moderate, the increase may be quite 

 rapid. Thus, he found that it would require 11,739 generations to increase 

 the proportion of a dominant gene from 0.000,001 to 0.000,002 (1 in a 

 million to 2 in a million) with a selection pressure of 0.001. But a change 

 of gene proportion from 0.00001 to 0.01 would require only 6920 gen- 

 erations; and the change from 0.01 to 0.50 would require only 4819 gen- 

 erations; but the change from 0.990 to 0.99999 would require 309,780 

 generations. It appears then that it is extremely difficult for mild selection 

 pressure, unaided by any other factor, to establish a new dominant gene 

 in a species, or to bring a well established dominant to 100 per cent inci- 

 dence ("fix" it) in a species. But such pressures may readily result in a 

 great increase in the relative proportion of an already well established 

 gene. If selection favors a recessive gene, then the process is similar, but 

 much slower. The initial step from 0.000,001 to 0.000,002 would now re- 

 quire 321,444 generations. 



Purely mathematical studies of this type, and studies on experimental 

 populations, are not uncommon, but their application to natural popula- 

 tions is more difficult. Kurten has, however, reviewed the history of a pair 

 of alleles over the past million years. In bears, the growth of the first 

 upper molar is allometric (see below), that is, growth in height is more 

 rapid than growth in length. The degree of allometry is genetically con- 

 trolled, being quite marked in some bears, rather moderate in others. In 

 any, the larger the bear, the higher the crown of the molar in relation to its 

 length (Figure 98). The more extreme allometry is found in modern 

 bears, Ursiis arctos, while a more moderate degree characterized the late 

 Pleistocene cave bear, U. spelaeus, now extinct. Both types of allometry 



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