THE ORIGIN OF SPECIES 



are present in a population in a definite proportion, and if random mating 

 and equal viability of all genotypes obtain, then the original proportions 

 will be maintained in all subsequent generations, unless it is upset by 

 some other factor, such as mutation or selection. In mathematical terms, 

 the proportion of one allele. A, may be taken as q, the other, a, as 1 — q, 

 thus making the sum of their proportions 1. Then, in the F2 and in all 

 subsequent generations, the proportions of the possible genotypes will be 

 q~AA:2q{l — q) Aa:(l — q)~oa, and the proportions of the genes will be 

 q in the case of A and 1 —q in the case of a. Let us substitute figures now 

 for a monohybrid cross. In the cross AA X aa, q will equal 0.5, and 1 — q 

 will also equal 0.5, the meaning being simply that the alleles are present 

 in equal numbers in the experimental population. The expansion of the 

 binomial [q + {1 — q)]~ gives the Fo coordinates, q~ + 2q(l — q) + 

 (l — q)^, as stated above. Substituting numbers and genotypes, this 

 would be (0.5A + 0.5a) ~ = 0.25 AA + 0.50Aa + 0.25aa. As this is the fa- 

 miliar 1:2:1 genotypic ratio of the F2 of a monohybrid Mendelian cross, 

 it is clear that the elementary Mendelian ratios comprise special appli- 

 cations of the Hardy-Weinberg Law. 



The Hardy-Weinberg Law is just as applicable to problems involving 

 initial gene ratios not bearing any special relationship to standard Men- 

 delian ratios. If the frequency of A is 0.8 and that of a is 0.2, the expanded 

 formula would read (0.8A + 0.2a) ~ = 0.64AA + 0.32Art + O.Oiaa. The 

 sum of the frequencies of the several genotypes still equals 1, which veri- 

 fies the calculation. Now if this represented 50 organisms, it would repre- 

 sent 100 genes. Of these, 80 (all 64 genes of the 32 AA organisms, plus 

 16 of the 32 genes of the 16 Aa organisms) would be A, while 20 (the 

 other 16 genes of the Aa organisms plus all 4 genes of the 2 aa organisms) 

 would be a. Thus it may be seen that the values q = 0.8 and 1 — q = 0.2 

 are maintained. 



If the assumptions are changed now only by adding nonrandom breed- 

 ing, such as self-fertilization, or preferential breeding so that organisms 

 of similar phenotype tend to mate, the result will be an increase of the 

 genotypes AA and aa at the expense of Aa. But the relative proportions 

 of the genes will remain unchanged. 



One can use this formula to determine gene proportions if character 

 proportions are known. Thus, about 16 per cent of native white New 

 Yorkers are Rh negative. As an Rh negative person is homozygous reces- 

 sive (rhrh), {I — q)~ = 0.16, and (1 — ^) = 0.4. The proportion of the 

 recessive gene is therefore 0.4 and that of the dominant alleles, 0.6. Sub- 

 stitution of these figures in the expression 2^(1 — q) easily gives the fre- 

 quency of heterozygotes as 0.48. Hence, the frequency of homozygous 

 dominant persons must be 0.36. Thus the Hardy-Weinberg formula is a 

 most effective tool for analyzing the genetic composition of populations. 

 Again, the sickle cell trait (see Chapter 15), which depends upon a heter- 

 ozygous genotype, Ss, is as frequent as 40 per cent in some equatorial 

 African tribes, while 60 per cent are normal (ss). Substitution in the for- 

 mula reveals similarity to the hypothetical case with A = 0.8 and a = 0.2. 



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