284 JAMES F. CROW 



creasing their number. Here only two gene pairs were involved, but it was 

 mentioned that similar systems might hold for more complex cases. 



A more general development of the dominance hypothesis was given dur- 

 ing the same year by Bruce (1910). He designated the frequencies of domi- 

 nant and recessive alleles as p and q in one breed and P and Q in the other. 

 The array of individuals in the two groups will then be {p'-DD -\- IpqDR -\- 

 q^RRY and {PWD + 2PQDR + Q-RRY, where D and R are the dominant 

 and recessive alleles and n is the number of factor pairs involved.' If these 

 two populations are crossed, the mean number of homozygous recessive loci 

 is nqQ, whereas the average number for the two parent populations is 

 n{q- -f Q~)/2. The former is the geometric mean of the two parental recessive 

 genotype frequencies while the latter is the arithmetic mean. Since the geo- 

 metric mean is always less than the arithmetic, the number of homozygous 

 recessive loci will always be less in the hybrid population than the mean 

 number in the two parent populations. If either or both the parent popula- 

 tions are inbred the decrease will be greater. 



Bruce then said: 



If, now, it be assumed that dominance is positively correlated with vigor, we have the 

 final result that the crossing of two pure breeds produces a mean vigor greater than the col- 

 lective mean vigor of the parent breeds. ... I am aware that there is no experimental evi- 

 dence to justify the assumption that dominance is correlated with a "blending" character 

 like vigor; but the hypothesis is not an extravagant one, and may pass until a better takes 

 the field. 



The average proportion of recessive homozygotes in the parents, which is 

 (9^ + '2")/2, may be rewritten as qQ -{- {q — Q)~/2. This is always larger 

 than qQ, the proportion in the hybrid, unless q and Q are equal. Although 

 Bruce didn't mention this, after one generation of random mating the propor- 

 tion of recessives in the hybrid population becomes {q + QY/'^ = (lQ'\~ 

 (q — QY/4-, which shows that half the gain in vigor is lost as soon as ran- 

 dom mating begins. 



Bruce concentrated his attention on the decrease of homozygous reces- 

 sive loci in the hybrid, and postulated a correlation between recessiveness 

 and deleterious effect. He could have used the same algebraic procedures to 

 show that crossing produces an increase in heterozygous loci, and thus based 

 a theory of hybrid vigor on overdominance. He showed remarkable foresight 

 in choosing the former, at a time when he had no evidence of a correlation 

 between dominance and beneficial effect. 



1. The notation used by Bruce implies equal frequency of dominant and recessive 

 alleles at all loci. This assumption is not at all necessary for the argument, and I think 

 that what Bruce really meant was 



Yl {p]DD-^2p,q,DR-^qfRR). 



