288 JAMES F. CROW 



4. The gene and phenotype frequencies of the parent population are at 

 their equilibrium values. 



5. Increased vigor results in, and can be measured in terms of, increased 

 selective advantage, though the selection may be natural or artificial. This 

 assumption restricts the discussion to those cases in which heterosis results 

 in changes in the same direction as selection had previously been acting. 

 Such an assumption appears to be valid for yield characters in field crops, 

 and for viability and fertility as is measured in Drosophila population 

 studies. It is highly questionable for such things as increase in size of hybrids 

 between wild varieties or species, where natural selection pressure may well 

 have been toward an intermediate size. 



Under this assumption the increase of vigor on hybridization depends di- 

 rectly on the number of loci which are homozygous recessives in the parent, 

 but which become heterozygous in the hybrid. The individual or population 

 of maximum vigor is one in which every allelic pair contains at least one domi- 

 nant. The actual attainable heterosis would be less than this in any particu- 

 lar case. 



Consider the case of complete dominance. The recessive phenotype is as- 

 sumed to have a selective disadvantage of s. That is, the dominant and re- 

 cessive phenotypes are surviving and reproducing in the ratio of 1 to 1 — 5. 

 The rate of mutation from A to a is u per gene per generation. Reverse muta- 

 tion will be ignored as it can be shown to have a negligible effect on the 

 equilibrium gene frequency attained. 



Genotype A A A a a a 



Frequency P 2Q R 



Selective value 1 1 1 — 5 



F-\-2Q-\-R= I 



Under these assumptions, the frequency of gene A will be P -j- ^, while 

 the frequency of a will he Q -\- R. With mutation from A to a at rate u, the 

 frequency of .4 will be reduced in one generation by u{P -f- Q) and the 

 frequency of a increased by the same amount. Likewise, due to the effect 

 of selection, the frequency of a will be decreased by sR. Therefore the gene 

 ratio, {P -j- Q)/(Q + R), will change in one generation due to the effects 

 of mutation and selection to 



iP + Q){l -u) 



{P-hQ)u-\-Q-\-R- sR' 



When equilibrium is reached the gene frequency will no longer change from 

 generation to generation which, stated algebraically, is 



P^^ ^ {P + Q){l-u) 



Q^R {P-\-Q)u-\-Q-\-R- sR- 



