THE COMPOSITION OF PLANT POPULATIONS 321 



Solving we obtain h = 0.27; in other words, the chances are only 

 about one in four that a plant selected from a population of this kind 

 will be heterozygous. If there are 100 pairs of factors and ten generations 

 of self-fertilization only 9 per cent, of the population will be heterozygous. 

 Thus we see how powerful is the tendency of self-fertilization to reduce 

 the population to a homozygous condition. 



The number of homozygous genotypes to which the population will 

 be reduced, it should be remembered, is given by the expression, 2 m , in 

 which m again is the number of pairs of heterozygous factors. If there 

 are 10 pairs of heterozygous factors in the original individual, then the 

 population will ultimately be reduced to 1024 different homozygous 

 genotypes; if there are 100 pairs of such factors, the number of different 

 kinds of genotypes is approximately 1,267,666 X 10 24 . 



We should always remember in working with formulae such as these 

 that they are only valid for conditions postulated in the premises. For 

 the above formulae the following conditions are assumed: roughly 

 equal viability of all genotypes, absence of any natural selection, and 

 independent segregation of factors. Obviously none of these condi- 

 tions is fulfilled in any even moderately complex population. We have 

 already considered many examples of different viability in diverse geno- 

 types, of which the many different Drosophila mutants provide the most 

 conspicuous examples. Similarly natural selection of necessity enters in 

 whenever any differences whatever exist in the ability of different geno- 

 types to survive and reproduce themselves under a given set of condi- 

 tions. In addition to these two obvious difficulties the universal 

 occurrence of linkage also profoundly disturbs the mathematical rela- 

 tions whenever any considerable number of factors is concerned in a 

 given cross. It would be a very rare occurrence for even ten different 

 pairs of factors to exhibit independent assortment in any plant species, 

 impossible in a species like wheat which has but eight pairs of 

 chromosomes. 



The biological significance of this mathematical discussion is merely 

 this: that it demonstrates that populations in which self-fertilization 

 is an invariable condition in seed formation must consist entirely of 

 pure lines, if left undisturbed for a very few generations. Mathematic- 

 ally the limiting condition is one in which all possible pure lines exist 

 in constant proportions in the population, but biologically the limiting 

 condition is one in which the population is composed only of the most 

 vigorous and productive pure lines. 



Populations as Affected by Crossing. When a certain amount of 

 natural crossing occurs the relations above described are somewhat 

 disturbed. The population, of course, tends to reach an equilibrium, 

 and for all practical purposes does reach one very soon, but the mathe- 



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