THE COMPOSITION OF PLANT POPULATIONS 321 
Solving we obtain A = 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”, 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 1074. 
We should always remember in working with formule such as these 
that they are only valid for conditions postulated in the premises. For 
the above formule 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- 
21 
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