Editing Quantitative genetics (section)

==== Homozygosity and heterozygosity====
In the sub-section on "The sample gamodemes – Genetic drift", a series of gamete samplings was followed, an outcome of which was an increase in homozygosity at the expense of heterozygosity. From this viewpoint, the rise in homozygosity was due to the gamete samplings. Levels of homozygosity can be viewed also according to whether homozygotes arose allozygously or autozygously. Recall that autozygous alleles have the same allelic origin, the likelihood (frequency) of which '''''is''''' the '''inbreeding coefficient (''f'')''' by definition. The proportion arising ''allozygously'' is therefore '''(1-f)'''. For the '''A'''-bearing gametes, which are present with a general frequency of '''p''', the overall frequency of those that are autozygous is therefore ('''f''' ''p''). Similarly, for '''a'''-bearing gametes, the autozygous frequency is ('''f''' ''q'').<ref>Remember that the issue of auto/allo -zygosity can arise only for ''homologous'' alleles (that is ''A'' and ''A'', or ''a'' and ''a''), and not for ''non-homologous'' alleles (''A'' and ''a''), which cannot possibly have the ''same allelic origin''.</ref> These two viewpoints regarding genotype frequencies must be connected to establish consistency.

Following firstly the ''auto/allo'' viewpoint, consider the ''allozygous'' component. This occurs with the frequency of '''(1-f)''', and the alleles unite according to the ''random fertilization'' quadratic expansion. Thus: 
<math display="block"> \left( 1-f \right) \left[ p_0 + q_0 \right] ^2 = \left( 1-f \right) \left[ p_0^2 + q_0^2 \right] + \left( 1-f \right) \left[ 2 p_0 q_0 \right] </math> Consider next the ''autozygous'' component. As these alleles '''are''' ''autozygous'', they are effectively '''selfings''', and produce either '''AA''' or '''aa''' genotypes, but no heterozygotes. They therefore produce <math display="inline"> f  p_0 </math> ''"AA"'' homozygotes plus <math display="inline"> f  q_0 </math>  ''"aa"'' homozygotes. Adding these two components together results in:
<math display="inline"> \left[ \left( 1-f \right) p_0^2 + f  p_0 \right] </math> for the '''AA''' homozygote; <math display="inline"> \left[ \left( 1-f \right) q_0^2 + f  q_0 \right] </math> for the '''aa''' homozygote; and <math display="inline"> \left( 1-f \right) 2 p_0 q_0 </math> for the '''Aa''' heterozygote.<ref name="Crow & Kimura"/>{{rp|65}}<ref name="Falconer 1996"/> This is the same equation as that presented earlier in the section on "Self fertilization – an alternative". The reason for the decline in heterozygosity is made clear here. Heterozygotes can arise '''''only''''' from the allozygous component, and its frequency in the sample bulk is just '''(1-f)''': hence this must also be the factor controlling the frequency of the heterozygotes.

Secondly, the ''sampling'' viewpoint is re-examined. Previously, it was noted that the decline in heterozygotes was <math display="inline"> f \left( 2 p_0 q_0 \right)</math>. This decline is distributed equally towards each homozygote; and is added to their basic ''random fertilization'' expectations. Therefore, the genotype frequencies are: <math display="inline"> \left( p_0^2 + f p_0 q_0 \right) </math> for the ''"AA"'' homozygote; <math display="inline"> \left( q_0^2 + f p_0 q_0 \right) </math> for the ''"aa"'' homozygote;  and <math display="inline"> 2 p_0 q_0 - f \left( 2 p_0 q_0 \right) </math> for the heterozygote.

Thirdly, the ''consistency'' between the two previous viewpoints needs establishing. It is apparent at once [from the corresponding equations above] that the heterozygote frequency is the same in both viewpoints.  However, such a straightforward result is not immediately apparent for the homozygotes. Begin by considering the '''AA''' homozygote's final equation in the ''auto/allo'' paragraph above:- <math display="inline"> \left[ \left( 1-f \right) p_0^2 + f  p_0 \right] </math>. Expand the brackets, and follow by re-gathering [within the resultant] the two new terms with the common-factor ''f'' in them. The result is: <math display="inline"> p_0^2 - f \left( p_0^2 - p_0 \right) </math>. Next, for the parenthesized " ''p<sup>2</sup><sub>0</sub>'' ", a ''(1-q)'' is substituted for a ''p'', the result becoming <math display="inline"> p_0^2 - f \left[ p_0 \left( 1-q_0 \right) - p_0 \right] </math>. Following that substitution, it is a straightforward matter of multiplying-out, simplifying and watching signs. The end result is <math display="inline"> p_0^2 + f p_0 q_0 </math>, which is exactly the result for '''AA''' in the ''sampling'' paragraph. The two viewpoints are therefore ''consistent'' for the '''AA''' homozygote. In a like manner, the consistency of the '''aa''' viewpoints can also be shown. The two viewpoints are consistent for all classes of genotypes.