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Quantitative genetics
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=== Contributions from ancestral genepools === In the section on "Pedigree analysis", <math display="inline"> \left( \tfrac {1}{2} \right)^n </math> was used to represent probabilities of autozygous allele descent over '''n''' generations down branches of the pedigree. This formula arose because of the rules imposed by sexual reproduction: '''(i)''' two parents contributing virtually equal shares of autosomal genes, and '''(ii)''' successive dilution for each generation between the zygote and the "focus" level of parentage. These same rules apply also to any other viewpoint of descent in a two-sex reproductive system. One such is the proportion of any ancestral gene-pool (also known as 'germplasm') which is contained within any zygote's genotype. Therefore, the proportion of an '''ancestral genepool''' in a genotype is: <math display="block"> \gamma_n = \left( \frac{1}{2}\right) ^n </math> where '''n''' = number of sexual generations between the zygote and the focus ancestor. For example, each parent defines a genepool contributing <math display="inline"> \left( \tfrac{1}{2} \right)^1 </math> to its offspring; while each great-grandparent contributes <math display="inline"> \left( \tfrac{ 1}{2} \right)^3 </math> to its great-grand-offspring. The zygote's total genepool ('''Ξ''') is, of course, the sum of the sexual contributions to its descent. <math display="block"> \begin{align} \Gamma & = \sum_{n=1} ^{2^n} {\gamma_n} \\ & = \sum_{n=1} ^{2^n} {\left( \frac{1}{2}\right)^n} \end{align} </math> ====Relationship through ancestral genepools==== Individuals descended from a common ancestral genepool obviously are related. This is not to say they are identical in their genes (alleles), because, at each level of ancestor, segregation and assortment will have occurred in producing gametes. But they will have originated from the same pool of alleles available for these meioses and subsequent fertilizations. [This idea was encountered firstly in the sections on pedigree analysis and relationships.] The genepool contributions [see section above] of their '''nearest common ancestral genepool'''(an ''ancestral node'') can therefore be used to define their relationship. This leads to an intuitive definition of relationship which conforms well with familiar notions of "relatedness" found in family-history; and permits comparisons of the "degree of relatedness" for complex patterns of relations arising from such genealogy. The only modifications necessary (for each individual in turn) are in Ξ and are due to the shift to "shared '''common''' ancestry" rather than "individual '''total''' ancestry". For this, define '''Ξ‘''' (in lieu of '''Ξ'''); ''' m = number of ancestors-in-common''' at the node (i.e. m = 1 or 2 only); and an "individual index" '''k'''. Thus: <math display="block"> \begin{align} \Rho_k & = \sum_{m=1} ^{1 , 2} {\gamma_n} \\ & = \sum_{m=1} ^{1 , 2} {\left( \frac{1}{2} \right) ^n} \end{align} </math> where, as before, ''n = number of sexual generations'' between the individual and the ancestral node. An example is provided by two first full-cousins. Their nearest common ancestral node is their grandparents which gave rise to their two sibling parents, and they have both of these grandparents in common. [See earlier pedigree.] For this case, ''m=2'' and ''n=2'', so for each of them <math display="block"> \begin{align} \Rho_k & = \sum_{m=1} ^{2} {\gamma_2} \\ & = \sum_{m=1} ^{2} { \left( \frac{1}{2} \right) ^2} \\ & = \frac{1}{2} \end{align} </math> In this simple case, each cousin has numerically the same Ξ‘ . A second example might be between two full cousins, but one (''k=1'') has three generations back to the ancestral node (n=3), and the other (''k=2'') only two (n=2) [i.e. a second and first cousin relationship]. For both, m=2 (they are full cousins). <math display="block"> \begin{align} \Rho_1 & = \sum_{m=1} ^{2} {\gamma_3} \\ & = \sum_{m=1} ^{2} {\left( \frac{1}{2}\right) ^3} \\ & = \frac{1}{4} \end{align} </math> and <math display="block"> \begin{align} \Rho_2 & = \sum_{m=1} ^{2} {\gamma_2} \\ & = \sum_{m=1} ^{2} {\left( \frac{1}{2}\right) ^2} \\ & = \frac{1}{2} \end{align} </math> Notice each cousin has a different Ξ‘ <sub>k</sub>. ====GRC β genepool relationship coefficient==== In any pairwise relationship estimation, there is one '''Ξ‘<sub>k</sub>''' for each individual: it remains to average them in order to combine them into a single "Relationship coefficient". Because each ''Ξ‘'' is a '''fraction of a total genepool''', the appropriate average for them is the ''geometric mean'' <ref>the square-root of their product</ref><ref name="Moroney 1956">{{cite book |last1=Moroney|first1=M.J. |title=Facts from figures|date=1956 |publisher=Penguin Books |location=Harmondsworth|edition=third}}</ref>{{rp|34β55}} This average is their '''Genepool Relationship Coefficient'''βthe "GRC". For the first example (two full first-cousins), their GRC = 0.5; for the second case (a full first and second cousin), their GRC = 0.3536. All of these relationships (GRC) are applications of path-analysis.<ref name="Li 1977">{{cite book|last1=Li|first1=Ching Chun|title=Path analysis - a Primer|date=1977|publisher=Boxwood Press|location=Pacific Grove|isbn=0-910286-40-X|edition=Second printing with Corrections}}</ref>{{rp|214β298}} A summary of some levels of relationship (GRC) follow. {| class="wikitable" |- ! GRC !! Relationship examples |- | 1.00 || full Sibs |- | 0.7071 || Parent β Offspring; Uncle/Aunt β Nephew/Niece |- | 0.5 || full First Cousins; half Sibs; grand Parent β grand Offspring |- | 0.3536 || full Cousins First β Second; full First Cousins {1 remove} |- | 0.25 || full Second Cousins; half First Cousins; full First Cousins {2 removes} |- | 0.1768 || full First Cousin {3 removes}; full Second Cousins {1 remove} |- | 0.125 || full Third Cousins; half Second Cousins; full 1st Cousins {4 removes} |- |0.0884 || full First Cousins {5 removes}; half Second Cousins {1 remove} |- | 0.0625 || full Fourth Cousins; half Third Cousins |}
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