Variance of Trait in an Inbred Population. Variance of Trait in an Inbred Population

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Σιληνός Ζαΐμης
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1 Varance of Trat n an Inbred Populaton Varance of Trat n an Inbred Populaton

2 Varance of Trat n an Inbred Populaton Revew of Mean Trat Value n Inbred Populatons We showed n the last lecture that for a populaton wth nbreedng coeffcent F (or a sngle ndvdual wth nbreedng coeffcent F ), the mean trat value s E(G F ) = µ GF = µ G + FH where µ G s the mean trat value n a non-nbreedng (or outbred) populaton and H = p δ were the δ terms are the domnance devaton for the non-nbreedng case. Inbred ndvduals are almost always less ft than progeny of nonrelatves The declne n the mean phenotype wth ncreasng homozygosty wthn populatons s known as nbreedng depresson H s the nbreedng depresson, and t s the change of outbred populaton mean value as a result of nbreedng.

3 Varance of Trat n an Inbred Populaton Trat Value Moments n Inbred Populatons We found the frst moment (the mean) for the trat value dstrbuton n an nbreed populaton. To obtan the trat varance, we need to obtan the second moment, whch s the expected value of the square of the lnear model for the trat values. Usng a smlar approach used to obtan the frst moment, we wll now obtan the second moment.

4 Varance of Trat n an Inbred Populaton Genetc Values Overvew As before, we denote the genetc value for genotype A A j as G j = µ G + α + α j + δ j where µ G = p p j G j = G.. j α = j p j G j µ G = G. G.. δ j = G j µ G α α j = G j G. G j. + G.. These mply that p α = 0 and p δ j = 0

5 Varance of Trat n an Inbred Populaton Genetc Varance of Trat Values wth Inbreedng For a random member of a populaton nbred to an extent F relatve to the reference populaton, the genotype frequences are P = p 2 + Fp (1 p ) So we have that E(G 2 F ) = = P j = 2p p j (1 F ) P j (µ G + α + α j + δ j ) 2 j [p 2 + Fp (1 p )](µ G + 2α + δ ) 2 + [p p j (1 F )](µ G + α + α j + δ j ) 2 j

6 Varance of Trat n an Inbred Populaton Genetc Varance of Trat Values wth Inbreedng = [p 2 (1 F ) + Fp ](µ 2 G + 4α2 + δ 2 + 4µ G α + 2µ G δ + 4α δ ) + j [p p j (1 F )](µ 2 G + α2 + α 2 j + δ 2 j + 2µ G α + 2µ G α j +2µ G δ j + 2α α j + 2α δ j + 2α j δ j ) = Fp (µ 2 G + 4α2 + δ 2 + 4µ G α + 2µ G δ + 4α δ ) + [p p j (1 F )](µ 2 G + α2 + αj 2 + δj 2 + 2µ G α + 2µ G α j j +2µ G δ j + 2α α j + 2α δ j + 2α j δ j )

7 Varance of Trat n an Inbred Populaton Genetc Varance of Trat Values wth Inbreedng Let s focus on the frst term nvolvng F We have that p α = 0 and p δ j = 0. So ( F µ 2 G + 4 p α 2 + p δ 2 + 4µ G p α +2µ G p δ + 4 ) p α δ = F ( µ 2 G + 4 p α 2 + p δ 2 + 2µ G p δ + 4 p α δ )

8 Varance of Trat n an Inbred Populaton Genetc Varance of Trat Values wth Inbreedng Now focus on the frst term nvolvng 1 F [p p j (1 F )](µ 2 G + α2 + αj 2 + δj 2 + 2µ G α + 2µ G α j +2µ G δ j + 2α α j + 2α δ j + 2α j δ j ) Smplfy ths term usng p α = 0 and p δ j = 0.

9 Varance of Trat n an Inbred Populaton Genetc Varance of Trat Values wth Inbreedng = (1 F ) µ 2 G + p p j α 2 + p p j αj 2 + p p j δj 2 + 2µ G p p j α + 2µ G p p j α j +2µ G p p j δ j + 2 p p j α α j + 2 p p j α δ j +2 p p j α j δ j

10 Varance of Trat n an Inbred Populaton Genetc Varance of Trat Values wth Inbreedng = (1 F ) µ 2 G + p j p α 2 + p p j αj 2 j + p p j δj 2 + 2µ G p j p α + 2µ G p p j α j j +2µ G p p j δ j + 2 p α p j α j + 2 p α p j δ j +2 p j α j p δ j j = (1 F ) µ 2 G + 2 p α 2 + p p j δ 2 j

11 Varance of Trat n an Inbred Populaton Genetc Varance of Trat Values wth Inbreedng Combnng the smplfed two terms together we have that ( E(GF 2 ) = F µ 2 G + 4 p α 2 + p δ+ 2 2µ G p δ + 4 ) p α δ +(1 F ) µ 2 G + 2 p α 2 + p p j δ 2 j = µ 2 G + 2(1 + F ) p α 2 + (1 F ) p p j δj 2 +F p δ 2 + 4F p α δ + 2F µ G p δ

12 Varance of Trat n an Inbred Populaton Genetc Varance of Trat Values wth Inbreedng So E(G 2 F ) = µ2 G + 2(1 + F ) +F p δ 2 + 4F p α 2 + (1 F ) p α δ + 2F µ G p δ Now calculate the varance of G F notng that E(G F ) = µ G + FH where H = p δ p p j δj 2 j

13 Varance of Trat n an Inbred Populaton Genetc Varance of Trat Values wth Inbreedng We have that Var(G F ) = E(GF 2 ) [E(G F )] 2 = (1 + F )2 p α 2 + (1 F ) p p j δj 2 +F p δ 2 + 4F p α δ F 2 H 2 = (1 + F )σ 2 A + (1 F )σ2 D + 4FD 1 + FD 2 + F (1 F )H 2 where H = p δ, D 1 = p α δ, and D 2 = p δ 2 H2. Note that D 2 s the varance of the δ s So the genetc varance for an nbreed populatons can be wrtten n terms of fve components For a non-nbreed populaton, F = 0 and the genetc varance s σa 2 + σ2 D, as we prevously showed.

14 Varance of Trat n an Inbred Populaton Termnology and Expressons for Varance of Trat The followng terms are commonly used for expressng the varances and covarances of trat values Components Mult-allelc B-allelc Addtve Varance σa 2 = 2 p α 2 σa 2 = 2pqα2 Domnance Varance σd 2 = j p p j δj 2 σd 2 = (2pqd)2 Inbreedng depresson H = p δ H=-2qpd Covarance of α s and δ s D 1 = p α δ D 1 = 2pqdα(p q) Varance of δ s D 2 = p δ 2 H2 D 2 = 4pq(1 4pq)d 2 where α = α 1 α 2 for the b-allelc case. For the b-allelc case, f the frequency of the two alleles are the same, what s D 1 and D 2? For the b-allelc case, f the trat s addtve, what s H, D 1, and D 2?

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