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PBG 650 Advanced Plant Breeding

PBG 650 Advanced Plant Breeding. Module 5: Quantitative Genetics Genetic variance: additive and dominance. Variance and Covariance - definition. The variance of a variable X is: V( X ) = E[( X i -  X ) 2 ] = E( X i 2 ) -  X 2 The covariance of variable X and variable Y is:

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PBG 650 Advanced Plant Breeding

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  1. PBG 650 Advanced Plant Breeding Module 5: Quantitative Genetics • Genetic variance: additive and dominance

  2. Variance and Covariance - definition • The variance of a variable X is: V(X) = E[(Xi- X)2] = E(Xi2) - X2 • The covariance of variable X and variable Y is: Cov(X,Y) = E[(X - X)(Y - Y)] = E(XY) - XY

  3. Properties of variances • The variance of a constant is zero V(c) = 0   V(c+X) = V(X) • The variance of the product of a variable and a constant is the constant squared times the variance of the variable V(cX) = c2V(X) • The variance of a sum of random variables is the sum of the variances plus twice the covariance between the variables V(X + Y) = V(X) + V(Y) + 2Cov(X,Y)

  4. Application to a genetic model P = G + E G = A + D + I P = A + D + I + E Gijkl =  + (i +j + ij) + (k +l + kl) + Iijkl Because there are no covariances among the components

  5. Additive genetic variance Variance of breeding values (No adjustment for the mean is necessary because the mean of breeding values is zero) When p=q=1/2 σA2 =(1/2)a2 When d=0 σA2 = 2pqa2

  6. Dominance Variance Variance of dominance deviations (No adjustment for the mean is necessary because the mean of dominance deviations is zero) When p=q=1/2 σD2 =(1/4)d2 When d=0, σD2 = 0

  7. Genetic variance For a single locus (It can be shown that the Cov(A,D) = 0)

  8. P=MP=midparent value Regression of genotypic values on allele number M Mean(X) = (ΣfiXi) = q2(0) + 2pq(1) + p2(2) = 2p(q+p) = 2p = p2(22) + 2pq(12) +q2(02) – (2p)2= 2pq

  9. Covariance of genotypic values and allele number M = p2(2)(P+a) + 2pq(1)(P+d) + q2(0)(P-a)– (2p)(P+a(p-q)+2pqd) = 2pq[a+d(q-p)] = 2pq Same result with scaled values (a, d, -a) or the adjusted genotypic values: = p2(2)(a-M) + 2pq(1)(d-M) +q2(0)(-a-M)-(2p)(0)  2pq

  10. Regression cont’d.

  11. Genetic Variances - Example p=0.6 q=0.4 • Options for estimating variances • Use formulas with known values of a and d • Calculate breeding values and dominance deviations, and estimate their variances • Regress observed values on number of Z1 alleles Example from Falconer & Mackay

  12. Option 1 – use formula P = (6+14)/2 = 10 a=14-10=4 d=12-10=2 p=0.6 q=0.4

  13. Option 2 – calculate variances directly σG2 = 0.16(-5.76)2+0.48(0.24)2+0.36(2.24)2-02 = 7.1424 σA2 = 0.16(-4.32)2+0.48(-0.72)2+0.36(2.88)2-02 = 6.2208 σD2 = 0.16(-1.44)2+0.48(0.96)2+0.36(-0.64)2-02 = 0.9216

  14. Option 3 – Regress values on allele number p=0.6 q=0.4 Mean(X) = 0.16(0) + 0.48(1) + 0.36(2) = 1.20 = 0.16(02) + 0.48(12) +0.36(22) – (1.20)2 = 0.4800 = 2pq

  15. Option 3 – Regress values on allele number = 0.16(0)(6) + 0.48(1)(12) + 0.36(2)(14)– (1.20)(11.76) =1.7280=2pq The result is the same if we use the adjusted genotypic values: = 0.16(0)(-5.76) + 0.48(1)(0.24) + 0.36(2)(2.24) – (1.20)(0) =1.7280

  16. Regression cont’d.

  17. genotypic value breeding value a = 3.6 1 Z1Z2 0 Z2Z2 2 Z1Z1 Regression of genotypic values on allele number Excel

  18. Magnitude of genetic variances • With no dominance, all genetic variance is additive and maximum genetic variance occurs when p=q=0.5 • With complete dominance • maximum additive genetic variance occurs when the unfavorable allele has a frequency of q=0.75 • maximum dominance variance occurs when q=0.5 • maximum genetic variance occurs when q2=0.5 (q=0.71)

  19. Effect of Inbreeding (selfing) on Variances Total genetic variance increases with selfing!! Hallauer, Carena and Miranda, 2010

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