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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

Module 5: Quantitative Genetics

  • Genetic variance: additive and dominance


Variance and covariance definition
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


Properties of variances
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)


Application to a genetic model
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


Additive genetic variance
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


Dominance variance
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


Genetic variance
Genetic variance

For a single locus

(It can be shown that the Cov(A,D) = 0)


Regression of genotypic values on allele number

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


Covariance of genotypic values and allele number
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



Genetic variances example
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


Option 1 use formula
Option 1 – use formula

P = (6+14)/2 = 10

a=14-10=4

d=12-10=2

p=0.6 q=0.4


Option 2 calculate variances directly
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


Option 3 regress values on allele number
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


Option 3 regress values on allele number1
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



Regression of genotypic values on allele number1

genotypic value

breeding value

a = 3.6

1

Z1Z2

0

Z2Z2

2

Z1Z1

Regression of genotypic values on allele number

Excel


Magnitude of genetic variances
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)


Effect of inbreeding selfing on variances
Effect of Inbreeding (selfing) on Variances

Total genetic variance increases with selfing!!

Hallauer, Carena and Miranda, 2010


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