1. Lecture 2: Fisher’s Variance Decomposition

2. Contribution of a locus to a trait
Basic model: P = G + E
Phenotypic value -- we will occasionally
also use z for this value
Genotypic value
Environmental value
G = average phenotypic value for that genotype
if we are able to replicate it over the universe
of environmental values, G = E[P]
Modifications: G - E covariance -- higher performing
Animals may be disproportionately rewarded
G x E interaction --- G values are different
across environments. Basic model now
becomes P = G + E + GE

3. Alternative parameterizations of Genotypic values
Q1Q1
Q2Q1
Q2Q2 C
C + a(1+k)
C + 2a C
C + a + d
C + 2a
C -a
C + d
C + a
d measures dominance, with d = 0 if the heterozygote
is exactly intermediate to the two homozygotes
d = ak =G(Q1Q2 ) - [G(Q2Q2) + G(Q1Q1) ]/2
2a = G(Q2Q2) - G(Q1Q1)
k = d/a is a scaled measure of the dominance

4. Example: Booroola (B) gene
Genotype bb Bb BB
Average Litter size
1.48
2.17
2.66
2a = G(BB) - G(bb) = > a = 0.59
ak =d = G(Bb) - [ G(BB)+G(bb)]/2 = 0.10
k = d/a = 0.17

5. Fisher’s Decomposition of G
One of Fisher’s key insights was that the genotypic value
consists of a fraction that can be passed from parent to
offspring and a fraction that cannot. G i j = b
Dominance deviations --- the difference (for genotype
AiAj) between the genotypic value predicted from the
two single alleles and the actual genotypic value, G i j = π + Æ π G = X i j f r e q ( Q )
Mean value, with
Average contribution to genotypic value for allele i
Since parents pass along single alleles to their
offspring, the ai (the average effect of allele i)
represent these contributions b G i j = π + Æ
The genotypic value predicted from the individual
allelic effects is thus

6. G = π + Æ ± Fisher’s decomposition is a Regression G = π + 2 Æ ( ° ) N
Residual error G i j = π + Æ
Predicted value
A notational change clearly shows this is a regression,
Independent (predictor) variable N = # of Q2 alleles
Regression slope
Intercept G i j = π + 2 Æ 1 ( ) N
Regression residual 2 Æ 1 + ( ) N = 8
>
< : f o r ; e . g , Q

7. Allele Q1 common, a2 > a1
Slope = a2 - a1
Allele Q2 common, a1 > a2
Both Q1 and Q2 frequent, a1 = a2 = 0 1 2 N G
G22
G11
G21

8. Æ = p a [ + k ( ° ) ] ± = G ° π Æ π = 2 p a ( 1 + k ) Genotype Q1Q1
Consider a diallelic locus, where p1 = freq(Q1)
Genotype
Q1Q1
Q2Q1
Q2Q2
Genotypic
value
a(1+k) 2a π G = 2 p a ( 1 + k )
Mean
Allelic effects Æ 2 = p 1 a [ + k ( ) ]
Dominance deviations i j = G π Æ

9. B V ( G ) = Æ + Average effects and Breeding Values i j B V = X ≥ Æ +
The a values are the average effects of an allele
Breeders focus on breeding value (BV) B V = n X k 1 ≥ Æ ( ) i + B V ( G i j ) = Æ +
Why all the fuss over the BV?
Consider the offspring of a QxQy sire mated
to a random dam. What is the expected value
of the offspring?

10. π ° = µ Æ + 2 ∂ B V ( S i r e ) G e n o t y p F r q u c V a l Q 1 / 4x w 1 / 4 π + Æ z
The expected value of an offspring is the expected value of
For random w and z alleles, this has an
expected value of zero ∂ ∂
For a random dam, these have expected value 0 π O = G + Æ x y 2 w z 4 π O G = Æ x + y 2 ∂ B V ( S i r e )
Hence,

11. B V ( S i r e ) = 2 π ° π ° = B V ( S i r e ) 2 + D a m G
We can thus estimate the BV for a sire by twice
the deviation of his offspring from the pop mean, B V ( S i r e ) = 2 π G
More generally, the expected value of an offspring
is the average breeding value of its parents, π G = B V ( S i r e ) 2 + D a m

12. æ = + Genetic Variances G = π + ( Æ ) ± 2 G A D æ ( G ) = X Æ + ± æ (j = π g + ( Æ ) æ 2 ( G ) = π g + Æ i j
As Cov(a,d) = 0 æ 2 ( G ) = n X k 1 Æ i + j
Dominance Genetic Variance
(or simply dominance variance)
Additive Genetic Variance
(or simply Additive Variance) æ 2 G = A + D

13. æ = ( p a k ) æ = p a [ + k ( ° ) ] æ = E [ Æ ] X p Q1Q1 Q1Q2 Q2Q2
a(1+k) a æ 2 A = E [ Æ ] m X i 1 p
Since E[a] = 0,
Var(a) = E[(a -ma)2] = E[a2]
One locus, 2 alleles:
Dominance effects
additive variance æ 2 A = p 1 a [ + k ( ) ]
When dominance present,
asymmetric function of allele
frequencies æ 2 D = E [ ] m X i 1 j p
Equals zero if k = 0
One locus, 2 alleles:
This is a symmetric function of
allele frequencies æ 2 D = ( p 1 a k )

14. Additive variance, VA, with no dominance (k = 0)
Allele frequency, p VA

15. Complete dominance (k = 1)VA VD
Allele frequency, p

16. Overdominance (k = 2) VA VD
Zero additive
variance
Allele frequency, p
Allele frequency, p

17. Epistasis æ = + 2 G A D G = π + ( Æ ) ± A D DD
Dominance value -- interaction
between the two alleles at a locus
Breeding value G i j k l = π + ( Æ ) A D DD
Additive x Additive interactions --
interactions between a single allele
at one locus with a single allele at another
Additive x Dominant interactions --
interactions between an allele at one
locus with the genotype at another, e.g.
allele Ai and genotype Bkj
Dominance x dominance interaction ---
the interaction between the dominance
deviation at one locus with the dominance
deviation at another.
These components are defined to be uncorrelated,
(or orthogonal), so that æ 2 G = A + D