1. The Basic Genetic Model
Phenotypic observations = environmental effects + genetic effects + residual effect
Yik=sk+gi+eik
where
Yik is the observed value of the ith individual in the kth
environment,
sk is the non-random (fixed environment effects
(locations, management, etc) effect,
gi is the sum of the additive (ga), dominance (gd), epistatic
(ge) genetic values of the genotype of the ith animal.
eik is the sum of random environmental effects affecting
the ith individual in the kth environment

2. Breeding value basic equation
Yik=sk+gi+eik
Yik=sk+gai+gdi+gei+eik
Yik=sk+gai+e*ik
Assumptions
Yik follows a MVN distribution, that is, the trait is determined by infinitely many additive genes of infinitesimal effects at unlinked loci (infinitesimal model).
 V(Y), V(g), and V(e*) are known
 Cov(gai,e*ik)=0 and Cov(e*ij,e*ik)

3. BREEDING VALUES
The additive values in g represents the average additive effects of genes one individual receives from both parents and it is named BREEDING VALUE.
Each parent contributes half of its genes to its progeny in a sample. This transmitting ability corresponds to ½ of its additive genetic value.
Then the breeding values of the progeny is the sum of the transmitting abilities of both parents

4. BREEDING VALUE= Parent average + Mendelian sampling
Then the breeding value of the ith individual is
ai=gai=1/2(am+af)+mi
where
am+af are the breeding values of the male and female parents. Parental average
mi is the deviation of the breeding values of individual ith from the average breeding value for both parents, that is, Mendelian sampling

5. Pedigree relationships
Captured in a matrix A – traditionally built using pedigree
Relationship between each pair of individuals
Range from 0 to 2
Inbred individuals have a relationship with themselves of 2
Pair of completely unrelated individuals have a coefficient of relationship or 0
Full sib have a relationship of 0.5
If parents are not related
Half sibs have a relationship of 0.25

6. Mendelian sampling
Inheritance is a sample process that implies that each parent passes to their progenies ½ of its genes
Therefore there is genetic variation between offspring of the same parents because all offspring do not receive exactly the same genes
Mendelian sampling can be seen as the deviation of the average effects of additive genes an individual receives from both parents from the average effects of genes from the parents common to all offspring

7. Mendelian sampling This is why I am different from my brother
One parent
Child 1
Child 2
Child 3
Child 4
This is why I am different from my brother
In theory you can have sibs that are genetically unrelated
Child 5
Child 6
And “hidden” relationships

8. Genomic Relationships
“some individuals are more equal than others”…….. even if the additive genetic relationship is the same
e.g. actual relationship between HS can vary between 0.2 and 0.3

9. Various ways of doing a genomic prediction (using A or using G)
Rather than normal BLUP (with pedigree A)
…use GBLUP (with G)
G = the genomic relationship matrix
G consists of estimated relationships between individuals based on all markers  genomic relationships

10. Genomic Prediction: GBLUP
Example:
Data on plant 1, with progeny plant 2 and plant 3, plant 4 is unrelated
Want to predict 5 who that is the progeny of plant 1
A-matrix (pedigree-based)
G-matrix (DNA-based)

11. Genomic Prediction: GBLUP
Example:
Data on plant 1, with progeny plant 2 and plant 3, plant 4 is unrelated
Want to predict 5 who that is the progeny of plant 1
A-matrix (pedigree-based)
G-matrix (DNA-based)

12. Genomic Prediction: GBLUP
Example:
Data on plant 1, with progeny plant 2 and plant 3, plant 4 is unrelated
Want to predict 5 who that is the progeny of plant 1
A-matrix (pedigree-based)
G-matrix (DNA-based)
BLUP uses: Family Info
GBLUP uses: Family Info
Segregation within family
Info on ‘unrelated’

13. Breeding value has a number of components
Breeding value = genetic merit of an individual
Twice the average deviation of its offspring from the population mean
50% due to parent average component
Best prediction at birth is the average of two parents breeding value
50% due to Mendelian sampling component
Sampling of parents genes
Reason for differences in:
a pair of full sibs at birth
a pair of F2 in a bi-parental
Can make no prediction at birth
Response to selection driven by
Accuracy of and time taken to estimate of Mendelian sampling term

14. How to get accurate evaluation of Mendelian sampling term?
Animals = progeny test
Evaluate 100 daughters of a bull
Massive cost and time
Plants = replicated field trials
Cost and time?
Some cases not possible
Genomic selection offers us an alternative
Can get accurate prediction of Mendelian sampling term at birth/seed
TIME AND MONEY – two potential benefits
Accurate estimates of breeding value available earlier in the cycle
Cheaper than phenotyping (some traits)
Cost and time per unit of genetic gain

15. Industrial implementation of genomic selection
ANIMAL INDUSTRY
Dairy industry
All developed world national breeding programs
International genetic evaluation
Pig and poultry sectors
Major companies = routine or advanced testing
Beef industry
Sparse use
Sheep industry
Australia and New Zealand = routine
SEED INDUSTRY
Major seed companies -- some have implemented in all crops

16. Genomic selection Complete coverage of genome with markers
All QTL in linkage disequilibrium with at least 1 marker
No QTL size thresholds needed
Accurate breeding values of individuals at birth
Individual marker effects are poorly estimated but the sum is well estimated (law of large numbers)
GWAS results are implicit

17. Genetic relationships
Captured in a matrix A – traditionally built using pedigree
Relationship between each pair of individuals
Range from 0 to 2
Inbred individuals have a relationship with themselves of 2
Pair of completely unrelated individuals have a coefficient of relationship or 0
Full sib have a relationship of 0.5
If parents are not related
Half sibs have a relationship of 0.25

18. Prediction of Breeding Value
Accurate prediction of the true breeding values is important in any breeding program.
Methods for predicting breeding values depends on the type and amount of information available on candidates of selection.
Information for predicting breeding values can come from phenotyic data, pedigree (parental average) and also from markers that measure the Mendelian sampling (differences of within family individuals)

19. Meuwissen, Hayes, and Goddard (2001)
Bernardo and Yu (2007)
marker effect
Yik=sk+gi+eik
Additive effect
of SNPs
-- linearity --
Effect of allele,
chromosome,
haplotype,etc.

20. The breeding value is the adjusted mean multiplied by the heritability
Prediction of Breeding values from the individual own performance (single record)
The breeding value âi for the ith individual is
âi=b(yi-sk) (where b is the regression of true breeding values on
phenotypic performance)
b=Cov(a,y)/Var(y)=
Cov(a,a+e)/Var(y)=
Cov(a,a)+Cov(a,e)/Var(y)=
Var(a)+0/Var(y)=
b=2a/2y=h2
The breeding value is the adjusted mean multiplied by the heritability

21. Accuracy of prediction
ra,y=correlation between the selection criteria -- phenotypic values-- and the true breeding value. Provides a mean of evaluating different selection criteria because the higher the correlation the better the criteria as predictor of the breeding value.
ra,y = Cov(a,y)/ay
= 2a /ay= a /y = h
(ra,y)2=reliability or repeatability – the squared correlation between the selection criterion and the true breeding values
(ra,y)2=h2=heritability

22. Expected response (R) and the variance of estimate breeding value
R= i (ra,y)2y=i h2y (i= intensity of selection)
Var(âi)=Var(by)=Var(h2y)
=h4 Var(y)
=(ra,y)2 h2 Var(y)
=(ra,y)2 2a
(because 2a/2y=h2 and 2y= 2a/h2 )

23. The Numerator Relationship Matrix (A)
The probability of identical genes by descent occurring in two individuals is the COP (or COC or COK) and the additive genetic relationship between two individuals is twice their coancestry. The matrix that indicates the additive relationship among individuals is matrix A that is symmetric and its diagonal elements for the individual ith is 1+Fi (where Fi is the inbreeding coefficient)

24. Diagonal and off diagonal elements of A
The diagonal elements represent twice the probability that two gametes taken at random from animal ith will carry identical alleles by descent.
The off diagonal elements equals the numerator of the coefficient of relationship between two different individuals. When multiplied by the additive genetic variance 2a then A2ais the covariance among breeding values.
Then if ai is the breeding value of the ith individual, Var(ai)= aii 2a = (1+Fi )2a

25. Use of matrix A There are different ways to calculate A
The prediction of breeding values requires the inverse of A
There are different ways to compute the inverse of matrix A