Alternative single-step type genomic prediction equations

(1)

Alternative single-step type genomic prediction

equations

N. Gengler *

1

_{, G. Nieuwhof}

2,3

_{, K. Konstantinov}

2,3

_{, M. Goddard}

3,4

1_{ULg - Gembloux Agro-Bio Tech, B-5030 Gembloux, Belgium} 2_{ADHIS, Bundoora, Australia}

3_{DPI, Victoria, Bundoora, Australia}

4_{University of Melbourne, Melbourne, Australia}

63rd Annual Meeting of the EAAP

Bratislava, Slovakia, August 27-31, 2012

(2)

Partitioning Genetic (Co)variances

 General Model for Genomic Prediction

• Two

sources of genetic (co)variances

1. Explained by genomic differences between animals

• Currently SNP based

• But can also be known QTLs, major gene effects or copy-number variant (CNV) based effects

2. Explained by pedigree  polygenic “residual”

• Logical choice

– Random mixed inheritance model – Jointly modelling and estimating:

• SNP (or similar) effects and • residual polygenic effects

(3)

• Expectation and variances

• Remarks:

– G, Q and D can have whatever structure needed

– G always function of A (pedigree based relationship) and polygenic (co)variances

– u*, G* where “*” indicates linked to polygenic AND SNP effects

– F strictly genomic (co)variance structure , .

F

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General Model

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e.g.,G  AG₀

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(4)

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(1)

(2)

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(5)

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(1)

(2)



Simpler MME?

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(6)

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(1)

(2)



Simpler MME?



Similarity to Genetic Groups!

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(7)

Alternative MME based on

Quaas-Pollack Transformation

• Please note

three advantages

:

1. Inverted G here based on inverted A, no genomic relationships!



Major advantage, usual method to set-up

2. Explicit equations for estimation of SNP effects (g)



Major advantage of multi-step genomic prediction (MS-GP)

3. Direct estimation of



Major advantage of single-step genomic prediction (SS-GP)

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* uˆ -1 A

(3)

(8)

General MME based on

Quaas-Pollack Transformation

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(9)

Identification Of Two Blocks in

Transformed MME – 1

st

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(10)

Identification Of Two Blocks in

Transformed MME – 2

nd

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(11)

Practical Considerations

• In practice

not all animals genotyped

– Non-genotyped animals = “1” – Genotyped animals = “2”

• Definition of

direct SNP contribution

to GEBV (dGV)

– For genotyped animals:

– For non-genotyped animals predicted from dGV of genotyped animals using selection index theory:

2 1 -22 12 1 2 -1 22 12 1 -1 22 12 1

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(12)

Not All Animals Are Genotyped

(System II: “SNP” System)

NB: 1 = non-genotyped, 2 = genotyped animals

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(13)

• First assume only genotyped animals have records

• Recovering genetic (co)variance not explained by

strictly polygenic effect by assuming proportionality

between polygenic and total (co)variance:

• Predicting animals without genotypes

Not All Animals Are Genotyped

(System I: “BLUP” System)

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(14)

Further Modification “BLUP” System

• Introducing

• Predicting animals without genotypes inside MME

(Henderson, 1976)

• Lifting restriction on records only for animals 2

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(15)

Further Modification “BLUP” System

• Using again , following MME are derived

• Please note similarity to

Bayesian procedures to

integrate external information into genetic evaluations

(Vandenplas and Gengler, 2012)

1

T

1 G







φ

1                                                                        g Q T 0 y R Z' y R X' u u β T T T T T Z R Z' X R Z' Z R X' X R X' 1 1 * * 1 1 1 1 ˆ ˆ ˆ ˆ 2 1 -22 2 1 1 -22 22 21 12 11 φ 1 1 φ 1

(16)

Reassembling Systems I and II









                                                                _          0 y R Z' y R X' g u β D Q T ' Q T ' Q 0 0 Q T 0 T T T T T Z R Z' X R Z' 0 Z R X' X R X' 1 1 * 1 1 1 1 1 ˆ ˆ ˆ φ φ 1 φ 1 φ 1 1 φ 1 2 1 -22 2 1 -22 2 2 1 -22 1 -22 22 21 12 11

(17)

Reassembling Systems I and II

Alternative MME using

strictly

genomic (co)variances F







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



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

                                                                _          0 y R Z' y R X' d u β F T T 0 0 T 0 T T T T T Z R Z' X R Z' 0 Z R X' X R X' 1 1 2 * 1 1 1 1 1 ˆ ˆ ˆ φ φ 1 φ 1 φ 1 1 φ 1 1 -22 1 -22 1 -22 1 -22 22 21 12 11

(18)

Equivalence with Single-Step MME









1 22 1 1 1 -22 1 22 1 -22 1 -φ φ 1 φ 1 φT₂₂ F  T  T T  F  T











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                                      y R Z' y R X' u β T F T T T T T Z R Z' Z R X' Z R X' X R X' 1 1 * 22 1 1 1 1 ˆ ˆ 1 -22 1 -22 21 12 11 φ



Equivalence derived from:

1. Absorb equations for d into those for u* 2. Apply rules inverse of sum of matrices:

(3)

(19)

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φ



(20)

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

Polygenic and Genomic (Co)Variances

Often called genomic (co)variance matrix (infact combined one with implicit weights)

(21)

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1 1 * 22 1 1 1 1

ˆ

1 -22 1 -22 21 12 11

φ



Polygenic and Genomic (Co)Variances

Definition of polygenic “residual” as part of total genetic (co)variance T

(22)



































































     

y

R

Z'

y

R

X'

u

β

T

F

T

Z

R

Z'

Z

R

X'

Z

R

X'

X

R

X'

1 1 * 22 1 1 1 1

ˆ

1 -22 1 -22 21 12 11

φ



Polygenic and Genomic (Co)Variances

F represents strictly genomic (co)variance matrix

(23)



































































     

y

R

Z'

y

R

X'

u

β

T

F

T

Z

R

Z'

Z

R

X'

Z

R

X'

X

R

X'

1 1 * 22 1 1 1 1

ˆ

1 -22 1 -22 21 12 11

φ



Polygenic and Genomic (Co)Variances

Weight here defined as constant across all traits, however equations

can be modified to allow different weights across traits

(24)

Most Useful MME  No F

-1

Needed









• Alternative Single Step Genomic Prediction (SS-GP)

– Allows combining advantages of SS-GP and MS-GP

• Different implementations, other advantages

– Setting-up as one systems (cf. above)  direct solving

– Setting-up two systems as seen before, some advantages: • Solving through parallel systems by updating RHS periodically • Alternative “SNP” Systems possible  alternative models, solvers • Excluding some u* (e.g., preferentially treated cows),

(25)

Conclusions

• Developed alternative genomic prediction equations have

many advantages:

– Explicit weighting of genomic (SNP) and polygenic effects – Direct estimation of SNP effects

• Better use of High-Density SNP panels

• Other genetic effects (e.g. CNV) can be accommodated – Direct estimation of GEBV effects

– Genomic relationship matrix never explicitly formed, stored or inversed – Implementation straight-forward

• Based on use of existing software

• System I and System II can run in parallel (updating of RHS)

• But

additional research

required:

(26)

Final Remarks

• General consensus

– Single-step methods combine all sources of information into accurate rankings for animals with and without genotypes

– Especially adapted for novel traits (e.g., milk fat composition) and more complex models (e.g., multitrait, random regression model) – With increasing number of genotyped animals equivalent models

not requiring inverting genomic relationship matrix

• Therefore currently

different research efforts

– To get these equivalent models

• This

development complementary approach

because

– Not based on (matrix of) relationship differences – But on partitioning of genetic (co)variances

(27)

Acknowledgments

• ADHIS and DPI - Victoria (“Visiting Fellows Project”)

– Support for stay of Nicolas Gengler in Australia in 2011

• National Fund for Scientific Research (Belgium)

– Support for Nicolas Gengler as former Senior Research Associate (until end of 2011) and as recipient of several research and travel grants

• European Commission, Directorate-General for Agriculture and

Rural Development, under Grant Agreement 211708

This study has been carried out with financial support from the Commission of the European Communities, FP7, KBBE-2007-1. It does not necessarily reflect its view and in no way anticipates the Commission's future policy in this area

• And many colleagues, in particular Jérémie Vandenplas

(28)

Thank you very much

for your attention