Extensionofsingle-stepGBLUPtomanygenotypedindividuals

IgnacyMisztalUniversityofGeorgia

Genomicselectionandsingle-step

• Simplicity– NoDYDorDP– Noindex– Nocomplexity

• Accuracy– Avoidsdoublecounting– Avoidsfixedindex– Accountsforpreselection bias

H-1 = A-1 +0 00 G-1 −A22

-1

⎡

⎣⎢⎢

⎤

⎦⎥⎥Aguilaretal.,2010

ChristensenandLund,2010

CurrentimplementationofSS

• GandA22 createdexplicitly• Quadraticmemoryandcubiccomputations• Costper100kgenotypes- 1.5hr (Aguilaretal.,2014)

é ùê úê ú-ë û

-1 -1

-1 -122

H = A +0 00 G A

Numberofgenotypesandimpendingproblem

>2MforHolsteins>400kforAngus

Genomicpre-selectionissue(Patry andDucrocq,2011;VanRaden etal.,2013)

– BLUPincreasinglybiased– Needalldataonpreselectionincluded

Unsymmetric equations

Misztaletal.,2009

NoconvergencewithoutgoodpreconditionerNoconvergencewithlargeHorA

NoGorA22 inversemodel

Legarra andDucrocq (2011)

SlowconvergencewithfewgenotypesDivergencewithmanygenotypes

SNPmodelforgenotypedanimals

Legarra andDucrocq,2011

Nosuccessfulprogramming

SNPmodelforgenotypedanimals

Liuetal,2014

SNPeffectsforallanimals(Fernandoetal.,2014)

centeredgenotypes

imputedgenotypes

CostofimputationRequiresnewtypeofprogrammingExtensiontocomplexmodelsunclear

CanregularssGBLUPbemademoreefficient?

ScalingupA22-1

𝐀𝟐𝟐#𝟏 = 𝐀𝟐𝟐 − 𝐀𝟐𝟏 𝐀𝟐𝟐#𝟏𝐀𝟏𝟐

• 𝐀𝟐𝟐#𝟏 dense(Fauxetal.,2014)

• ForPCGiteration(Stranden etal.,2014)

𝐀𝟐𝟐#𝟏𝐪 = 𝐀𝟐𝟐𝐪 − 𝐀𝟐𝟏 𝐀𝟐𝟐#𝟏

𝐀𝟏𝟐𝐪

• Secondsfor500kanimalswithgoodprogramming(Masudaetal.,2017)

Isdimensionalityofgenomicinformationlimited?

• RegularGnotpositivedefinitepast~5k– BlendingwithA(VanRaden,2008)

• DimensionalityofSNPBLUPsmall(Maciotta etal.,2013)

• Successofimputation

• Manhattanplotsnoisyuntilaveragedby300k-10Mb(dependingonspecies)

…………

Heterogeneticandhomogenic tractsingenome(Stam,1980)

E(#tracts)=4NeL(Stam,1980)Ne– effectivepopulationsizeL–lengthofgenomeinMorgans

Holsteins:Ne≈100L=30Me=12,000

InversionviaSVD/eigenvaluedecompositionAssume1millionanimalsgenotypedwith60kchip

𝐆 = 𝐙𝐙= = 𝐔𝐃𝐔= Eigenvalue decomposition (1M x 1M)

𝐆# = 𝐔𝐃#𝐔= Generalized inverse (1M x 1M)

Z = 𝐔𝐒𝐕 = 𝐔𝐃B.C𝐕 - SVD decomposition (1M x 60k)10h for 720k animals (Masuda, 2017)

t - index for non-negligible eigenvalues, say 10k𝐆# = 𝐔D 𝐃D#E𝐔D= = 𝐔D 𝐒F#E𝐒F#E𝐔D= = 𝐔∗ 𝐔∗

For PCG iteration𝐆#E𝐪 = 𝐔∗ 𝐔∗ 𝐪 - only 1 M x 10k elements

InversebyWoodburyformula

𝐆 = 𝐙𝐙′ + 𝐈𝛆, Woodbury formula𝐆#𝟏 = 𝟏

𝛆𝐈 − 𝟏

𝛆𝐙(𝟏

𝛆𝐙′𝐙 + 𝐈)#𝟏𝐙′ 𝟏

𝛆Z’Z 60k x 60k

For PCG iteration:

𝐆#𝟏𝐪 =𝟏𝛆 𝐈 − 𝐙(𝐔𝐃𝐔′)#𝟏𝐙′ 𝒒 =

𝟏𝛆 𝐈 − 𝐒𝐒′ 𝒒

𝐒 = 𝐙𝐔′𝐃#𝟏/𝟐

With reduced rank 𝐒 = 𝐙𝐔𝒕= 𝐃𝒕#𝟏𝟐 (1M x 10k)

Ostersen etal.,2017

Mantysaari etal.,2017

IfGhaslimiteddimensionality,canG-1

besparselikeA-1?

UseofalaHenderson’srules?

UseofrelativesforG-1

AccuraciesnotgoodenoughTheorynotclear

Assumptionoflimiteddimensionality

= +u Ts eBreedingvalue

s – nx1vectorcontainingadditiveinformationofpopulation(haplotypes,chromosomesegments,LDblocks)?

1 c c-»s T uIfuc containsnanimals:

Verysmallerror

Breedingvaluesofanynanimalscontainsalladditiveinformation

é ùé ù é ù= ê úê ú ê ú

ë û ë ûë ûc c

ncn n

I 0u uP Iu ε

é ùé ù é ù= ê úê ú ê ú

ë ûë û ë û0cncc

nc nn

I 0 I PG 0G

P I 0 IM

--

-

- é ùé ù é ù= ê úê ú ê ú-ë ûë û ë û

11

1cccn

ncnn

I 0G 0I PG

P I0 M0 I

Choosecore“c”andnoncore“n”animals

e= +n nc c nu P u=c cu u

=var( )n nnε M

HowtoestimateP andinv(G)?2var c cc cnu

n nc nn

sæ öé ù é ù

=ç ÷ê ú ê úë û ë ûè ø

u G Gu G G

cc cc cnnc cc

é ù é ùé ùê ú ê ú ë û

ë û ë û

-1 -1-1 -1 -1G 0 G GG = + M G 'G I

0 0 I

1 1| ,n c nc cc c nc cc- -= =u u G G u P G G

Gis“true”relationshipmatrix

APYalgorithm(AlgorithmforProvenandYoung)

PropertiesofAPYalgorithm

Cost:Almostlinearmemoryandcomputations

è

G G-1

G

è

APYG-1

Cost:Quadraticmemoryandcubiccomputations

EAAPmeeting2016

Reliabilities– Holsteins(77k)

Finalscore

regularG-1

4.5k 8k14k 19k77kNeL 2NeL 4NeL

Pocrnic etal.,2016b

Distributionofsegments/haplotypes/..

Assumeddimensionality

Reliability

100%

≈4NeL

Costswith720kgenotypedanimals

• 30MHolsteins• 50Mrecords• 764k60kgenotypes

Item BLUP ssGBLUPAPY G - 7hA22-1 - 10minrounds 402 464Time/round 51s 83sTotaltime 6h 17h

Masudaetal.,2017

Which core animals in APY?Bradfordetal.(2017)

§ Simulatedpopulations(QMSim;Sargolzaei andSchenkel,2009)

§ Ne=40§ #genotypedanimals=50,000

§ Coreanimals:§ Randomgen6||gen7||gen8||gen9||gen

10(y)§ Randomallgenerations

27

Which core animals in APY?

00.10.20.30.40.50.60.70.80.91

98% 95% 90%

Correlation(GEB

V,TBV

)

PercentofvariationexplainedinG

Accuracy

Gen6 Gen7 Gen8 Gen9 Gen10 Random

G-1

28

Bradfordetal.(2016)

4NeL NeL

Persistenceovergenerations

Generations

Reliability

1.0

BLUP

GBLUP– smallpopulation

BayesB – smallpopulation

GBLUP– verylargepopulation

GBLUP– verylargepopulation90%genome

Verylarge– equivalentto4NeLanimalswith99%accuracyAreSNPeffectsfromHolsteinnationalpopulationsconverging

TheoryoflimiteddimensionalityNumberofhaplotypes:4NeLNewithineach¼Morgansegment

¼Morgan

QTLsGenomehaplotypes

Dimensionalityof¼Morgancase:Ne

Fragomeni etal.,2018

ornumberofidentifiedQTLsèReduceddimensionalitywithweightedGRM

ssGBLUP accuraciesusingSNP60Kand100QTNs– simulationstudy

0 10 20 30 40 50 60 70 80 90 100

BLUP

ssGBLUP- unweightedSNP60k

unweightedSNP60k+100QTN

SNP60k+100QTNweightedbyGWAS

SNP60K+100QTNwith"true"variance

plusbyAPY

only100QTNunweightedbyAPY

Fragomeni etal.(2017)

Rank

19k

5k

98

Multitrait ssGBLUP orSNPselection?

• SNPselection/weighting(BayesB,etc.)– Largeimpactwithfewgenotypes– Littleornoimpactwithmany

Variancecomponents

• BasedonSNP– limitations

• REMLbasedonrelationships– Equationsnolongersparse– YAMSsparsematrixpackage–upto100timesspeedup(Masudaetal.,2017)

– APYforREML• MethodR(Legarra andReverter,2017)

Extratopics

• Matchingpedigreesandgenomicrelationships• Missingpedigrees• Crossbreeding• CausativeSNP

• Haplotypesforcrossbreds(Christensenetal.,2016)

• Metafounders (Legarra etal.,2016)• Approximationofreliabilities

Conclusions

• Limiteddimensionalityofgenomicinformationduetolimitedeffectivepopulationsize

• ssGBLUP suitableforanydatasetandmodel

• WithlargedatasetsforHolsteins:– Goodpersistenceofpredictions– Convergenceofpredictionsfromdifferentcountries

Acknowledgements

YutakaMasuda

ShogoTsuruta

DanielaLourenco

BrenoFragomeni

IgnacioAguilar

AndresLegarra

IvanPocrnic

HeatherBradford

TomLawlor,HolsteinAssocPaulVanRaden,AGILUSDA

TheoryforAPY

• Breedingvaluesofcoreanimalslinearfunctionsof:– Independentchromosomesegments(Me)– IndependenteffectiveSNP

• E(Me)=4NeL(Stam,1980;VanRaden,2008)Ne–effectivepopulationsizeL– lengthofgenomeinMorgans

Me=4(Ne=100)(L=30)=12,000

QTL

Accuracy and distance from markers to QTL

38

Fragomenietal.(2017)