Crosses. Computation APY Sherman-Woodbury «hybrid» model. Unknown parent groups Need to modify H to include them (Misztal et al., 2013) Metafounders

Transcription

1 Details in ssgblup

2 Details in SSGBLUP Storage Inbreeding G is not invertible («blending») G might not explain all genetic variance («blending») Compatibility of G and A22 Assumption p(u 2 )=N(0,G) If there is selection, mean is not 0 («tuning» solves it: see Vitezica later) Same genetic variance in genotyped and ungenotyped animals Unknown parent groups Need to modify H to include them (Misztal et al., 2013) Metafounders Crosses Computation APY Sherman-Woodbury «hybrid» model 2

3 Details in SSGBLUP Storage Inbreeding G is not invertible («blending») G might not explain all genetic variance («blending») Compatibility of G and A22 Assumption p(u 2 )=N(0,G) If there is selection, mean is not 0 («tuning» solves it: see Vitezica later) Same genetic variance in genotyped and ungenotyped animals Unknown parent groups Need to modify H to include them (Misztal et al., 2013) Metafounders Crosses Computation APY Sherman-Woodbury «hybrid» model 3

4 Storage! "# = % "# + ' ' ' ( "# "+ % ** % "# is very sparse (9 elements /animal) ( "# % "+ ** is very dense (number of animals 2 ) Efficient storage and handling using hash/ija/yams When ( "# % "+ ** is very big, use APY or similar methods Manech Tete Rouse sheep: 3000 animals (rams) genotyped 500,000 animals pedigree. % "# ~ 36 Mb RAM! "# ~ 108 Mb Angus beef cattle: 500,000 animals genotyped 11M animalspedigree. % "# ~ 800 Mb RAM! "# has / elements ~ 2800 Gb!

5 Details in SSGBLUP Storage Inbreeding G is not invertible («blending») G might not explain all genetic variance («blending») Compatibility of G and A22 Assumption p(u 2 )=N(0,G) If there is selection, mean is not 0 («tuning» solves it: see Vitezica later) Same genetic variance in genotyped and ungenotyped animals Unknown parent groups Need to modify H to include them (Misztal et al., 2013) Metafounders Crosses Computation APY Sherman-Woodbury «hybrid» model 5

6 Inbreeding Inbreeding! " is useful to: Monitor genetic diversity Obtain accuracies as #$$ " = 1 ()* +,-. + Obtaining inbreeding in / is easy 0 /" = / "" 1 (e.g. Meuwissen and Luo 1992) Obtaining inbreeding in 1 is easy 0 1" = 1 "" 1 = Obtaining inbreeding in 9 is very complicated!!

7 Inbreeding It is easy to get the diagonal of! "# = % "# + ' ' ' ( "# % ** It is not easy to get the diagonal of "+, = "+ +*- ** ' '... / - **.. - "+ ** - *+ ' '. If you form the matrices explicitly, it becomes very big Still an open question

8 2 2! " 1 y2, From Eq. (1), we obtain z1 = y1 + A12 G 1 A22 whereas from Eq. (2)! " 1 1 G 1 A22 y2 = A22 d2, leading to: we obtain Obtaining overall measures of diversity 1 d1 = A12 A22 d2. the pres+ Gx 2 ssion" 1 2 A21 If w tions ative uence quite (4) Global measures of diversity (e.g. average relationship of all young y1 "through the Finally, y1 = z1 + d1. Then, computing " bulls) can be obtained as! #! =! (#!) indirect method is as simple as for y2, in total contrast with Obtaining is very easy using the algorithm by Colleau the direct'! method. Hx: To summarize, in to compute yto=obtain Modification oforder the algorithm #! Compute z = Ax using [4],! " 1 z = G A Compute y2 = GA z2, Compute d2 = y2 z2, 1 d2, Compute d1 = A12 A22 Compute y1 = z1 + d1. This is the final step. Efficient solving 1 z A G Product GA 1 can be obtained as times vector z2, Colleau et al. Genet Sel Evol (2017) 49:87 RESEARCH ARTICLE Ge n e t i c s Se l e c t i o n Ev o l u t i o n Open Access A fast indirect method to compute functions of genomic relationships concerning genotyped and ungenotyped individuals, for diversity management Jean-Jacques Colleau1, Isabelle Palhière2, Silvia T. Rodríguez-Ramilo2 and Andres Legarra2* Abstract Background: Pedigree-based management of genetic diversity in populations, e.g., using optimal contributions, involves computation of the Ax type yielding elements (relationships) or functions (usually averages) of relationship

9 Details in SSGBLUP Storage Inbreeding G is not invertible («blending») G might not explain all genetic variance («blending») Compatibility of G and A22 Assumption p(u 2 )=N(0,G) If there is selection, mean is not 0 («tuning» solves it: see Vitezica later) Same genetic variance in genotyped and ungenotyped animals Unknown parent groups Need to modify H to include them (Misztal et al., 2013) Metafounders Crosses Computation APY Sherman-Woodbury «hybrid» model 9

10 Details in SSGBLUP Storage Inbreeding G is not invertible («blending») G might not explain all genetic variance («blending») Compatibility of G and A22 Assumption p(u 2 )=N(0,G) If there is selection, mean is not 0 («tuning» solves it: see Vitezica later) Same genetic variance in genotyped and ungenotyped animals Unknown parent groups Need to modify H to include them (Misztal et al., 2013) Metafounders Crosses Computation APY Sherman-Woodbury «hybrid» model 10

11 Blending and compatibility These are two different things Many people don t understand this compatibility tries to put G and A in the same scale blending : assigns part of the genetic variance to pedigree not markers at the same time used to have an invertible G.

12 Details in SSGBLUP Storage Inbreeding G is not invertible («blending») G might not explain all genetic variance («blending») Compatibility of G and A22 Assumption p(u 2 )=N(0,G) If there is selection, mean is not 0 («tuning» solves it: see Vitezica later) Same genetic variance in genotyped and ungenotyped animals Unknown parent groups Need to modify H to include them (Misztal et al., 2013) Metafounders Crosses Computation APY Sherman-Woodbury «hybrid» model 12

13 Compatibility of marker and pedigree relationships Populations evolve with time, but genotypes came years after pedigree started Genomic Predictions are shifted from Pedigree Predictions This makes them not directly comparable Underlying hypothesis false: Christensen & Lund (base allelic frequencies known) Legarra et al. (average genetic value does not change) 13

14 U.S. dairy population and milk yield Bull genotyping starts Massive genotyping starts Pedigree start RL meeting, Aug. 15 (14) Wiggans, 2013

15 EVOLUTION GENETIQUE ET DE MILIEU 2017 LAIT en première lactation (exprimé en équivalent adulte ) MANECH TETE NOIRE DP annuel : 4,9 litres DG annuel : 2,7 litres genotyping starts Lait (litres) Pedigree starts This Δ" from pedigree start to genotype starts needs to be considered lait index lait troupeau Année de production

16 Compatibility of marker and pedigree relationships The population for which average & = 0 and for which the genetic variance is defined is called the genetic base Founders of the pedigree in classical A Whole set of genotyped animals in most typical G Typically, genotyped animals come after pedigreestarts e.g. Lacaune sheep pedigree go back to 1960 but genotypes start in 1995 Drift (and selection) causes : Average genetic values drift (in particular in small populations) Genetic variance reduces 16

17 Reduction of genetic variance Long-term selection experiments (Weber, 1996) Two populations of Drosophila selected for performance in a wind tunnel with effective sizes and selected proportion of 4.5%.

18 Cut data For practical purposes, you only need a few years of data Simplest thing: cut old data and pedigree Then there is no problem of selection and! "#$%! '())%*+ Lourenco (2014) did this with good results Many breeds are reluctant because they feel that they loose information

19 Force G to be similar to A This Δ" from pedigree start to genotype starts needs to be considered genotyping starts # $ % = '(), +, - % ) Lait (litres) Setting both to 0 does not make sense Pedigree starts # $ = '(), /, - % ) lait index lait troupeau Année de production

20 Force G to be similar to A Vitezica et al included the Δ" explicitely 220 Δ" is random because Δ" = $ % = & ' ( ( ) * + %% ) 200 Δ" has variance Var Δ" = / = % = & ' ( ( ) * + %% ) for typical G genotyping starts 4 $ % = 5()7, 9: ; % ) Lait (litres) Setting both to 0 does not make sense Pedigree starts 4 $ = 5(=, +: ; % ) lait index lait troupeau Année de production

21 Force G to be similar to A You can include explicitly: 2 3 XkX XkZ 0 4 ZkX ZkZ+H x 1 l x H x 1 Ql 5 0 x QkH x 1 l QkH x 1 Ql+a x 1 l b Xky r 4 u 5= 4 Zky5: m 0 Or implicitly (equivalent model) XkX ZkX XkZ ZkZ+H #x1 l b = Xky, u Zky where H #x1 = A11 A 12 A 21 A 22 +(G+11ka) x1 : xa x1 22

22 Force G to be similar to A The method has an interesting genetic interpretation Using!! + $$ % & forces G to yield same average relationship than ' (( But we forgot something There is reduction in the genetic variance This reduction is contained in the inbreeding coefficients Thus, we should have diag!./01(' (( )

23 Force G to be similar to A Vitezica et al. (2011) and Christensen et al. (2012) provided an unbiased method that forces the same genetic base across G and A :! = $ + &! $ accounts for old relationships among non genotyped ancestors & accounts for reduction in the genetic variance $ + & '! = '( )) $ + & *+$,(!) = *+$, ( // 23

24 Force G to be similar to A! = $ + &! This is because we use current allele frequencies ' ()**+,- If knew base allele frequencies './0+ But instead we use So: 1. = / / './0+: ;./0+: 1 ()**+,- = ()**+, ()**+,- 7 2 ' ()**+,-: ; ()**+,-: & 2 ' ()**+,- : ; ()**+,-: 2 './0+: ;./0+: $ 2=476 >?@A 2=476 >?@A B 2=476 FGHHAIJ 2=476 FGHHAIJ C D >?@A: E >?@A: C D FGHHAIJ: E FGHHAIJ: B 24

25 Force G to be similar to A Recipe (default in blupf90) Compute! with current allele frequencies Compute " ## Solve equations $ + & '! = '" ##, $ + & )*$+(!) = )*$+ ".. Get new! = $ + &! Build final 0 12 = " ! " ##

26 Does actually G resemble A? If pedigree is good and genotyping is good, yes Usually!"# $ %% &', ) &' 0.8!"#. /01&2300&, &7& 0.5 Useful for quality control

27 Does actually G resemble A? Differences between genomic-based and pedigree-based relationships in a chicken population, as a function of quality control and pedigree links among individuals H. Wang 1, I. Misztal 2 & A. Legarra 3 Table 2 Statistics for coefficient differences between genomic (G) and numerator (A) relationship matrices for genotyped chickens Quality control level G A coefficient measure Number of animal pairs Minimum Maximum Mean Standard deviation Strong 2 Diagonals Off-diagonals Parent-progeny pairs Full-sib pairs Half-sib pairs

28 Force A to be similar to G Christensen (2012) suggests fitting A to G instead of the opposite A dependson pedigree completion Good for chicken, bad for the rest Ancestral relationships that can be seen in G go undetected in A Christensen analitically integrates out! " (=allele frequencies) in a model that uses! = 0.5 as reference in ALL loci and builds ' () uses a relationship matrix * + with related founders The parameter, is the relationship across founders such that we see current genomic relationships 28

29 Relationship across founders Classically we assume! = Christensen changes this into: 1 + ( ( ( ( % & = ( 1 + ( ( ( 2 ( ( 1 + ( ( 2 ( ( ( 1 + ( 2 29

30 Blending Many people claim that SNPs do not explain all genetic variance We can fit two genetic effects Due to markers:! ", #$%! " = '( + )*, explains 1. = / 0* 1 of the 1 / 1 0 * 2/ 0 3 / 1 0 * 2/ 0 3 total genetic variance Due to pedigree:! 4, #$%! 4 = 5( )3, explains. = / 03 of the total genetic variance This is a bit inconvenient Define instead! =! " +! 4 with #$%! = '( + + )* + 5( )3

31 Blending How can we invert!"# $ = &' * () +,' * (- for the mixed model equations? We do not need to just need to modify G (again) Recipe2 Take previous. = " + 0. Put an amount 1 of genomic relationships in G:. = , ** = 1 1 " , ** Actually blupf90 does kind of the opposite First blending, then compatibility Negligible difference in practice

32 Blending How to estimate! = # $% &? # & $ ' (# & $ % REML: fit two genetic effects ) * and ) + Not realistic for big data files Many people do by cross-validation to get unbiased estimates (I am not sure that this is a good idea) Blupf90 uses a default! = # $% & # & $ ' (# & $ % the variance) = 0.05 (markers explain 95% of

33 Blending for inversion Our first attempts used non-blended, non-compatible G People using GBLUP do not need to modify G But G is not always invertible system is computationally singular: reciprocal condition number = e-22 Trick 1:! = 1 &! + &( )) is invertible Trick 2:! =! + *+ is invertible with * = 0.01

34 Blending Blending is not needed for compatibility! =! + %& is invertible with % = 0.01 does not include pedigree information

35 Details in SSGBLUP Storage Inbreeding G is not invertible («blending») G might not explain all genetic variance («blending») Compatibility of G and A22 Assumption p(u 2 )=N(0,G) If there is selection, mean is not 0 («tuning» solves it: see Vitezica later) Same genetic variance in genotyped and ungenotyped animals Unknown parent groups Need to modify H to include them (Misztal et al., 2013) Metafounders Crosses Computation APY Sherman-Woodbury «hybrid» model 35

36 In the 70 s there was a massive export of US Holstein to European Friesian U.S. dairy population and milk yield RL meeting, Aug. 15 (36) Wiggans, 2013

37 Unknown Parent Groups US Holstein had no data in European countries But treating them as equal to European cows was unfair European Genetic evaluations included effect of origin This effect mutated to Unknown Parent Groups

38 Unknown Parent Groups (Thompson 1979, Quaas 1988): Regression on % of origin computed from pedigree E.g. one cow is 15% US, 80% European, 5% New Zealand Final EBV = portions of UPG + random part!" = $!% + '" $ contains fractions Use of unknown parent groups is essential to get unbiased estimates across origins (UY vs US) and years (2000 vs. 2008) 38

39 Unknown Parent Groups Unknown Parent Groups are used extensively to model: Missing parentship, as in sheep (father is often unknown). Genetic Groups are often defined by year of birth to model genetic progress. Importations, or introduction of foreign material (as in pig companies). Genetic Groups are often defined by country of origin. Crosses (e.g. Angus x Gelbvieh). Genetic Groups are often defined by breed.

40 Unknown Parent Groups in Single Step GBLUP Things get complicated! " # = % &', )* + #! " # = %,, -* + # Contradictions Reports of problems in SSGBLUP with complex UPG structure Unknown-parent groups in single-step genomic evaluation I. Misztal 1, Z.G. Vitezica 2, A. Legarra 3, I. Aguilar 4 & A.A. Swan 5 40

41 Unknown Parent Groups in Single Step GBLUP Still open problem Current options Simplify your model!!! Truncate pedigree and data Approximate UPGs! = $ ' () () $ ++ $ includes UPG using existing theory $ ++ is constructed as if UPG don t exist, which is an approximation Default in blupf90 Fitting UPG as covariates, = -. + / with! () = $ () ' () () $ ++ Final EBVs / Fitting exact UPGs Equivalent to Fitting UPG as covariates Still not quite perfect The fancyest solution is «metafounders» 41

42 The G matrix Is exact, independently of pedigree depth Breeds/UPGs were considered unrelated, but they ARE related if we look at markers We may need to adjust the UPG theory to match A to G instead of viceversa 42

43 To condensate: Things are easier if we define pseudo-individuals (metafounders) that represent pools of founder individuals These pools have self-relationships and across-relationships contained in a matrix!. For instance! "#$%&'() = *'+%', Holstein is more variable than, and related to, Jersey Build A from! following tabular rules 43

44 INTRODUCTION METHODS RESULTS FINAL COMMENTS Metafounder relationships RELATIONSHIPS Across founders within the population A SINGLE METAFOUNDER Across founders acrossthe populations TWO OR MORE METAFOUNDERS Pedigree It has self-relationship A 11 =! so F =!-1. If! = 0 then we have regular relationships. All A and A -1 methods work. Pedigree Algorithms change but they are still easy. 44

45 Compatibility of G and A using metafounders ü Extension of Christensen (2012) ü Write asmany metafounders asbase populations ü These metafounders are related by a matrix of additive relationships! ü Estimate " using markers and pedigree (and maybe data) ü Define # as crossproduct # = %&'( %&'( ) * + with P containing 0.5 ü Then combine everything into one H matrix for all animals,!-. = /! # &2! / /!-. : first invert!, then use Henderson s rules This is the best compatibility of G and A 45

46 Technical note: Genomic evaluation for crossbred performance in a single-step approach with metafounders 1 T. Xiang,* 2 O. F. Christensen,* and A. Legarra Re-analyses of exact same data as previous paper: Application of single-step genomic evaluation for crossbred performance in pig 1 T. Xiang,* 2 B. Nielsen, G. Su,* A. Legarra, and O. F. Christensen* 46

47 Landrace Some are genotyped (7700 L, 7700 Y, 5500 F1) Yorkshire F1 TNB was recorded in 293,339 LL, 180,112 YY, and 10,974 crossbred. 332,929 LL, 210,554 YY, and 10,974 crossbreds were in the pedigree 47

48 Landrace x Yorkshire = F1 (Tao Xiang) Single Step Genotypes and phenotypes in purebreds and crosses Old method: two SSGBLUPs separate for each origin (Xiang 2016 J Anim Sci) New method: metafounders Two populations Landrace and Yorkshire! = #$ % #$ %,' #$ %,' #$ ' = estimated by GLS 48

49 Landrace x Yorkshire = F1 (Tao Xiang) One H matrix for all animals (Landrace, Yorkshire, or F1)! "#$ = & & & ' () " + #$ + +"#$,,, Three trait model (L,Y, F1) depending on which population the trait was recorded The three trait model accommodates interactions GxG and GxE. 49

50 Landrace x Yorkshire = F1 (Tao Xiang) The results were as good as the more complex method in the previous paper But much easier 50