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Genetic model
2
Non additive genetic effects: dominance + epistasis
A D EIP = + + +
Additive (or breeding) value Dominance deviation
Phenotype
Epistatic deviation
P EG= +
3
Before the genomic era
Genetic model
4
Fisher (1918) explained that the substitution effect of one allele is the regression of phenotype on genotype
26
Gen
otyp
ic V
alue
N = # Copies of Allele 20 1 2
G11
G21
G22
µ + 2!1
µ + !1 + !2
µ + 2!2
"12
"11
"22
Slope = ! = !2 - !1
1
!
11 21 22Genotypes
This decomposition is a regression of G
! = #$# %&#$', # = )*&+
Walsh B., 2013
One locus , = - + / + 0
Genetic model
5
26
Gen
otyp
ic V
alue
N = # Copies of Allele 20 1 2
G11
G21
G22
µ + 2!1
µ + !1 + !2
µ + 2!2
"12
"11
"22
Slope = ! = !2 - !1
1
!
11 21 22Genotypes
This decomposition is a regression of G
Walsh B., 2013
! = # + % + &
• Dominance deviations are essentially residuals.
• It is the difference for a genotype (e.g. 11, in red) between the genotypic value and its prediction from 2 alleles.
• This difference is named “interaction” between 2 alleles
Genetic model
6
! = # + %
&'((!*, !,) = .*,/01 + 2*,/31
Under random mating
Coefficients of identity15 possible configurations of IBD
Dominant relationships involve the probability of identical genotypes of two individuals at one locus
7
Individual X
Coefficient of fraternity !"#It is the probability that two alleles in individual X (genotype of X) are IBD to two alleles in individual Y (genotype in Y).
Individual Y
8
AA
aa
Aa!" +$ 2&' −2&")2!& ) & − ! ' 2!&)&" −$ 2!' −2!")
Genotype Frequency Genotypic Additive Dominance value value deviation
One locus model without inbreeding
' = $ + & − ! )+," = 2!&'"
+-" = 2!&) "
Before the genomic era
9
Dominance in outbred populations
• Additive or breeding values u and dominance deviations vare uncorrelated in a non-inbred population
• ! = #$ + &' + &( + )
• *+, '( = -./01 2
2 3.401
• D= “dominant relationships”
• The dominance matrix D can be obtained from A.
• 567 = 89 +:;:< +=;=<+ +:;=< +=;:< (Cockerham, 1954)
• Using this rule for inbred populations is wrong.
10
AA
aa
Aa!" + !$% +& 2$( −2$"* !" !
2!$ − 2!$% * $ − ! ( 2!$* 2!$ 0$" + !$% −& 2!( −2!"* $" $
Genotype Frequency Genotypic Additive Dominance F=0 F=1 value value deviation
One locus model with inbreeding
Inbreeding may reduce the mean phenotypic value of a population
Inbreeding depression (Falconer, 1981)∆= −2!$*%
In crosses → ”hybrid vigour” or heterosis
11De Boer & Hoeschele (1993)
!"# $%, $' = )%'*+,- + /0%'*1,- + /2%'*13- + 4%'*+15 + ⋯
One locus model with inbreeding
Computation of the genetic covariance between two relatives
12
One locus model with inbreeding
! = # + % − ' ()*+, = 2'%!,
).+, = 2'%( ,
)./, = 4'% '1 + %1 (, − 2'%( ,
)*./ = 4'% ' − % !(
∆= −2'%(3
There are TWO estimates of additive/dominant covariance in livestock (Hoeschele-Vollema1993; Fernandez et al., 2017)
(R) for the base and (I) for a completely inbred population
Computation of the genetic covariance between two relatives
De Boer & Hoeschele (1993)
Dominance in inbred populations• Breeding values u and dominance deviations v are correlated • ! = #$ + &'( + &) + &* + +
• ,-. )* = /0123 40156
470156 820523 + 860563
• 82= “dominant relationships without inbreeding” (probability of identical genotypes at the locus excluding inbred relationships)
• 86= “dominant relationships in inbreeding” (probability of identical genotypes at the locus including only inbred relationships)
• 4=“dominant – additive relationships” (probability that 3 alleles out of 4 are identical)
Details: DeBoer and Hoeschele 1993
&'( = inbreeding depression
13
• Genetic evaluation is difficult• Matrix D is complex to obtain• The most interesting case (small closed populations) is
extremely complex• Not much accuracy in estimation of dominance values
unless we have MANY full-sibs• Difficult use of dominant values in practice (assortative
matings)
14
Why do not we run genetic evaluation for dominance, even for non inbred populations?
15
Estimation of non-additive effects is important ?
Estimates of dominance variance (Misztal et al., 1998; van Tassell et al., 2000; etc. )
Ratio of dominance and additive genetic variances !"/ℎ"
16
In the genomic era
Dominance and epistasis are much easier to deal with markers !!!
We do not deal with probabilities of identical genotypes but with observations of heterozygote states
All those ugly covariances disappear because we refer to the current genotyped population
Genomic evaluations have renewed the interest in non-additive effects (e.g., Toro and Varona, 2010; Wellmann and Bennewitz, 2012; Su et al., 2012; Nishio & Satoh, 2014; Jiang & Reif, 2015; Aliloo et al. 2016)
17
Estimation of non-additive effects
18
AA
aa
Aa!" +$ (2 − 2!)) −2*"+2!* + 1 − 2! ) 2!*+*" −$ 2!) −2!"+
Genotype Frequency Genotypic Additive Dominance value value deviation
If we could know a and d…
) = $ + * − ! +./" = 2!*)"
.0" = 2!*+ "
In the genomic era
A model with substitution effects and dominance deviations
Additive or “breeding” values (u) of individuals are generated by substitution effects (α) (Falconer, 1981)
The α involve both “biological” additive (a) and dominant (d) effects of the markers and the allelic frequency p
Dominance deviations (v) only include part of the biological dominant effects of the markers
19
( )a d q pa = + -
Breeding values
20
Substitution effect of the SNP
The breeding value (u) for an individual is
Dominant effect of the SNP
Additive effect of the SNP
1 1(2 2 )( ( )) (2 2 )A Au p a d q p p a= - + - = -
1 2(1 2 )( ( )) (1 2 )A Au p a d q p p a= - + - = -
2 2( 2 )( ( )) ( 2 )A Au p a d q p p a= - + - = -
1 1
1 2
2 2
(2 2 )(1 2 ) for genotypes
2i
p A Az p A A
p A A
-ì ìï ï= -í íï ï-î î
VanRaden (2008)
! = #$
Dominant deviations
21
Also, the dominant deviation (v) of an individual is
So, the dominant deviations of a set of individuals are
Dominant effect of the SNP
1 1
22A Av q d= -
1 22A Av pqd=
2 2
22A Av p d= -
21 1
1 22
2 2
22 for genotypes 2
i
q A Aw pq A A
p A A
ì- ìï ï= í íï ï- îî
! = #$
Vitezica et al., 2013
Additive and dominance variances
22
The partition of the total variance
If a and d are considered random, the covariance of breedingvalues, u, is
SNP variances for additive and dominant components
with
VanRaden, 2008
G is the genomic additive relationship matrixGenetic “additive” (breeding values) variance
!"# $ = &&'2∑*+,+
-./ = 0-./
-./ = 2∑*+,+-1/ + 2∑*+,+ ,+ − *+ /-4/
-5/ = -6/ + -7/ = -./ + -8/
-./ = 2*,9/ -8/ = 2*,: /
Genomic dominant relationship matrix
23
Also, the covariances across dominant deviations (v) are
As we have
D is the dominant genomicrelationship matrix
SNP variance for dominant component
Use in Mixed Model: GD-BLUP
Genetic “dominant” (dominance deviations) varianceVitezica et al., 2013
!"# $ = &&'
∑ 2*+,+ - ./- = 0./-
./- = ∑ 2*+,+ -.1-
24
Misunderstanding about models with dominance
!∗ = $%Additive genotypic values, not breeding values
Dominant genotypic values
&∗ = '(
Additive Dominant Additive Dominant
)*)* 2 − 2- −2./ 2 − 2- −2-.)*)/ 1 − 2- 2-. 1 − 2- 1 − 2-.)/)/ −2- −2-/ −2- −2-.
Su et al. 2012Vitezica et al. 2013
! = $1 & = 2(
Breeding values
Dominance deviations
Remarks
25
The “classical” model (Vitezica et al., 2013) gives breeding values and dominant deviations
The “genotypic” model (Su et al., 2012) predicts additive and dominant genotypic effects
Both models are able to explain the data but their results and interpretation is different.
26
These variances are different
The covariances are also different
Misunderstanding about models with dominance
Additive Dominant Additive Dominant
!"!" 2 − 2% −2&' 2 − 2% −2%&!"!' 1 − 2% 2%& 1 − 2% 1 − 2%&!'!' −2% −2%' −2% −2%&
) = +, )∗ = .,
/01 ) = ++2
∑45"6768 2%4&4 ' 9:' = ;9:' /01 )∗ = ..2
∑<5"6768 %4&4 1 − 2%4&4
9:∗'
Su et al. 2012Vitezica et al. 2013
More remarks
27
Only the variances estimated with the “classical” model (breeding values and dominance deviations) are useful in selection and comparable with pedigree-based estimates.
The variances estimated from the “genotypic” model are not comparable with pedigree-based estimates.
Using variance components estimated from the “genotypic” model is misleading because they underestimate the importance of additive component and overestimate the importance of dominance.
The genotypic parameterization (G*D* model) proposed by Su et al. (2012) underestimatesadditive variance and overestimates dominance variance
28Vitezica et al. 2013
29
RESEARCH ARTICLE Open Access
Genomic heritability estimates in sweetcherry reveal non-additive genetic varianceis relevant for industry-prioritized traitsJulia Piaskowski1* , Craig Hardner2, Lichun Cai3, Yunyang Zhao4, Amy Iezzoni3 and Cameron Peace1
Abstract
Background: Sweet cherry is consumed widely across the world and provides substantial economic benefits inregions where it is grown. While cherry breeding has been conducted in the Pacific Northwest for over half acentury, little is known about the genetic architecture of important traits. We used a genome-enabled mixed modelto predict the genetic performance of 505 individuals for 32 phenological, disease response and fruit quality traitsevaluated in the RosBREED sweet cherry crop data set. Genome-wide predictions were estimated using a repeatedmeasures model for phenotypic data across 3 years, incorporating additive, dominance and epistatic variancecomponents. Genomic relationship matrices were constructed with high-density SNP data and were used toestimate relatedness and account for incomplete replication across years.
Results: High broad-sense heritabilities of 0.83, 0.77, and 0.76 were observed for days to maturity, firmness, and fruitweight, respectively. Epistatic variance exceeded 40% of the total genetic variance for maturing timing, firmnessand powdery mildew response. Dominance variance was the largest for fruit weight and fruit size at 34% and 27%,respectively. Omission of non-additive sources of genetic variance from the genetic model resulted in inflation ofnarrow-sense heritability but minimally influenced prediction accuracy of genetic values in validation. Predictedgenetic rankings of individuals from single-year models were inconsistent across years, likely due to incompletesampling of the population genetic variance.
Conclusions: Predicted breeding values and genetic values revealed many high-performing individuals for use asparents and the most promising selections to advance for cultivar release consideration, respectively. This studyhighlights the importance of using the appropriate genetic model for calculating breeding values to avoid inflationof expected parental contribution to genetic gain. The genomic predictions obtained will enable breeders toefficiently leverage the genetic potential of North American sweet cherry germplasm by identifying high qualityindividuals more rapidly than with phenotypic data alone.
Keywords: GBLUP, Sweet cherry, Prunus, Genomic selection, Non-additive genetic variation
BackgroundSweet cherry (Prunus avium L.) is a lucrative fresh markethorticultural crop whose monetary worth is directly andindirectly determined by several horticultural and fruittraits. Worldwide, more than 2.8 million tons of sweetcherry fruit were produced in 2014 [1]. In 2015, the U.S.was the second largest producer of cherries, producing
338.6 kt of fruit valued at $703 million, of which 60% weregrown in Washington State [2, 3].Sweet cherry cultivars must garner a positive critical
reception among growers, market intermediaries (acategory which includes packers, shippers, andmarketers), and consumers to succeed commercially.The U.S. sweet cherry industry and consumers havepreviously prioritized which fruit trait thresholds areessential for a successful cultivar. Sweet cherry producersidentified fruit size, flavor, firmness, and powdery mildewresistance as trait priorities in a survey conducted in2011 [4]. Powdery mildew (causative agent Podosphaera
* Correspondence: [email protected] of Horticulture, Washington State University, Pullman, WA99164-6414, USAFull list of author information is available at the end of the article
© The Author(s). 2018 Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, andreproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link tothe Creative Commons license, and indicate if changes were made. The Creative Commons Public Domain Dedication waiver(http://creativecommons.org/publicdomain/zero/1.0/) applies to the data made available in this article, unless otherwise stated.
Piaskowski et al. BMC Genetics (2018) 19:23 https://doi.org/10.1186/s12863-018-0609-8
progeny. These individuals were used in the model-building and prediction steps for all models except forcross validation.
SNP dataThe SNP data were obtained from the RosBREEDproject using the RosBREED cherry 6 K SNP array v1(an Illumina Infinium® II array) [55]. The SNP curationpipeline was described in Cai et al. [56]. Missing datawere imputed with Beagle as implemented in SynBreed[57, 58] using the hidden Markov model and a minor al-lele frequency of 0.05. Individuals or SNPs missing morethan 25% data were removed from analysis. In total, agenome-wide set of 1615 SNPs was used.
Statistical modelingVariance components were estimated with R-ASReml 3.0 [59], and additional statistical analyses were conductedin R v3.4 [60]. The following model was used for initialestimates of genetic effects for a single trait, Y:
Y¼Xbþ Z1aþ Z2dþ Z3iþ Z4aY þ Z5dY þ Z6iY þ e
where a, d, i, aY, dY and iy are the random variables foradditive effects, dominance effects, effects from additive-by-additive epistatic, additive-by-year effects, dominance-by-year effects, and epistasis-by-year effects, respectively.Variables Z1 ‐ 3 and Z4 ‐ 6 are design matrices for main ef-fects and interaction terms, respectively. Dimensions ofZ1 ‐ 3 are nY × Y and Z4 ‐ 6 are nY × nY, where n is thenumber of individuals and Y is the number of years withtrait data for an individual. Year was treated as a fixed ef-fect, where X is the design matrix relating observations toyears and b is a vector of fixed effects due to year. In apreliminary analysis, the effect of location was evaluatedas a fixed effect using a Wald test. Location did not have asignificant effect on the focus traits (p-value > 0.10) andwas omitted from the model. Random variables wereassumed to follow a normal distribution:
a # N 0;Gaσ2a
! ";d # N 0;Dσ2
d
! "; i # N 0;Gaaσ2aa
! ";
aY # N 0; IY $ Gaσ2aY! "
;dY # N 0; IY $ Dσ2dY! "
;
iY # N 0; IY $ Gaaσ2aaY
! "e # N 0;Rð Þ
The covariance structure for year was modeled as arepeated measure: R = IIndividual⊗ eY where IIndividual isan identity matrix of individuals included in the studyand eY is a 3 × 3 matrix of year error terms using ageneral correlation structure implemented in ASReml.The genomic additive relationship matrix was computedwith R/rrBLUP [61] using the VanRaden method [62]:
Ga ¼HHT
2X
jpjð1−pjÞ
where pj is frequency of the positive allele for a singlemarker column, and H was computed as equal tocentered marker data, {H}ij= {M}ij− 2(pj− 0.5). M is ann xm marker matrix with n individuals and m markersexpressed as (− 1,0,1) frequency. The dominance rela-tionship matrix was computed using normalized matri-ces described by Su et al. [63] and implemented using acustom R program [64]:
D ¼ ZZTPj2pjð1−pjÞð1−2pjð1−pjÞÞ
where the Z matrix is a transformation of the markermatrix, M:
fZgi j¼ f −2pjð1−pjÞ i fmij¼ −1
1−2pjð1−pjÞ i fmij¼ 0
−2pjð1−pjÞ i fmij¼ 1
The epistatic relationship matrix for additive-by-additive effects was computed by taking the Hada-mard product between Ga, the additive genomic rela-tionship matrix, and itself: Gaa =Ga ∘Ga.When a relationship matrix was not positive definite, a
constant of 1e− 6 was added to the first eigenvector, andthe matrix was inverted.The full model included additive, dominance, and
epistatic main effects and their interactions with yearand is also called the “ADI model” in this paper. Modelfit was assessed by checking for model convergence,examining studentized residuals for each trait-by-yearcombination, and examining the extended hat matrix forinfluential observations. The default model convergencecriteria for ASReml were used, in which the finaliteration must satisfy the following conditions: a changelog likelihood less than 0.002 * previous log likelihood,and the variance parameters estimates change less than1% from the previous iteration. The extended hat matrixfor linear mixed models is:
WC−1WT
Where C ¼ WTR−1Wþ
0 0
0 G−1
!
and W= [X Z].
Influential data points were those with a value greaterthan 2 times the average value of the diagonal of the hatmatrix excluding zeros.The statistical significance of main effects and interac-
tions were tested by first generating reduced models andthen performing log-likelihood ratio tests between full
Piaskowski et al. BMC Genetics (2018) 19:23 Page 4 of 16
1
J. Dairy Sci. 96 :1–10http://dx.doi.org/ 10.3168/jds.2013-6969 © American Dairy Science Association®, 2013 .
ABSTRACT Computerized mating programs using genomic in-
formation are needed by breed associations, artificial-insemination organizations, and on-farm software providers, but such software is already challenged by the size of the relationship matrix. As of October 2012, over 230,000 Holsteins obtained genomic predictions in North America. Efficient methods of storing, com-puting, and transferring genomic relationships from a central database to customers via a web query were developed for approximately 165,000 genotyped cows and the subset of 1,518 bulls whose semen was available for purchase at that time. This study, utilizing 3 breeds, investigated differences in sire selection, methods of as-signing mates, the use of genomic or pedigree relation-ships, and the effect of including dominance effects in a mating program. For both Jerseys and Holsteins, selec-tion and mating programs were tested using the top 50 marketed bulls for genomic and traditional lifetime net merit as well as 50 randomly selected bulls. The 500 youngest genotyped cows in the largest herd in each breed were assigned mates of the same breed with lim-its of 10 cows per bull and 1 bull per cow (only 79 cows and 8 bulls for Brown Swiss). A dominance variance of 4.1 and 3.7% was estimated for Holsteins and Jerseys using 45,187 markers and management group deviation for milk yield. Sire selection was identified as the most important component of improving expected progeny value, followed by managing inbreeding and then inclu-sion of dominance. The respective percentage gains for milk yield in this study were 64, 27, and 9, for Holsteins and 73, 20, and 7 for Jerseys. The linear programming method of assigning a mate outperformed sequential selection by reducing genomic or pedigree inbreeding by 0.86 to 1.06 and 0.93 to 1.41, respectively. Use of ge-
nomic over pedigree relationship information provided a larger decrease in expected progeny inbreeding and thus greater expected progeny value. Based on lifetime net merit, the economic value of using genomic rela-tionships was >$3 million per year for Holsteins when applied to all genotyped females, assuming that each will provide 1 replacement. Previous mating programs required transferring only a pedigree file to customers, but better service is possible by incorporating genomic relationships, more precise mate allocation, and domi-nance effects. Economic benefits will continue to grow as more females are genotyped. Key words: mating program , genomic relationship , dominance , genotype
INTRODUCTION Phenotypic performance, animal viability, and dairy
farm profitability can be affected negatively by decreased heterozygosity and increased frequency of harmful recessives that result from inbreeding. Computerized mating programs have helped breeders reduce pedigree inbreeding by identifying matings between animals with fewer than average ancestors in common (Weigel, 2001). In the genomic era, dense SNP markers across the whole genome have been widely used for genomic selection. Use of genomic relationships is the best way to reduce progeny homozygosity, even for other SNP that are not genotyped directly (Pryce et al., 2012; Sonesson et al., 2012). Breeders should use genomic relationships to control genomic inbreeding when selection is based on genomic EBV, just as pedigree-based relationships were used to control inbreeding when selection was based on traditional EBV computed from pedigree and phenotypic performance (Sonesson et al., 2012). Use of genomic rather than pedigree relationships in mating plans resulted in almost twice the reduction in progeny homozygosity compared with random mating; this ad-ditional reduction in genomic inbreeding of 1 to 2% was worth $5 to $10 for Australian Profit Ranking (Pryce et al., 2012). New programs to minimize genomic in-breeding by comparing genotypes of potential mates are needed by the dairy industry.
Mating programs including genomic relationships and dominance effects 1 C. Sun ,*2 P. M. VanRaden ,† J. R. O’Connell ,‡ K. A. Weigel ,§ and D. Gianola § * National Association of Animal Breeders, Columbia, MO 65205 † Animal Improvement Programs Laboratory, Agricultural Research Service, US Department of Agriculture, Beltsville, MD 20705-2350 ‡ School of Medicine, University of Maryland, Baltimore 21201 § Department of Dairy Science, University of Wisconsin–Madison, Madison 53706
Received April 29, 2013. Accepted August 16, 2013. 1 The use of trade, firm, or corporation names in this publication is
for the information and convenience of the reader. Such use does not constitute an official endorsement or approval by the US Department of Agriculture or the Agricultural Research Service of any product or service to the exclusion of others that may be suitable.
2 Corresponding author: [email protected]
Inbreeding depression252 INBREEDING AND CROSSBREEDING: I [Chap. 14
linear with respect to F, and this might be taken as evidence that
epistatic interaction between loci is not of great importance. Thereare, however, several practical difficulties that stand in the way of
drawing firm conclusions from observations of the rate of inbreeding
depression. One is that as inbreeding proceeds and reproductive
capacity deteriorates, it soon becomes impossible to avoid the loss of
Fig. 14. i. Examples of inbreeding depression affecting fertility.
(a) Litter-size in mice (original data). Mean number born alive in
1 st litters, plotted against the coefficient of inbreeding of the litters.
The first generation was by double-first-cousin mating; thereafter
by full-sib mating. No selection was practised, (b) Fertility in
Drosophila subobscura. Mean number of adult progeny per pair perday, plotted against the inbreeding coefficient of the parents.
Consecutive full-sib matings. (Redrawn from Hollingsworth &Smith, I955-)
some lines. The survivors are then a selected group to which the
theoretical expectations no longer apply. Thus precise measurementof the rate of inbreeding depression can generally be made only over
the early stages, before the inbreeding coefficient reaches high levels.
Another difficulty, met with particularly in the study of mammals,arises from maternal effects. Maternal qualities are among the mostsensitive characters to inbreeding depression. The effect of inbreed-
ing on another character that is influenced by maternal effects is
therefore two-fold: part being attributable to the inbreeding of the
individuals measured and part to the inbreeding in the mothers. Sothe relationship between the character measured and the coefficient
of inbreeding cannot be depicted in any simple manner. In conse-
(Lynch & Walsh, 1998)
Inbreeding depression is the decline in biological fitness (viability, fertility, …) as a consequence of inbreeding
• This phenomenon may be explained by directional dominance.• Directional dominance, e.g. higher percentage of positive than negative
functional dominant effect (d)
(Falconer, 1981)
• The models assume that ! and " have zero means
• Not true with directional dominance
• So we can posit # " = # %&' = %&' > )• Define *∗ = * − E * , then E *∗ = )• Genomic model is
. = /0 + 23 +4 *∗ + 5 * + 6
. = /0 + 23 +4*∗ +4%&' + 6
Inbreeding depression
! = #$ + &' +()∗ +(+,- + .(+,- = /,-
• / is row-sum of (, individual heterozygosities
• Genomic inbreeding coefficient is 0 = + − //3/ = + − 0 3
/,- = + − 0 3,- = +3,- + 0 −3,- = +3,- + 0b! = #$∗ + 0b + &' +()∗ + .
Genomic inbreeding coefficient explains directional dominance
Inbreeding depression
Xiang et al., 2016
33
Genomic inbreeding
One measure of genomic inbreeding can be the within-individual average homozygosity ("#$) across all SNPs.For individual i
"#$& =()) + (++
()) + ()+ + (++
where ()), ()+ and (++ refer to the numbers of SNPs that are classified as AA, Aa, and aa, respectively
(Silió et al., 2013)
34
Inbreeding and its effect on variance
Without inbreeding, dominance variance is overestimated
20
Figure 3. Estimates (boxplots of posterior distributions) of additive, dominance and epistatic 414
genetic variances for A+D+AA+AD+DD evaluation model including (GDIF) or not (GDI) 415
genomic inbreeding. The A+D+AA+AD+DD model involves additive, dominance, additive-416
by-additive, additive-by-dominance, and dominance-by-dominance effects. 417
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Additive Dominance Epistasis
GDI GDIF GDI GDIF GDI GDIF
0.25
0.50
0.75
0.00
0.25
0.50
0.75
0.4
0.6
0.8
1.0
1.2
Model
Variance Model
GDI
GDIF
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
20
Figure 3. Estimates (boxplots of posterior distributions) of additive, dominance and epistatic 414
genetic variances for A+D+AA+AD+DD evaluation model including (GDIF) or not (GDI) 415
genomic inbreeding. The A+D+AA+AD+DD model involves additive, dominance, additive-416
by-additive, additive-by-dominance, and dominance-by-dominance effects. 417
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Additive Dominance Epistasis
GDI GDIF GDI GDIF GDI GDIF
0.25
0.50
0.75
0.00
0.25
0.50
0.75
0.4
0.6
0.8
1.0
1.2
Model
Variance Model
GDI
GDIF
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
Classical model + consideration of inbreeding (F) in pig data for litter size
Genomic inbreeding must be included as a covariate in the model for obtaining correct estimation of dominance variance
This has long been known for pedigree analysis (e.g. DeBoer and Hoeschele, 1993).
35
Same “classical” model + consideration of inbreeding (H) in beef cattle for body yearling weight
These results were confirmed in dairy cattle data by Aliloo et al. (2017).
Inflation in dominance variance estimates
Inbreeding and its effect on variance
Raidan et al., 2017
36
Xiang et al. Genet Sel Evol (2016) 48:92 DOI 10.1186/s12711-016-0271-4
RESEARCH ARTICLE
Genomic evaluation by including dominance effects and inbreeding depression for purebred and crossbred performance with an application in pigsTao Xiang1,2* , Ole Fredslund Christensen1, Zulma Gladis Vitezica3 and Andres Legarra2
Abstract Background: Improved performance of crossbred animals is partly due to heterosis. One of the major genetic bases of heterosis is dominance, but it is seldom used in pedigree-based genetic evaluation of livestock. Recently, a trivari-ate genomic best linear unbiased prediction (GBLUP) model including dominance was developed, which can distin-guish purebreds from crossbred animals explicitly. The objectives of this study were: (1) methodological, to show that inclusion of marker-based inbreeding accounts for directional dominance and inbreeding depression in purebred and crossbred animals, to revisit variance components of additive and dominance genetic effects using this model, and to develop marker-based estimators of genetic correlations between purebred and crossbred animals and of correla-tions of allele substitution effects between breeds; (2) to evaluate the impact of accounting for dominance effects and inbreeding depression on predictive ability for total number of piglets born (TNB) in a pig dataset composed of two purebred populations and their crossbreds. We also developed an equivalent model that makes the estimation of variance components tractable.
Results: For TNB in Danish Landrace and Yorkshire populations and their reciprocal crosses, the estimated propor-tions of dominance genetic variance to additive genetic variance ranged from 5 to 11%. Genetic correlations between breeding values for purebred and crossbred performances for TNB ranged from 0.79 to 0.95 for Landrace and from 0.43 to 0.54 for Yorkshire across models. The estimated correlation of allele substitution effects between Landrace and Yorkshire was low for purebred performances, but high for crossbred performances. Predictive ability for crossbred animals was similar with or without dominance. The inbreeding depression effect increased predictive ability and the estimated inbreeding depression parameter was more negative for Landrace than for Yorkshire animals and was in between for crossbred animals.
Conclusions: Methodological developments led to closed-form estimators of inbreeding depression, variance components and correlations that can be easily interpreted in a quantitative genetics context. Our results confirm that genetic correlations of breeding values between purebred and crossbred performances within breed are positive and moderate. Inclusion of dominance in the GBLUP model does not improve predictive ability for crossbred animals, whereas inclusion of inbreeding depression does.
© The Author(s) 2016. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The Creative Commons Public Domain Dedication waiver (http://creativecommons.org/publicdomain/zero/1.0/) applies to the data made available in this article, unless otherwise stated.
Open Access
Ge n e t i c sSe lec t ionEvolut ion
*Correspondence: [email protected] 1 Department of Molecular Biology and Genetics, Center for Quantitative Genetics and Genomics, Aarhus University, 8830 Tjele, DenmarkFull list of author information is available at the end of the article
Inbreeding and its effect on crosses
Landrace Yorkshire LYEffect of 100%
inbreeding ("# ) -9.731 -1.878 -5.055
There is genomic inbreeding in the crosses and it does have an effectInbreeding depression leads to a loss in the number of piglets born also in the crosses
37
Dominance and accuracy
Inclusion of dominance does not increase the accuracy of prediction of breeding values (Ertl et al., 2014; Xiang et al, 2016;
Esfandyari et al., 2016; Moghaddar and van der Werf, 2017) with the exception of Aliloo et al. (2016) (for fat yield in Holstein)
Inclusion of inbreeding depression effect increased predictive ability (Xiang et al., 2016)
38
Epistatic genomic relationship matrices
A D EIP = + + +
Epistatic deviation
Epistasis
Henderson, 1984
With pedigrees
Hadamard products (#)
Chapter 29Non-Additive Genetic Merit
C. R. Henderson
1984 - Guelph
1 Model for Genetic Components
All of the applications in previous chapters have been concerned entirely with additivegenetic models. This may be a suitable approximation, but theory exists that enablesconsideration to be given to more complicated genetic models. This theory is simple fornon-inbred populations, for then we can formulate genetic merit of the animals in a sampleas
g =X
i
gi.
g is the vector of total genetic values for the animals in the sample. gi is a vectordescribing values for a specific type of genetic merit. For example, g1 represents additivevalues, g2 dominance values, g3 additive ⇥ additive, g4 additive by dominance, etc. In anon-inbred, unselected population and ignoring linkage
Cov(gi,g0j) = 0
for all pairs of i 6= j.
V ar(additive) = A�2a,
V ar(dominance) = D�2d,
V ar(additive ⇥ additive) = A#A�2aa,
V ar(additive ⇥ dominance) = A#D�2ad,
V ar(additive ⇥ additive ⇥ dominance) = A#A#D�2aad, etc.
The # operation on A and D is described below. These results are due mostly to Cock-erham (1954). D is computed as follows. All diagonals are 1. dkm(k 6= m) is computedfrom certain elements of A. Let the parents of k and m be g, h and i, j respectively. Then
dkm = .25(agiahj + agjahi). (1)
In a non-inbred population only one at most of the products in this expression can begreater than 0. To illustrate suppose k and m are full sibs. Then g = i and h = j.
1
Chapter 29Non-Additive Genetic Merit
C. R. Henderson
1984 - Guelph
1 Model for Genetic Components
All of the applications in previous chapters have been concerned entirely with additivegenetic models. This may be a suitable approximation, but theory exists that enablesconsideration to be given to more complicated genetic models. This theory is simple fornon-inbred populations, for then we can formulate genetic merit of the animals in a sampleas
g =X
i
gi.
g is the vector of total genetic values for the animals in the sample. gi is a vectordescribing values for a specific type of genetic merit. For example, g1 represents additivevalues, g2 dominance values, g3 additive ⇥ additive, g4 additive by dominance, etc. In anon-inbred, unselected population and ignoring linkage
Cov(gi,g0j) = 0
for all pairs of i 6= j.
V ar(additive) = A�2a,
V ar(dominance) = D�2d,
V ar(additive ⇥ additive) = A#A�2aa,
V ar(additive ⇥ dominance) = A#D�2ad,
V ar(additive ⇥ additive ⇥ dominance) = A#A#D�2aad, etc.
The # operation on A and D is described below. These results are due mostly to Cock-erham (1954). D is computed as follows. All diagonals are 1. dkm(k 6= m) is computedfrom certain elements of A. Let the parents of k and m be g, h and i, j respectively. Then
dkm = .25(agiahj + agjahi). (1)
In a non-inbred population only one at most of the products in this expression can begreater than 0. To illustrate suppose k and m are full sibs. Then g = i and h = j.
1
39
Can epistatic genomic relationship matrices be built using Hadamard products (#) of additive and dominance genomic relationship matrices ?
Additive-by-dominant
"#$ %&' = )& # )'*+,-. = )&'*+,-.
Epistatic genomic relationship matrices
This is wrong.
)& and )'
Genomic additive relationship matrix
Genomic dominantrelationship matrix
| GENOMIC SELECTION
Orthogonal Estimates of Variances for Additive,Dominance, and Epistatic Effects in Populations
Zulma G. Vitezica,*,†,1 Andrés Legarra,† Miguel A. Toro,‡ and Luis Varona§,***Institut National Polytechnique, École Nationale Supérieure Agronomique de Toulouse, Université de Toulouse, and †InstitutNational de la Recherche Agronomique, UMR 1388 Génétique, Physiologie et Systèmes d’Elevage, F-31326 Castanet-Tolosan,
France, ‡Escuela Técnica Superior de Ingenieros Agrónomos, Universidad Politécnica de Madrid, 28040, Spain, §Departamento deAnatomía, Embriología y Genética, Universidad de Zaragoza, and **Instituto Agroalimentario de Aragón, 50013 Zaragoza, Spain
ABSTRACT Genomic prediction methods based on multiple markers have potential to include nonadditive effects in prediction andanalysis of complex traits. However, most developments assume a Hardy–Weinberg equilibrium (HWE). Statistical approaches forgenomic selection that account for dominance and epistasis in a general context, without assuming HWE (e.g., crosses or homozygouslines), are therefore needed. Our method expands the natural and orthogonal interactions (NOIA) approach, which builds incidencematrices based on genotypic (not allelic) frequencies, to include genome-wide epistasis for an arbitrary number of interacting loci ina genomic evaluation context. This results in an orthogonal partition of the variances, which is not warranted otherwise. We alsopresent the partition of variance as a function of genotypic values and frequencies following Cockerham’s orthogonal contrastapproach. Then we prove for the first time that, even not in HWE, the multiple-loci NOIA method is equivalent to construct epistaticgenomic relationship matrices for higher-order interactions using Hadamard products of additive and dominant genomic orthogonalrelationships. A standardization based on the trace of the relationship matrices is, however, needed. We illustrate these results withtwo simulated F1 (not in HWE) populations, either in linkage equilibrium (LE), or in linkage disequilibrium (LD) and divergent selection,and pure biological dominant pairwise epistasis. In the LE case, correct and orthogonal estimates of variances were obtained usingNOIA genomic relationships but not if relationships were constructed assuming HWE. For the LD simulation, differences were smaller,due to the smaller deviation of the F1 from HWE. Wrongly assuming HWE to build genomic relationships and estimate variancecomponents yields biased estimates, inflates the total genetic variance, and the estimates are not empirically orthogonal. The NOIAmethod to build genomic relationships, coupled with the use of Hadamard products for epistatic terms, allows the obtaining of correctestimates in populations either in HWE or not in HWE, and extends to any order of epistatic interactions.
KEYWORDS GenPred; shared data resource; genomic selection; genetic variance components; dominance; epistasis; genomic models; NOIA approach
DOMINANCE and epistasis may play an important role inthe genetic determinism of complex traits of interest,
suchashumanhealthoreconomic traits in livestockandcrops.The existence of interactions within and across loci is sup-ported by classic quantitative genetic studies, QTL mapping,and the wide application of crossbreeding as a breedingstrategy. Nowadays, genomics provides tools to understandthe effects of the genes and their interactions and to offer new
directions for genetic improvement (Mäki-Tanila and Hill2014). In quantitative genetics, the partition of the variancein statistical components due to additivity, dominance, andepistasis does not reflect the biological (or functional) effectof the genes but it is most useful for prediction, selection, andevolution (Huang and Mackay 2016).
In livestock populations, one of the main reasons whydominance or higher-order interaction terms have not beenconsidered in genetic evaluations is that pedigree relation-ships arenot informative enough.However, genomic selectionmethods are beginning to demonstrate their potential to in-clude nonadditive effects in evaluation models. Inclusion ofdominant or/and epistatic effects in genomic evaluation hasbeen proposed by several authors (Toro and Varona 2010;Su et al. 2012; Vitezica et al. 2013; Nishio and Satoh 2014;Jiang and Reif 2015). Most epistatic models only consider
Copyright © 2017 by the Genetics Society of Americadoi: https://doi.org/10.1534/genetics.116.199406Manuscript received December 20, 2016; accepted for publication May 11, 2017;published Early Online May 18, 2017.Supplemental material is available online at www.genetics.org/lookup/suppl/doi:10.1534/genetics.116.199406/-/DC1.1Corresponding author: Institut National de la Recherche Agronomique, UMR1388 Génétique, Physiologie et Systèmes d’Elevage, 24 Chemin de Borde Rouge,31326 Castanet-Tolosan Cedex, France. E-mail: [email protected]
Genetics, Vol. 206, 1297–1307 July 2017 1297
40
| GENOMIC SELECTION
Orthogonal Estimates of Variances for Additive,Dominance, and Epistatic Effects in Populations
Zulma G. Vitezica,*,†,1 Andrés Legarra,† Miguel A. Toro,‡ and Luis Varona§,***Institut National Polytechnique, École Nationale Supérieure Agronomique de Toulouse, Université de Toulouse, and †InstitutNational de la Recherche Agronomique, UMR 1388 Génétique, Physiologie et Systèmes d’Elevage, F-31326 Castanet-Tolosan,
France, ‡Escuela Técnica Superior de Ingenieros Agrónomos, Universidad Politécnica de Madrid, 28040, Spain, §Departamento deAnatomía, Embriología y Genética, Universidad de Zaragoza, and **Instituto Agroalimentario de Aragón, 50013 Zaragoza, Spain
ABSTRACT Genomic prediction methods based on multiple markers have potential to include nonadditive effects in prediction andanalysis of complex traits. However, most developments assume a Hardy–Weinberg equilibrium (HWE). Statistical approaches forgenomic selection that account for dominance and epistasis in a general context, without assuming HWE (e.g., crosses or homozygouslines), are therefore needed. Our method expands the natural and orthogonal interactions (NOIA) approach, which builds incidencematrices based on genotypic (not allelic) frequencies, to include genome-wide epistasis for an arbitrary number of interacting loci ina genomic evaluation context. This results in an orthogonal partition of the variances, which is not warranted otherwise. We alsopresent the partition of variance as a function of genotypic values and frequencies following Cockerham’s orthogonal contrastapproach. Then we prove for the first time that, even not in HWE, the multiple-loci NOIA method is equivalent to construct epistaticgenomic relationship matrices for higher-order interactions using Hadamard products of additive and dominant genomic orthogonalrelationships. A standardization based on the trace of the relationship matrices is, however, needed. We illustrate these results withtwo simulated F1 (not in HWE) populations, either in linkage equilibrium (LE), or in linkage disequilibrium (LD) and divergent selection,and pure biological dominant pairwise epistasis. In the LE case, correct and orthogonal estimates of variances were obtained usingNOIA genomic relationships but not if relationships were constructed assuming HWE. For the LD simulation, differences were smaller,due to the smaller deviation of the F1 from HWE. Wrongly assuming HWE to build genomic relationships and estimate variancecomponents yields biased estimates, inflates the total genetic variance, and the estimates are not empirically orthogonal. The NOIAmethod to build genomic relationships, coupled with the use of Hadamard products for epistatic terms, allows the obtaining of correctestimates in populations either in HWE or not in HWE, and extends to any order of epistatic interactions.
KEYWORDS GenPred; shared data resource; genomic selection; genetic variance components; dominance; epistasis; genomic models; NOIA approach
DOMINANCE and epistasis may play an important role inthe genetic determinism of complex traits of interest,
suchashumanhealthoreconomic traits in livestockandcrops.The existence of interactions within and across loci is sup-ported by classic quantitative genetic studies, QTL mapping,and the wide application of crossbreeding as a breedingstrategy. Nowadays, genomics provides tools to understandthe effects of the genes and their interactions and to offer new
directions for genetic improvement (Mäki-Tanila and Hill2014). In quantitative genetics, the partition of the variancein statistical components due to additivity, dominance, andepistasis does not reflect the biological (or functional) effectof the genes but it is most useful for prediction, selection, andevolution (Huang and Mackay 2016).
In livestock populations, one of the main reasons whydominance or higher-order interaction terms have not beenconsidered in genetic evaluations is that pedigree relation-ships arenot informative enough.However, genomic selectionmethods are beginning to demonstrate their potential to in-clude nonadditive effects in evaluation models. Inclusion ofdominant or/and epistatic effects in genomic evaluation hasbeen proposed by several authors (Toro and Varona 2010;Su et al. 2012; Vitezica et al. 2013; Nishio and Satoh 2014;Jiang and Reif 2015). Most epistatic models only consider
Copyright © 2017 by the Genetics Society of Americadoi: https://doi.org/10.1534/genetics.116.199406Manuscript received December 20, 2016; accepted for publication May 11, 2017;published Early Online May 18, 2017.Supplemental material is available online at www.genetics.org/lookup/suppl/doi:10.1534/genetics.116.199406/-/DC1.1Corresponding author: Institut National de la Recherche Agronomique, UMR1388 Génétique, Physiologie et Systèmes d’Elevage, 24 Chemin de Borde Rouge,31326 Castanet-Tolosan Cedex, France. E-mail: [email protected]
Genetics, Vol. 206, 1297–1307 July 2017 1297
!"# =!" ⊙ !#
&' !" ⊙ !# /)
A standardization based on the trace of the relationship matrices is needed.
Additive-by-dominant
Use in Mixed Model: GDI-BLUP
Genomic additive relationship matrix
Genomic dominantrelationship matrix
Epistatic genomic relationship matrices
41
Orthogonality of classical model
A D EIP = + + +
The components are defined to be uncorrelated (or orthogonal) in LE
Lynch & Walsh, 1998
42
Orthogonality of classical model
A A+D A+D+AA A+D+AA+AD A+D+AA+AD+DD
Gen
etic
varia
nce
0.0
0.2
0.4
0.6
0.8
1.0
1.2
Var.AVar.DVar.AAVar.ADVar.DD
Variance component estimates should not change much among models
Litter size in pigs
Vitezica et al., 2018
43
Dominance and epistasis is much easier with markers.
Classical model using genomic additive and dominant relationships can estimate variance components correctly
Genomic inbreeding must be included as a covariate in the model.
In our experience, accurate variance estimation is difficult in epistatic models.
Some conclusions
Thank you for your attention!
44
