1. Whole-genome prediction of complex traits
using kernel methods
Gota Morota
May 12, 2014
1 / 31

2. Outline
• Part 1: Kernel-based whole-genome prediction method
• Part 2: Application to dairy cattle and wheat data
• Part 3: Application to dairy cow health traits
• Part 4: Application to broiler chicken data using genome
annotation
• Part 5: Ph.D.-wide Conclusions
2 / 31

3. Quantitative genetics – related individuals
3 / 31

4. Prediction of complex traits from genotypes
1) GWAS-based prediction
• select influential markers first and then predict
• single marker regression (2001)
• marker assisted BLUP (MABLUP, 1989)
2) Whole-genome prediction
• use all available markers simultaneously
• Whole-genome marker regression
• Kernel-based regression
4 / 31

5. How to parameterize response variable y ?
1. prediction of additive genetic effects
• y = E + a +
2. prediction of total genetic effects parametrically
• y = E + a + d + a ∗ a + a ∗ d + d ∗ d
g
+
3. prediction of total genetic effects non-parametrically
• y = E + g +
5 / 31

6. Whole-genome marker regressions
50K SNP panel (n << p)
1. additive model
• 50K RV to predict
2. additive + dominance model
• 50K + 50K RV to predict!
3. additive + dominance + second-order epistasis model
• 50K + 50K RV + 50K(50K-1)/2 RV to predict!!
⇓
Overparameterization!!!!!
6 / 31

7. Kernel methods
Regression of a phenotype on a n × n symmetric positive (semi)
definite matrix K
⇓
a) Parametric
• A: Pedigree kernel
• G: Additive genomic kernel
• D: Dominance genomic kernel
b ) Non-parametric
• GK: Gaussian kernel
• MK: Mat´ern kernel
• DK: Diffusion kernel
⇓
In non-parametric kernels, choice of the metric determines
characteristics of a kernel
7 / 31

8. Example of kernel methods: Genomic BLUP
y = g + (where g = Xβ) (1)
BLUP(ˆg) = [I + (XXT
)−1 σ2
σ2
β
]−1
y. (2)
Under common marker variance assumption,
σ2
β =
σ2
g
2 j pj(1 − pj)
(3)
Then, BLUP(ˆg) = [I + G−1 σ2
σ2
g
]−1
y (4)
where G = XXT
2 j pj(1−pj)
→ Is this the best kernel??
8 / 31

9. Euclidean space – Gaussian Kernel
Euclidean distance is a metric on a metric space called Euclidean
space
Figure 2 : 3-dimensional Euclidean
space. −∞ ≤ (X, Y, Z) ≤ ∞
Suppose, we observed two
individuals with 3 SNP
genotypes.
• ID1 = x1 = (0,2,2)
• ID2 = x2 = (2,1,0)
Euclidean distance (genetic distance) on R3
||x1 − x2|| = (0 − 2)2 + (2 − 1)2 + (2 − 0)2 = 3
9 / 31

10. Diffusion on 3 dimensional graph (Morota et al., (2013))
0 1 2
012
0
1
2
1st Genotype
2ndGenotype
3rdGenotype
(2,1,2)
(2,0,1)
(0,1,2)
(0,2,0)
(0,1,0)
(0,1,1)
(1,0,0) (2,0,0)
(1,1,0) (2,1,0)
(1,2,0) (2,2,0)
(1,0,1)
(1,1,1) (2,1,1)
(0,2,1) (1,2,1) (2,2,1)
(0,2,2) (1,2,2) (2,2,2)
(1,0,2)
(1,1,2)
Figure 3 : SNP codes are viewed as coordinates of genotypes in
p-dimensional space. → spatial distance
10 / 31

11. Bayesian kernel ridge regression
A quantitative genetics decomposition is
yi = g(xi) + i (5)
y − g 2
+ λ g 2
H (6)
• The representer theorem is used to find the optimal g.
(α) = y − Kα 2
+ λ Kα 2
H (7)
Here, g = Kα, is the function that minimizes (7).
• Kα 2
H
= α Kα, so that the function to be minimized is
(α) = (y − Kα) (y − Kα) + λα Kα. (8)
11 / 31

12. Kernel Averaging (Multiple Kernel Learning)
• Fit three kernels simultaneously: K1, K2, and K3
K = K1
σ2
K1
˜σ2
K
+ K2
σ2
K2
˜σ2
K
+ K3
σ2
K3
˜σ2
K
• Example 1: Three Gaussian kernels: GK1, GK2, and GK3
K = GK1
σ2
GK1
˜σ2
K
+ GK2
σ2
GK2
˜σ2
K
+ GK3
σ2
GK3
˜σ2
K
• Example 2: Three parametric kernels: G, D, and G#D
K = G
σ2
G
˜σ2
K
+ D
σ2
D
˜σ2
K
+ (G#D)
σ2
GD
˜σ2
K
12 / 31

13. Part 2: Application (Morota et al., 2013)
Holstein
• 7,902 Holstein bulls (USDA-ARS AIPL)
• 43,382 SNPs
• PTA of productive life (PL)
Wheat
• 599 inbred lines
• 1,279 binary markers
• average grain yield
Methods
• Bayesian kernel ridge regression (RKHS method)
• Cross-validation
13 / 31

14. Averages of kernel elements and their predictive
correlations for the Holstein data
Kernel θ k(xi, xj) Cor(ˆytest
, yPTA
)
DK 10 0.138 0.727
11 0.483 0.745
11.5 0.644 0.739
12 0.765 0.739
13 0.907 0.734
14 0.966 0.729
GK 5 × 10−5
0.237 0.721
2 × 10−5
0.551 0.736
1 × 10−5
0.749 0.742
5 × 10−6
0.866 0.736
3 × 10−6
0.917 0.734
1 × 10−6
0.971 0.729
G1 NA -0.000126 0.729
G2 NA -0.000113 0.730
14 / 31

15. Averages of kernel elements and their predictive
correlations for the wheat data
Kernel θ k(xi, xj) Cor(ˆytest
, ytrain
)
Diffusion 3 0.136 0.586
3.25 0.289 0.580
3.5 0.466 0.577
4 0.752 0.547
5 0.962 0.522
Gaussian 0.005 0.134 0.582
0.003 0.290 0.579
0.002 0.434 0.562
0.001 0.655 0.558
0.0005 0.809 0.556
G1 NA -0.003 0.518
G2 NA -0.003 0.521
15 / 31

16. Part 3: Application (Morota et al., 2014)
Data
• 4,482 dairy cows (Zoetis)
• 41,266 SNPs
• EBV and PCP of six health traits
Two steps approach
1. variance components estimation using parametric kernels
2. predict total genetic values using parametric and
non-parametric kernels
Methods
• Bayesian kernel ridge regression (RKHS method)
• Cross-validation
16 / 31

17. Estimated ratios of variance components (weights)
Traits Types Variance Components
VG/VP VD/VP VGD/VP
KET
PCP 0.09 0.13 0.14
EBV 0.25 0.03 0.01
DA
PCP 0.06 0.09 0.25
EBV 0.39 0.04 0.30
RP
PCP 0.05 0.09 0.35
EBV 0.27 0.03 0.07
LAME
PCP 0.06 0.07 0.39
EBV 0.39 0.03 0.27
METR
PCP 0.06 0.07 0.21
EBV 0.31 0.05 0.42
CM
PCP 0.06 0.07 0.26
EBV 0.36 0.02 0.16
17 / 31

18. Scatter plots of relationships among G, D, and GD
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−0.05 0.05
−0.040.000.04
animal1
Additive relationship
Dominancerelationship
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−0.05 0.10 0.25
−0.040.020.06
animal2
Additive relationship
Dominancerelationship
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−0.05 0.10
−0.020.020.06
animal3
Additive relationship
Dominancerelationship
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0.0 0.2
−0.040.020.06
animal4
Additive relationship
Dominancerelationship
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−0.05 0.05
0.0000.0040.008
animal1
Additive relationship
Additivebydominancerelationship
qq
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−0.05 0.10 0.25
0.0000.0100.020
animal2
Additive relationship
Additivebydominancerelationship
q qqq
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qqq qq qqq
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qq
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−0.05 0.10
0.0000.010
animal3
Additive relationship
Additivebydominancerelationship
qq
q
qqqqqqq
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q qq qq qqqqqq qqq qqq qqqqqqqq
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qqqqqq qqqq qqqqqqqqqqqqqq qqqqqqqqq q
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qqqqqqqqqqq qqq qq qqqqqq qq q
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0.0 0.2
0.0000.0100.020
animal4
Additive relationship
Additivebydominancerelationship
qqq
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−0.04 0.02 0.06
0.0000.0040.008
animal1
Dominance relationship
Additivebydominancerelationship
qq
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−0.04 0.02 0.08
0.0000.0100.020
animal2
Dominance relationship
Additivebydominancerelationship
q qqq
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qqq qqq qq
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qqqq qqq q
q
q
qq
qqqq q
q
q
m m
18 31

19. Predictive correlations for six health traits
Traits Types Kernels
G GKA GKD GKALL ALL
KET
PCP 0.16 0.18 0.16 0.19 0.18
EBV 0.85 0.86 0.84 0.87 0.86
DA
PCP 0.07 0.08 0.07 0.08 0.07
EBV 0.59 0.61 0.53 0.59 0.60
RP
PCP 0.03 0.05 0.05 0.06 0.05
EBV 0.65 0.67 0.60 0.66 0.65
LAME
PCP 0.07 0.08 0.04 0.07 0.05
EBV 0.64 0.66 0.58 0.65 0.64
METR
PCP 0.05 0.07 0.04 0.05 0.05
EBV 0.48 0.52 0.43 0.50 0.49
CM
PCP 0.07 0.08 0.05 0.07 0.07
EBV 0.72 0.74 0.68 0.73 0.73
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20. Part 4: High-density genotyping chips
• Cattle
Figure 5 : 50K SNP array (2007)
Figure 6 : 778K SNP array (2010)
• Chicken
Figure 7 : 60K SNP array (2011)
Figure 8 : 600K SNP array (2013)
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21. International Sequencing Consortiums
Figure 9 : Human (2001) Figure 10 : Chicken (2004)
Figure 11 : Bovine (2009) Figure 12 : Swine (2012)
Coding DNA sequences cover only tiny fraction of the entire
genomes. The role of non-coding sequences?
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22. Functional?
Figure 13 : FANTOM (2000∼) Figure 14 : ENCODE (2003∼)
Figure 15 : Evolutionary genetics Figure 16 : Quantitative genetics??
Which genomic regions are influential in the context of quantitative
genetics?
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23. Methods
• Data
• 1,351 chickens (Aviagen Ltd. )
• body weight at 35 days (BW), ultrasound area of breast meat
(BM) and hen house production (HHP)
• Affymetrix 600K chips (580,954 SNPs)
• Kernel-based Bayesian ridge regression
• (α|λ) = ||y − Kα||2
+ λ||α||2
• 600K → n
• 10 fold CV
Purpose of this study
Which genomic regions play an important role in prediction of
complex traits?
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24. Genomic regions
Physical positions of SNP were mapped to Gallus gallus 4.0
assembly
• genic vs. intergenic regions (IGR)
Figure 17 : Illustration of intergenic DNA (Wikipedia)
• 5’ and 3’ UTR, exon, intron
Figure 18 : Gene structure (Wikipedia)
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25. Annotation
Table 1 : Numbers of SNPs assigned to each genomic
region.
Annotation # of SNPs annotated after filtering
IGR 299,498 193,970
Genes ± 1kb 281,455 184,047
Genes 266,947 183,768
Exons 29,764 19,511
CDS 21,975 14,416
Genomic region specific kernels
• All → K0
• IGR → K1
• Genes±1kb → K2
• Genes → K3
• Exons → K4
• CDS → K5
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26. Body Weight (BW)
All CDS Exons Genes Genes1kb
q
q
q
0.16
0.18
0.20
All
CDS
CDS−IGR
Exons
Exons−IGR
Genes
Genes−IGR
Genes1kb
Genes1kb−IGR
Genomic regions
Predictivecorrelation
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27. Breast Meat (BM)
All CDS Exons Genes Genes1kb
q
q
0.25
0.26
0.27
0.28
0.29
All
CDS
CDS−IGR
Exons
Exons−IGR
Genes
Genes−IGR
Genes1kb
Genes1kb−IGR
Genomic regions
Predictivecorrelation
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28. Hen House Production (HHP)
All CDS Exons Genes Genes1kb
q
q
q
0.19
0.20
0.21
0.22
0.23
0.24
All
CDS
CDS−IGR
Exons
Exons−IGR
Genes
Genes−IGR
Genes1kb
Genes1kb−IGR
Genomic regions
Predictivecorrelation
28 / 31

29. 3D Chromosome Structure
Figure 19 : Lieberman-Aiden et al. (2009) Science.
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30. Ph.D.-wide Conclusions
No single approach stands out (traits dependent)
Figure 20 : Methods Figure 21 : Kernels
⇓
Functional genomics + quantitative genetics
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31. Acknowledgements
Advisor
• Daniel Gianola
Committee
• Corinne Engelman
• Guilherme Rosa
• Grace Wahba
• Kent Weigel
Data
• USDA-AIPL
• CIMMYT
• Aviagen
• Zoetis
Animal Genetics group
• 4th floor
• 6th floor
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