• Teaching philosophy
• Background
• Teaching experiences
• Intro to mixed models

In a world in transition, students and teachers both need to teach themselves one essential skill– learning how to learn. - Carl Sagan, Demon-haunted world

• Two-way learning
• Learn by doing
• Constant feedback

• B.Sc (Tribhuvan University) : Agriculture

  Plant Breeding and Genetics

• M.S (University of Georgia) : Horticulture

  Comparative Genomics, Molecular Biology, Apomixis

• Ph.D (Clemson University) : Plant and Environmental Sciences

  Genomics-Assisted Breeding in Sorghum

• Research Scientist (Clemson University): Advanced Plant Technology Program

  Genomics-Assisted Selection, High Throughput Phenotying

University of Georgia

• Introductory Horticulture (HORT 2000): Spring 2013

Clemson University

• Molecular Biology (GEN 3040): Fall 2015

• Population and Quantitative Genetics (GEN 4110): Spring 2016, 2017

• Molecular Genetics and Gene Regulation (GEN 4210): Fall 2016

• Quantitative genetics for breeding
• Advanced plant breeding (guest lectures)
• Workshops and special topics (based on demand and schedule)

Syllabus - outline

I. Plant breeding and genetics of population

• Genetic constitution of a population
• Changes in gene frequency
• Small populations
• Inbreeding and crossbreeding
• Introduction to mixed models

II. Means of genotypes and breeding populations

• Phenotypic and genotypic values
• Selecting parents for mean performance
• Mapping quantitative trait loci

Syllabus - outline

III. Variation in breeding populations

• Phenotypic and genetic variance
• Estimating genetic variances
• Genotype \(\times\) environment \(\times\) management

IV. Selection in breeding populations

• Response and its prediction
• Information from relatives
• Best linear unbiased prediction
• Whole genome prediction
• Selection of multiple traits

Approach

• Some lectures and concepts
• Study some experimental examples
• Work with datasets in R

Part 1: Concepts

• History of mixed models
• Mendelians vs. Biometricians: A lesson on history
• Mixed linear models
• Mixed models in plant breeding

Part 2: Applications

• Selection models
• Practical examples
• Variance components
• Ridges and Kernels

Francis Galton - 1886: Regression and heritability

Ronald Fisher - 1918: Infinitesimal model (P = G + E)

Sewall Wright - 1922: Genetic relationship

Charles Henderson - 1968: BLUP using relationship

Which famous scientist was Francis Galton related to?

“… the ordinary genealogical course of a race consists in a constant outgrowth from its centre, a constant dying away at its margins, and a tendency of the scanty remains of all exceptional stock to revert to that mediocrity, whence the majority of their ancestors originally sprang.” - Galton, Typical Laws of Heredity

When did Mendel publish his laws of inheritance?

When did Mendel publish his laws of inheritance?

1866

When was Mendelism rediscovered?

When was Mendelism rediscovered?

1900

Which of these was not the one to rediscover Mendelism?

Hugo De Vries

William Bateson

Carl Correns

Erich von Tschermak

Which of these was not the one to rediscover Mendelism?

Hugo De Vries

William Bateson

Carl Correns

Erich von Tschermak

Vigorous debates (sometimes on personal level) ensued

Vigorous debates (sometimes on personal level) ensued

G. Udny Yule (1902): New Phytologist 1 (9):226-27.

“Mendel’s Laws, so far from being in any way inconsistent with the Law of Ancestral Heredity, lead directly to a special case of that law.”

• All model terms assumed to be fixed effect

Combination of fixed and random terms

\(y=Xb+Zu+e\)

• y = vector of observations (n)
• X = design matrix of fixed effects (n x p)
• Z = design (or incidence) matrix of random effects (n x q)
• u = vector of random effect coefficients (q)
• b = vector of fixed effect coefficients (p)
• e = vector of residuals (n)

Fixed effect

• Assumed to be invariable (often you cannot recollect the data)
• Inferences are made upon the parameters
• Example: Overall mean and environmental effects

Fixed effect

• Assumed to be invariable (often you cannot recollect the data)
• Inferences are made upon the parameters
• Example: Overall mean and environmental effects

Random effects

• You may not have all the levels available
• Inference are made on variance components
• Prior assumption: coefficients are normally distributed
• Example: Genetic effects

\(y=Xb+Zu+e\)

Assuming: \(u \sim N(0,K\sigma^{2}_{u})\) and \(e \sim N(0,I\sigma^{2}_{e})\)

• K = random effect correlation matrix (q x q)
• \(\sigma^{2}_{u}\) = random effect variance (1)
• \(\sigma^{2}_{e}\) = residual variance (1)

\(y=Xb+Zu+e\)

Assuming: \(u \sim N(0,K\sigma^{2}_{u})\) and \(e \sim N(0,I\sigma^{2}_{e})\)

• K = random effect correlation matrix (q x q)
• \(\sigma^{2}_{u}\) = random effect variance (1)
• \(\sigma^{2}_{e}\) = residual variance (1)

Summary:

• We know (data): \(x=\{y,X,Z,K\}\)
• We want (parameters): \(\theta=\{b,u,\sigma^{2}_{u},\sigma^{2}_{e}\}\)
• Estimation based on Gaussian likelihood: \(L(x|\theta)\)

Henderson’s equation

\[ \left[\begin{array}{rr} X'X & Z'X \\ X'Z & Z'Z+\lambda K^{-1} \end{array}\right] \left[\begin{array}{r} b \\ u \end{array}\right] =\left[\begin{array}{r} X'y \\ Z'y \end{array}\right] \] \(\lambda=\sigma^{2}_{e}/\sigma^{2}_{u}\) (Regularization parameters) (1)

The mixed model can also be notated as follows

\[y=Wg+e\]

Solved as

\[[W'W+\Sigma] g = [W'y]\]

Where

\(W=[X,Z]\)

\(g=[b,u]\)

\(\Sigma = \left[\begin{array}{rr} 0 & 0 \\ 0 & \lambda K^{-1} \end{array}\right]\)

1 Genetic values

• BLUPs or BLUEs from replicated trials
• Captures additive and non-additive genetics together

1 Genetic values

• BLUPs or BLUEs from replicated trials
• Captures additive and non-additive genetics together

2 Breeding values

• Use pedigree information to create \(K\)
• Captures additive genetics (heritable)
• Trials not necessarily replicated

1 Genetic values

• BLUPs or BLUEs from replicated trials
• Captures additive and non-additive genetics together

2 Breeding values

• Use pedigree information to create \(K\)
• Captures additive genetics (heritable)
• Trials not necessarily replicated

3 Genomic Breeding values

• Genotypic information replaces pedigree
• Any signal: additivity, dominance and epistasis

• Example 1: Balanced data, no kinship
• Example 2: Balanced data, with kinship

Data:

##    Env Gen Phe
## 1   E1  G1  47
## 2   E1  G2  51
## 3   E1  G3  46
## 4   E1  G4  58
## 5   E2  G1  52
## 6   E2  G2  46
## 7   E2  G3  52
## 8   E2  G4  54
## 9   E3  G1  53
## 10  E3  G2  48
## 11  E3  G3  58
## 12  E3  G4  52

Model: \(Phenotype = Environment_{(F)} + Genotype_{(R)}\)

\[y=Wg+e\] Design matrix \(W=[X,Z]\):

##    EnvE1 EnvE2 EnvE3 GenG1 GenG2 GenG3 GenG4
## 1      1     0     0     1     0     0     0
## 2      1     0     0     0     1     0     0
## 3      1     0     0     0     0     1     0
## 4      1     0     0     0     0     0     1
## 5      0     1     0     1     0     0     0
## 6      0     1     0     0     1     0     0
## 7      0     1     0     0     0     1     0
## 8      0     1     0     0     0     0     1
## 9      0     0     1     1     0     0     0
## 10     0     0     1     0     1     0     0
## 11     0     0     1     0     0     1     0
## 12     0     0     1     0     0     0     1

Mixed-model: \[[W'W+\Sigma] g = [W'y]\] \(W'W\):

WW = crossprod(W)
print(WW)

##       EnvE1 EnvE2 EnvE3 GenG1 GenG2 GenG3 GenG4
## EnvE1     4     0     0     1     1     1     1
## EnvE2     0     4     0     1     1     1     1
## EnvE3     0     0     4     1     1     1     1
## GenG1     1     1     1     3     0     0     0
## GenG2     1     1     1     0     3     0     0
## GenG3     1     1     1     0     0     3     0
## GenG4     1     1     1     0     0     0     3

Mixed-model: \[[W'W+\Sigma] g = [W'y]\]

Left-hand side (\(W'W+\Sigma\)):

##       EnvE1 EnvE2 EnvE3 GenG1 GenG2 GenG3 GenG4
## EnvE1     4     0     0  1.00  1.00  1.00  1.00
## EnvE2     0     4     0  1.00  1.00  1.00  1.00
## EnvE3     0     0     4  1.00  1.00  1.00  1.00
## GenG1     1     1     1  3.17  0.00  0.00  0.00
## GenG2     1     1     1  0.00  3.17  0.00  0.00
## GenG3     1     1     1  0.00  0.00  3.17  0.00
## GenG4     1     1     1  0.00  0.00  0.00  3.17

Assuming independent individuals: \(K=I\)

Regularization: \(\lambda=\sigma^{2}_{e}/\sigma^{2}_{u}=1.64/9.56=0.17\)

Mixed-model: \[[W'W+\Sigma] g = [W'y]\]

Right-hand side (\(W'y\)):

##       [,1]
## EnvE1  202
## EnvE2  204
## EnvE3  211
## GenG1  152
## GenG2  145
## GenG3  156
## GenG4  164

Mixed-model: \[[W'W+\Sigma] g = [W'y]\]

We can find coefficients through least-square solution

\(g=(LHS)^{-1}(RHS)=(W'W+\Sigma)^{-1}W'y\)

##        [,1]
## EnvE1 50.50
## EnvE2 51.00
## EnvE3 52.75
## GenG1 -0.71
## GenG2 -2.92
## GenG3  0.55
## GenG4  3.08

\(BLUE=\frac{w'y}{w'w}=\frac{sum}{n}\) = simple average

\(BLUP=\frac{w'y}{w'w+\lambda}=\frac{sum}{n+\lambda}\) = biased average = \(BLUE\times h^2\)

Note:

• More observations = less shrinkage
• Higher heritability = less shrinkage: \(\lambda = \frac{1- h^{2}}{h^2}\)

If we know the relationship among individuals:

##       GenG1 GenG2 GenG3 GenG4
## GenG1  1.00  0.64  0.23  0.48
## GenG2  0.64  1.00  0.33  0.67
## GenG3  0.23  0.33  1.00  0.31
## GenG4  0.48  0.67  0.31  1.00

Then we estimate \(\lambda K^{-1}\)

##       GenG1 GenG2 GenG3 GenG4
## GenG1  0.15 -0.09  0.00 -0.01
## GenG2 -0.09  0.22 -0.02 -0.10
## GenG3  0.00 -0.02  0.10 -0.02
## GenG4 -0.01 -0.10 -0.02  0.17

Regularization: \(\lambda=\sigma^{2}_{e}/\sigma^{2}_{u}=1.64/17.70=0.09\)

And the left-hand side becomes

##       EnvE1 EnvE2 EnvE3 GenG1 GenG2 GenG3 GenG4
## EnvE1     4     0     0  1.00  1.00  1.00  1.00
## EnvE2     0     4     0  1.00  1.00  1.00  1.00
## EnvE3     0     0     4  1.00  1.00  1.00  1.00
## GenG1     1     1     1  3.15 -0.09  0.00 -0.01
## GenG2     1     1     1 -0.09  3.22 -0.02 -0.10
## GenG3     1     1     1  0.00 -0.02  3.10 -0.02
## GenG4     1     1     1 -0.01 -0.10 -0.02  3.17

We can find coefficients through least-square solution

\(g=(LHS)^{-1}(RHS)=(W'W+\Sigma)^{-1}W'y\)

##        [,1]
## EnvE1 51.05
## EnvE2 51.55
## EnvE3 53.30
## GenG1 -1.32
## GenG2 -3.34
## GenG3  0.03
## GenG4  2.45

Genetic coefficients shrink more: Var(A) < Var(G)

Expectation-Maximization REML (1977)

\(\sigma^{2}_{u} = \frac{u'K^{-1}u}{q-\lambda tr(K^{-1}C^{22})}\) and \(\sigma^{2}_{e} = \frac{e'y}{n-p}\)

Bayesian Gibbs Sampling (1993)

\(\sigma^{2}_{u} = \frac{u'K^{-1}u+S_u\nu_u}{\chi^2(q+\nu_u)}\) and \(\sigma^{2}_{e} = \frac{e'e+S_e\nu_e}{\chi^2(n+\nu_e)}\)

Kernel methods:

• Genetic signal is captured by the relationship matrix \(K\)
• Random effect coefficients are the breeding values (BV)
• Efficient to compute BV when \(markers \gg individuals\)
• Easy use and combine pedigree, markers and interactions

Ridge methods:

• Genetic signal is captured by the design matrix \(M\)
• Random effect coefficients are the marker effects
• Easy way to make predictions of unobserved individuals
• Enables to visualize where the QTLs are in the genome

Kernel

\(y=Xb+Zu+e\), \(u\sim N(0,K\sigma^2_u)\)

Ridge

\(y=Xb+Ma+e\), \(a\sim N(0,I\sigma^2_a)\)

Where

• \(M\) is the genotypic matrix, \(m_{ij}=\{0,1,2\}\)
• \(K=\alpha MM'\)
• \(u=Ma\)
• \(\sigma^2_a=\alpha\sigma^2_u\)

Kernel model

\[ \left[\begin{array}{rr} X'X & Z'X \\ X'Z & Z'Z+K^{-1}(\sigma^2_e/\sigma^2_u) \end{array}\right] \left[\begin{array}{r} b \\ u \end{array}\right] =\left[\begin{array}{r} X'y \\ Z'y \end{array}\right] \]

Ridge model

\[ \left[\begin{array}{rr} X'X & M'X \\ X'M & M'M+I^{-1}(\sigma^2_e/\sigma^2_a) \end{array}\right] \left[\begin{array}{r} b \\ a \end{array}\right] =\left[\begin{array}{r} X'y \\ M'y \end{array}\right] \]

Both models capture same genetic signal (de los Campos 2015)

K = tcrossprod(M)/ncol(M)
GBLUP = reml(y=y,K=K); Kernel_GEBV = GBLUP$EBV
RRBLUP = reml(y=y,Z=M); Ridge_GEBV = M%*%RRBLUP$EBV
plot(Kernel_GEBV,Ridge_GEBV, main='Comparing results')

• Variance components and heritability (Johnson and Thompson 1994)
• Trait associations (Gianola and Sorensen 2014)
• Estimation of genetic and breeding values (Piepho et al 2008)
• Prediction of unphenotyped lines (de los Campos et al 2013)
• Selection index (Wientjes et al. 2016)
• Genome-wide association analysis (Yang et al 2014)
• All sorts of inference (Robinson 1991)
• One hundred years of statistical developments (Gianola and Rosa 2014)