In charlotte-ngs/lbgfs2021: Livestock Breeding and Genomics (ETHZ) - Fall Semester 2021

knitr::opts_chunk$set(echo = FALSE, results = "asis")
knitr::knit_hooks$set(hook_convert_odg = rmdhelp::hook_convert_odg)

Introduction

• Proposed in 2001
• Widely adopted in 2007/2008
• Costs of breeding program reduced due to shorter generation intervals
• In cattle: young sire selection versus selection based on sire proofs
• In pigs: early selection among full sibbs
• Inbreeding must be considered

Terminology

• Genomic Selection: use of genomic Information for selection decisions
• Genomic Information is used to predict genomic breeding values

Benefits in Cattle

#rmdhelp::use_odg_graphic(ps_path = "odg/fig-benefit-geno-sel-cattle.odg")
knitr::include_graphics(path = "odg/fig-benefit-geno-sel-cattle.png")

Benefits in Pigs

#rmdhelp::use_odg_graphic(ps_path = "odg/fig-benefit-geno-sel-pig.odg")
knitr::include_graphics(path = "odg/fig-benefit-geno-sel-pig.png")

Genetic Model

• Recall: BLUP animal model is based on infinitesimal model
• Prediction of genomic breeding values is based on polygenic model
• In polygenic model: Single Nucleotide Polymorphisms (SNP) are used as markers
• Marker genotypes are expected to be associated with genotypes of Quantitative Trait Loci (QTL)

Polygenic Model

#rmdhelp::use_odg_graphic(ps_path = "odg/fig-polygenic-model.odg")
knitr::include_graphics(path = "odg/fig-polygenic-model.png")

Statistical Models

Two types of models are used

1. marker-effect models (MEM)
2. genomic-breeding-value based models (BVM)

MEM

• marker effects ($a$-values) are fitted using

  • a simple linear model $\rightarrow$ marker effects are fixed
  • a linear mixed effects model $\rightarrow$ marker effects are random

• Problem of finding which markers are associated to QTL

• With high number of SNP compared to number of genotyped animals: very large systems of equations to solve

BVM

• genomic breeding values as random effects
• similar to animal model
• genomic relationship matrix ($G$) instead of numerator relationship matrix ($A$)

MEM versus BVM

#rmdhelp::use_odg_graphic(ps_path = "odg/mem-vs-bvm.odg")
knitr::include_graphics(path = "odg/mem-vs-bvm.png")

Logistic Procedures

• Two Step:

  • use reference population to get marker effects using MEM
  • use marker effects to get to genomic breeding values

• Single Step

  • MEM or BVM in a single evaluation
  • difficulty how to combine animals with and without genotypes

Two Step Procedure

#rmdhelp::use_odg_graphic(ps_path = "odg/two-step-gs.odg")
knitr::include_graphics(path = "odg/two-step-gs.png")

Single Step GBLUP

• Use a mixed linear effect model
• Genomic breeding values $g$ are random effects

$$y = Xb + Zg + e$$ with

• $E(e) = 0$, $var(e) = I * \sigma_e^2$
• $E(g) = 0$, $var(g) = G * \sigma_g^2$
• Genomic relationship matrix $G$

Solution Via Mixed Model Equations

• All animals have genotypes and observations

$$\left[ \begin{array}{ll} X^TX & X^TZ \ Z^TX & Z^TZ + \lambda * G^{-1} \end{array} \right] \left[ \begin{array}{c} \hat{b} \ \hat{g} \end{array} \right] = \left[ \begin{array}{c} X^Ty \ Z^Ty \end{array} \right] $$ with $\lambda = \sigma_e^2 / \sigma_g^2$.

Animals Without Observations

• Young animals do not have observations
• Partition $\hat{g}$ into
  • $\hat{g}_1$ animals with observations and
  • $\hat{g}_2$ animals without observations
• Resulting Mixed Model Equations are (assume $\lambda = 1$)

$$\left[ \begin{array}{lll} X^TX & X^TZ & 0 \ Z^TX & Z^TZ + G^{(11)} & G^{(12)} \ 0 & G^{(21)} & G^{(22)} \end{array} \right] \left[ \begin{array}{c} \hat{b} \ \hat{g}_1 \ \hat{g}_2 \end{array} \right] = \left[ \begin{array}{c} X^Ty \ Z^Ty \ 0 \end{array} \right] $$

Predicted Genomic Breeding Values

• Last line of Mixed model equations

$$ G^{(21)} \cdot \hat{g}_1 + G^{(22)} \cdot \hat{g}_2 = 0$$

Solutions

• Solving for $\hat{g}_2$

$$\hat{g}_2 = - (G^{(22)})^{-1} \cdot G^{(21)} \cdot \hat{g}_1$$

Genomic Relationship Matrix

• Breeding value model uses genomic breeding values $g$ as random effects
• Variance-covariance matrix of $g$ are proposed to be proportional to matrix $G$

$$var(g) = G * \sigma_g^2$$

where $G$ is called genomic relationship matrix (GRM)

Properties of $G$

• genomic breeding values $g$ are linear combinations of $q$
• $g$ as deviations, that means $E(g) = 0$
• $var(g)$ as product between $G$ and $\sigma_g^2$ where $G$ is the genomic relationship matrix
• $G$ should be similar to $A$

Change of Identity Concept

• $A$ is based on identity by descent
• $G$ is based on identity by state (including ibd), assuming that the same allele has the same effect
• IBS can only be observed with SNP-genotype data

Identity

#rmddochelper::use_odg_graphic(ps_path = "odg/ibdvsibs.odg")
knitr::include_graphics(path = "odg/ibdvsibs.png")

Linear Combination

• SNP marker effects ($a$ values) from marker effect model are in vector $q$
• Genomic breeding values from breeding value model are determined by

$$g = U \cdot q$$

• Matrix $U$ is determined by desired properties of $g$

Deviation

• Genomic breeding values are defined as deviation from a certain basis

$\rightarrow$ $E(g) = 0$

• How to determine matrix $U$ such that $E(g) = 0$

Equivalence Between Models

Decomposition of phenotypic observation $y_i$ with

• Marker effect model

$$y_i = w_i^T \cdot q + e_i$$

• Breeding value model

$$y_i = g_i + e_i$$

• $g_i$ and $w_i^T \cdot q$ represent the same genetic effects and should be equivalent in terms of variability

Expected Values

• Required: $E(g_i) = 0$
• But: $E(w_i^T \cdot q) = q^T \cdot E(w_i)$
• Take $q$ as constant SNP effects
• Assume $w_i$ to be the random variable with:

\begin{equation} w_i = \left{ \begin{array}{lll} 1 & \text{ with probability } & p^2 \ 0 & \text{ with probability } & 2p(1-p) \ -1 & \text{ with probability } & (1-p)^2 \end{array} \right. \notag \end{equation}

$\rightarrow E(w_i):$ For a single locus

$$E(w_i) = 1 * p^2 + 0 * 2p(1-p) + (-1)(1-p)^2 = p^2 - 1 + 2p - p^2 = 2p - 1 \ne 0$$

Specification of $g$

• Set

$$g_i = (w_i^T - s_i^T) \cdot q$$ with $s_i = E(w_i) = 2p-1$

• Resulting in

$$g = U \cdot q = (W-S)\cdot q$$ with matrix $S$ having columns $j$ with all elements equal to $2p_j-1$ where $p_j$ is the allele frequency of the SNP allele associated with the positive effect.

Genetic Variance

• Requirement: $var(g) = G * \sigma_g^2$
• Result from Gianola et al. (2009):

$$\sigma_g^2 = \sigma_q^2 * \sum_{j=1}^k(1-2p_j(1-p_j))$$

• From earlier: $g = U \cdot q$

$$var(g) = var(U \cdot q) = U \cdot var(q) \cdot U^T = UU^T \sigma_q^2$$

• Combining

$$var(g) = UU^T \sigma_q^2 = G * \sigma_q^2 * \sum_{j=1}^k(1-2p_j(1-p_j))$$

Genomic Relationship Matrix

$$G = \frac{UU^T}{\sum_{j=1}^k(1-2p_j(1-p_j))}$$

How To Compute $G$

• Read matrix $W$
• For each column $j$ of $W$ compute frequency $p_j$
• Compute matrix $S$ and $\sum_{j=1}^k(1-2p_j(1-p_j))$ from $p_j$
• Compute $U$ from $W$ and $S$
• Compute $G$

charlotte-ngs/lbgfs2021 documentation built on Dec. 19, 2021, 3:01 p.m.