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)

Background

• Central Dogma of Molecular Biology

$\rightarrow$ Genotypes are the basis for phenotypic expression

• Start with simple model

$\rightarrow$ one locus that affects quantitative trait

Population

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

Terminology

• alleles: variants occuring at a given genetic Locus
• bi-allelic: only two alleles, e.g., $G_1$ and $G_2$ at a given locus $G$ in population
• genotype: combination of two alleles at locus $G$ in an individual
• homozygous: genotypes $G_1G_1$ and $G_2G_2$ where both alleles identical
• heterozygous: genotype $G_1G_2$ different alleles

Frequencies in Example Population

• genotype frequencies \vspace{-2ex} \begin{align} f(G_1G_1) &= \frac{4}{10} = 0.4 \notag \ f(G_1G_2) &= \frac{3}{10} = 0.3 \notag \ f(G_2G_2) &= \frac{3}{10} = 0.3 \notag \end{align}

• allele frequencies \vspace{-2ex} \begin{align} f(G_1) &= f(G_1G_1) + {1\over 2}f(G_1G_2) = 0.55 \notag \ f(G_2) &= f(G_2G_2) + {1\over 2}f(G_1G_2) = 0.45 \notag \end{align}

Hardy-Weinberg Equilibrium

• allele frequencies \begin{equation} f(G_1) = p \text{, } f(G_2) = q = 1-p \notag \end{equation}

• genotype frequencies

df_genfreq <- data.frame(Alleles = c("$G_1$", "$G_2$"),
                         G1 = c("$f(G_1G_1) = p^2$", "$f(G_1G_2) = p*q$"),
                         G2 = c("$f(G_1G_2) = p*q$", "$f(G_2G_2) = q^2$")) 
names(df_genfreq) <- c("Alleles", "$G_1$", "$G_2$")
knitr::kable(df_genfreq,
  booktabs = TRUE,
  align = "c",
  escape = FALSE
)

\begin{equation} f(G_1G_1) = p^2 \text{, } f(G_1G_2) = 2pq \text{, } f(G_2G_2) = q^2 \notag \end{equation}

Genotypic Values

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

Population Mean

• Expected value of genotypic value $V$ as discrete random variable

\begin{align} \mu &= V_{11} * f(G_1G_1) + V_{12} * f(G_1G_2) + V_{22} * f(G_2G_2) \notag \ &= a * p^2 + d *2pq + (-a) * q^2 \notag \ &= (p-q)a + 2pqd \notag \end{align}

Breeding Values Definition

The breeding value of an animal $i$ is defined as two times the difference between the mean value of offsprings of animal $i$ and the population mean.

Derivation of Breeding value for $G_1G_1$

\begin{center} {\renewcommand{\arraystretch}{1.7} \renewcommand{\tabcolsep}{0.2cm} \begin{tabular}{|c|c|c|} \hline & \multicolumn{2}{|c|}{Mates of $S$} \ \hline & $f(G_1) = p$ & $f(G_2) = q$ \ \hline Parent $S$ & & \ \hline $f(G_1) = 1$ & $f(G_1G_1) = p$ & $f(G_1G_2) = q$\ \hline \end{tabular}} \end{center}

Computation of Breeding value for $G_1G_1$

\begin{equation} \mu_{11} = pa + qd \notag \end{equation}

The breeding value $BV_{11}$ corresponds to

\begin{align} BV_{11} &= 2*(\mu_{11} - \mu) \notag \ &= 2\left(pa + qd - \left[(p - q)a + 2pqd \right] \right) \notag\ &= 2\left(pa + qd - (p - q)a - 2pqd \right) \notag\ &= 2\left(qd + qa - 2pqd\right) \notag \ &= 2\left(qa + qd(1 - 2p)\right) \notag \ &= 2q\left(a + d(1 - 2p)\right) \notag \ &= 2q\left(a + (q-p)d\right) \notag \end{align}

Computation of Breeding value for $G_2G_2$

\begin{equation} \mu_{22} = pd - qa \notag \end{equation}

The breeding value $BV_{22}$ corresponds to

\begin{align} BV_{22} &= 2*(\mu_{22} - \mu) \notag \ &= 2\left(pd - qa - \left[(p - q)a + 2pqd \right] \right) \notag \ &= 2\left(pd - qa - (p - q)a - 2pqd \right) \notag \ &= 2\left(pd - pa - 2pqd\right) \notag \ &= 2\left(-pa + p(1-2q)d\right) \notag \ &= -2p\left(a + (q - p)d\right) \notag \end{align}

Computation of Breeding value for $G_1G_2$

\begin{equation} \mu_{12} = 0.5pa + 0.5d - 0.5qa = 0.5\left[(p-q)a + d \right] \notag \end{equation}

The breeding value $BV_{12}$ corresponds to

\begin{align} BV_{12} &= 2*(\mu_{12} - \mu) \notag \ &= 2\left(0.5(p-q)a + 0.5d - \left[(p - q)a + 2pqd \right] \right) \notag \ &= 2\left(0.5pa - 0.5qa + 0.5d - pa + qa - 2pqd \right) \notag \ &= 2\left(0.5(q-p)a + (0.5 - 2pq)d \right) \notag \ &= (q-p)a + (1-4pq)d \notag \ &= (q-p)a + (p^2 + 2pq + q^2 -4pq)d \notag \ &= (q-p)a + (p^2 - 2pq + q^2)d \notag \ &= (q-p)a + (q - p)^2d \notag \ &= (q-p)\left[a + (q-p)d \right] \notag \end{align}

Summary of Breeding Values

tbl_sum_bv <- tibble::tibble(Genotype = c("$G_1G_1$", "$G_1G_2$", "$G_2G_2$"),
                             `Breeding Value` = c("$2q\\alpha$", "$(q-p)\\alpha$", "$-2p\\alpha$"))
knitr::kable(tbl_sum_bv,
             booktabs = TRUE,
             align = "c",
             escape = FALSE)

with $\alpha = a + (q-p)d$

Allele Substitution

\begin{align} BV_{12} - BV_{22} &= (q-p)\alpha - \left( -2p\alpha \right) \notag \ &= (q-p)\alpha + 2p\alpha \notag \ &= (q-p+2p)\alpha \notag \ &= (q+p)\alpha \notag \ &= \alpha \notag \end{align}

\begin{align} BV_{11} - BV_{12} &= 2q\alpha - (q-p)\alpha \nonumber \ &= \left(2q - (q-p)\right)\alpha \nonumber\ &= \alpha \notag \end{align}

Dominance Deviation I

\begin{align} V_{11} - BV_{11} &= a - 2q \alpha \notag \ &= a - 2q \left[ a + (q-p)d \right] \notag \ &= a - 2qa -2q(q-p)d \notag \ &= a(1-2q) - 2q^2d + 2pqd \notag \ &= \left[(p - q)a + 2pqd\right] - 2q^2d \notag \ &= \mu + D_{11} \notag \end{align}

Dominance Deviation II

\begin{align} V_{12} - BV_{12} &= d - (q-p)\alpha \notag \ &= d - (q-p)\left[ a + (q-p)d \right] \notag \ &= \left[(p-q)a + 2pqd\right] + 2pqd \notag \ &= \mu + D_{12} \notag \end{align}

\begin{align} V_{22} - BV_{22} &= -a - (-2p\alpha) \notag \ &= -a + 2p\left[ a + (q-p)d \right] \notag \ &= \left[(p-q)a + 2pqd\right] - 2p^2d \notag \ &= \mu + D_{22} \notag \end{align}

Summary of Values

\begin{tabular}{|c|c|c|c|} \hline Genotype & genotypic value & Breeding Value & Dominance Deviation \ $G_iG_j$ & $V_{ij}$ & $BV_{ij}$ & $D_{ij}$ \ \hline $G_1G_1$ & $a$ & $2q\alpha$ & $-2q^2d$ \ \hline $G_1G_2$ & $d$ & $(q-p)\alpha$ & $2pqd$ \ \hline $G_2G_2$ & $-a$ & $-2p\alpha$ & $-2p^2d$ \ \hline \end{tabular}

Decomposition of Genotypic Values

\begin{align} V_{ij} &= \mu + BV_{ij} + D_{ij} \notag \end{align}

Variances

\begin{equation} Var\left[X\right] = \sum_{x_i \in \mathcal{X}} (x_i - \mu_X)^2 * f(x_i) \notag \end{equation}

\vspace*{1ex} \begin{tabular}{p{1cm}p{1cm}p{6cm}} where & $\mathcal{X}$: & set of all possible $x$-values\ & $f(x_i)$ & probability that $x$ assumes the value of $x_i$ \ & $\mu_X $ & expected value $E\left[X\right]$ of $X$ \end{tabular}

Variance Computation

\begin{align} \sigma_G^2 = Var\left[V\right] &= (V_{11} - \mu)^2 * f(G_1G_1) \notag \ & +\ (V_{12} - \mu)^2 * f(G_1G_2) \notag \ & +\ (V_{22} - \mu)^2 * f(G_2G_2) \notag \end{align}

where $\mu = (p - q)a + 2pqd$ the population mean.

Simplification

\begin{align} \sigma_G^2 = Var\left[V\right] &= (BV_{11} + D_{11})^2 * f(G_1G_1) \notag \ & +\ (BV_{12} + D_{12})^2 * f(G_1G_2) \notag \ & +\ (BV_{22} + D_{22})^2 * f(G_2G_2) \notag \end{align}

Result

\begin{align} \sigma_G^2 &= 2pq\alpha^2 + \left(2pqd \right)^2 \notag\ &= \sigma_A^2 + \sigma_D^2 \notag \end{align}

Two and more Loci

• Two loci $G$ and $H$ having an effect on the same quantitative trait.
• Effect of one locus can have an influence on the effect of the other locus

$\rightarrow$ Interaction between loci.

• Interaction is quantified by

$$I_{GH} = V - V_G - V_H$$ where $V$ is the total genotypic value, $V_G$ and $V_H$ correspond to the genotypic values due to loci $G$ and $H$, respectively

Decomposition and Collecting Terms

• Genotypic values can be decomposed as

$$V_G = \mu_G + BV_G + D_G$$ $$V_H = \mu_H + BV_H + D_H$$

• Collecting terms leads to

$$V = V_G + V_H + I_{GH} = \mu + U + D + I$$

with $\mu = \mu_G + \mu_H$, $U = BV_G + BV_H$, $D = D_G + D_H$ and $I = I_{GH}$

• Can be generalized to more than two loci.

More Than Two Loci

• Genotypic value $V$ influenced by an unknown number of loci: $A, B, C, \ldots$,
• Decomposition of $V$

$$V = V_A + V_B + V_C + \ldots + I_{ABC...}$$

where $I_{ABC...}$ is a generic Interaction term which we do not specify further here.

Decomposition

• Genotypic value of single loci

$$V_{A_iA_j} = \mu_A + BV_{A_iA_j} + D_{A_iA_j} $$ $$V_{B_iB_j} = \mu_B + BV_{B_iB_j} + D_{B_iB_j} $$ $$V_{C_iC_j} = \mu_C + BV_{C_iC_j} + D_{C_iC_j}$$

Collecting Terms

• for a given animal $k$, $BV$ and $D$ terms are collected

$$U_k = BV_k = BV_{A_iA_j} + BV_{B_iB_j} + BV_{C_iC_j} + \ldots $$

$$D_k = D_{A_iA_j} + D_{B_iB_j} + D_{C_iC_j} + \ldots$$

Phenotype

• Including phenotypic observations $y$
• Central Dogma of Molecular Biology
• Decomposition

$$y = V + E$$ where $V$ is the genotypic value and $E$ is the non-genetic or environmental rest

• Insert decomposition of $V$ as shown above

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