In charlotte-ngs/lbgfs2020: Livestock Breeding and Genomics (Fall 2020)

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

Assumption

• Variance of breeding value $u_i$: $var(u_i) = (A)_{ii} \sigma_u^2$
• Diagnoal elements of numerator relationship matrix $A$

$$(A)_{ii} = (1+F_i)$$

\begin{tabular}{llp{9cm}} where & & \ & $F_i$ & inbreeding coefficient of animal $i$ \ & $\sigma_u^2$ & genetic additive variance \end{tabular}

• The higher $F_i$, the higher the similarity, the higher the variance

$\rightarrow$ Contradiction?

Variance and Inbreeding

• Relationship between variance and inbreeding
• How does inbreeding affect the genetic variance?
• How do we get inbreeding in a population?

$\rightarrow$ Population separates into different lines

Population with Inbreeding

#rmdhelp::use_odg_graphic(ps_path = 'odg/ideal-pop.odg')
knitr::include_graphics(path = "odg/ideal-pop.png")

Assumptions

• $N$ individuals
• self-fertilizing, shedding constant rate of eggs and sperm
• at a given locus: alleles in base population are non-identical by descent
• $N$ individuals produce $2N$ gametes (eggs and sperm) at constant rate
• probability that a pair of gametes taken at random carry identical alleles: $1/2N$
• probability corresponds to the inbreeding coefficient ($F$)

Inbreeding Coefficient

• In generation 1:

$$F_1 = {1\over 2N}$$

• second generation
  • either de-novo match of alleles or
  • the same alleles from generation 1

$$F_2 = {1\over 2N} + (1-{1\over 2N})*F_1$$

• new variable $\Delta F = {1\over 2N}$, then

$$F_2 = \Delta F + (1-\Delta F) * F_1$$

Inbreeding Coefficient II

• Generation $t$: $F_t = \Delta F + (1-\Delta F) * F_{t-1}$
• Solving for $\Delta F$

\begin{equation} \Delta F = \frac{F_t - F_{t-1}}{1-F_{t-1}} \notag \end{equation}

• Panmicitic Index $P = 1-F$

$$\frac{P_t}{P_{t-1}} = 1 - \Delta F$$ $$P_t = (1 - \Delta F)^t * P_0 \text{ with } P_0 = 1$$ $$F_t = 1 - (1 - \Delta F)^t$$

Variance of Gene Frequency

• Allele frequencies ($p$ and $q$) no longer constant in lines $\rightarrow$ variation
• Variance of change of $q$ (same for $p$)

\begin{equation} \sigma_{\Delta q}^2 = \frac{p_0q_0}{2N} = p_0q_0 \Delta F \notag \end{equation}

• Variance of $q$ (same for $p$)

\begin{equation} \sigma_{q}^2 = p_0q_0 F \notag \end{equation}

Genotype Frequencies

• Average genotype frequency of homozygotes across all lines: $\bar{q^2}$
• Definition of variance

$$\sigma_{q}^2 = \bar{q^2} - \bar{q}^2$$ where $\bar{q}$ is the mean allele frequency across all lines and hence is the same as $q_0$ in the base population

• Therefore

$$\bar{q^2} = q_0^2 + \sigma_{q}^2 = q_0^2 + p_0q_0 F$$

Genotype Frequencies II

tbl_genofreq <- tibble::tibble(Genotype = c("$A_1A_1$", "$A_1A_2$", "$A_2A_2$"),
                               `Original Frequencies` = c("$p_0^2$", "$2p_0q_0$", "$q_0^2$"),
                               `Changes due to inbreeding` = c("$+p_0q_0 F$", "$-2p_0q_0 F$", "$+p_0q_0 F$"))
knitr::kable(tbl_genofreq,
             booktabs = TRUE, 
             caption = "Genotype Frequencies for a bi-allelic locus, expressed in terms of inbreeding coefficient $F$",
             escape = FALSE)  

Changes of Mean Value

tbl_genovalue <- tibble::tibble(Genotype = c("$A_1A_1$", "$A_1A_2$", "$A_2A_2$"),
                                Frequency = c("$\\bar{p}^2 + \\bar{p}\\bar{q}F$", "$2\\bar{p}\\bar{q} - 2\\bar{p}\\bar{q}F$","$\\bar{q}^2 + \\bar{p}\\bar{q}F$"),
                                Value = c("$a$", "$d$", "$-a$"),
                                Product = c("$(\\bar{p}^2 + \\bar{p}\\bar{q}F)a$","$(2\\bar{p}\\bar{q} - 2\\bar{p}\\bar{q}F)d$","$-(\\bar{q}^2 + \\bar{p}\\bar{q}F)a$"))
knitr::kable(tbl_genovalue,
             booktabs = TRUE, 
             caption = "Derivation of Inbreeding Depression",
             escape = FALSE) 

Inbreeding Depression

\begin{align} M_F &= (\bar{p}^2 + \bar{p}\bar{q}F)a + (2\bar{p}\bar{q} - 2\bar{p}\bar{q}F)d - (\bar{q}^2 + \bar{p}\bar{q}F)a \notag \ &= a(\bar{p} - \bar{q}) + 2d\bar{p}\bar{q} - 2d\bar{p}\bar{q}F \notag \ &= a(\bar{p} - \bar{q}) + 2d\bar{p}\bar{q}(1-F) \notag \ &= M_0 - 2d\bar{p}\bar{q}F \notag \end{align}

Changes of Variance

• Only additive locus
• Within line: as before

\begin{align} V_{\bar{G}} &= 2(\bar{pq})a^2 \notag \ &= 2p_0q_0(1-F) \notag \ &= V_G(1-F) \notag \end{align}

• New variance component: between line

\begin{equation} var(M) = \sigma_M^2 = 4a^2 \sigma_q^2 = 4a^2p_0q_0F = 2FV_G \notag \end{equation}

Summary

tbl_gen_anova <- tibble::tibble(Source = c("Between lines", "Within lines","Total"),
                                Variance = c("$2FV_G$", "$(1-F)V_G$", "$(1+F)V_G$"))

knitr::kable(tbl_gen_anova,
             booktabs = TRUE, 
             caption = "Partitioning of the variance in a population with inbreeding coefficient F",
             escape = FALSE) 

charlotte-ngs/lbgfs2020 documentation built on Dec. 20, 2020, 5:39 p.m.