class: center, middle, inverse, title-slide # Ch7. Dominance and Interaction ### Jinliang Yang ### Nov. 1st, 2019 --- # Dominance deviation (D) | Genotype | Value as deviated from `\(M\)` | Breeding Value | Dominance Deviation | | :-------: | :-------: | :-----------: | :-----------: | :-------: | :-------: | | `\(A_1A_1\)` | `\(2q(\alpha - qd)\)` | `\(2q\alpha\)` | `\(-2q^2d\)` | | `\(A_1A_2\)` | `\((q-p)\alpha + 2pqd\)` | `\((q-p)\alpha\)` | `\(2pqd\)` | | `\(A_2A_2\)` | `\(-2p(\alpha + pd)\)` | `\(-2p\alpha\)` | `\(-2p^2d\)` | - The mean of the BV is **zero** - It follows that the mean dominance deviation is **zero**. The mean dominance deviation: `\begin{align*} = & p^2 \times (-2q^2d) + 2pq \times 2pqd + q^2 \times (-2p^2d) \\ = & - 2p^2q^2d + 4p^2q^2d - 2p^2q^2d \\ = & 0 \end{align*}` --- # A Linear Regression Perspective ### G = A + D - **A** repsents the breeding value (i.e., A = `\(\alpha_i + \alpha_j\)`) - **D** represents the dominance deviation -- Further breakdown `\(A\)`: `\begin{align*} G = & \alpha_1N_1 + \alpha_2N_2 + \delta \end{align*}` where, - `\(\alpha_i\)` is the average effect of allele `\(i\)` and `\(\alpha = \alpha_1 - \alpha_2\)` - `\(N_i\)` is the number of allele `\(i\)` carried by the genotype - `\(N \in \{0, 1, 2\}\)` for a bi-allelic locus and `\(N_1 + N_2 = 2\)` -- Therefore, `\begin{align*} G = & \alpha_1N_1 + \alpha_2N_2 + \delta = \alpha_1N_1 + \alpha_2(2 - N_1) + \delta \\ = & 2\alpha_2 + (\alpha_1 - \alpha_2)N_1 + \delta \\ = & (2\alpha_2 + \delta) + \alpha N_1 \end{align*}` --- # Graphical Representation ```r a = 1; d = 3/4*a p = 4/5; q = 1 - p alpha <- a + d*(q - p) a1a1 <- 2*alpha*q a1a2 <- (q-p)*alpha a2a2 <- -2*p*alpha plot(c(0, 1, 2), c(-a, d, a), xlab="Genotype",ylab="", cex.lab=1.5, xaxt="n", pch=17, cex=2, col="red"); axis(1, at=c(0, 1, 2), labels=c("A2A2", "A1A2", "A1A1")); mtext("Breeding Value", side = 4, line = 1, cex=1.5, col="blue"); mtext("Genotypic Value", side = 2, line = 2, cex=1.5, col="red") points(c(0, 1, 2), c(a2a2, a1a2, a1a1), cex=2, col="blue") lines(c(0, 1, 2), c(a2a2, a1a2, a1a1), lwd=2, col="blue") ``` <img src="Chapter7_c3_files/figure-html/linear-1.png" width="35%" style="display: block; margin: auto;" /> --- # Graphical Representation `\begin{align*} G = & (2\alpha_2 + \delta) + \alpha N_1 \end{align*}` .pull-left[ <div align="center"> <img src="Chapter7_c2_files/figure-html/linear-1.png" height=350> </div> ] .pull-right[ - The slope is the average effect of allele substitution. As `\(A_1\)` is substituted by `\(A_2\)`, the breeding value increaes at a rate equal to `\(\alpha\)`. - Dominance deviation are the differences between the genotypic values and breeding values. - Dominance can be seen as residuals from the fitted regression line. ] --- # Interaction deviation (epistasis) - Epistasis = interaction of alleles at **different** loci - Also causes deviation of genotypic value from additive genetic value (or BV) -- Imagine a trait controlled by two loci, A and B. Then `\begin{align*} G = & G_A + G_B + I_{AB} \\ = & (\alpha_i + \alpha_j + \delta_{ij}) + (\alpha_k + \alpha_l + \delta_{kl}) + I_{AB} \end{align*}` -- Where `\begin{align*} I_{AB} = & \alpha_i\alpha_k + \alpha_i\alpha_l + \alpha_j\alpha_k + \alpha_j\alpha_l \\ & + \alpha_i\delta_{kl} + \alpha_j\delta_{kl} + \alpha_k\delta_{ij} + \alpha_l\delta_{ij} \\ & + \delta_{ij}\delta_{kl} \end{align*}` --- # Interaction deviation (epistasis) - Interaction effect can arise through interaction between the average allele effects at each locus, between average effect and dominance deviation, and between dominance deviations - As # of loci increases, so does # of interactions (exponentially) -- - Summed to contribute to the epistatic deviation ( `\(I\)` ) `\begin{align*} G = \sum_{i=1}^k(A_i + D_i) + I \end{align*}`