class: center, middle, inverse, title-slide # Ch11: Selection ### Jinliang Yang ### Nov. 9th, 2020 --- # The improvement through breeding .pull-left[ <div align="center"> <img src="yield.png" height=300> </div> <div align="center"> <img src="milk-per-cow.jpg" height=200> </div> ] -- .pull-right[ ## Response to selection ### 1. Why has a character changed over generations? ### 2. Can we predict a change? especially one that we wish for in a direction, through actions such as imposing artificial selection? ] --- # Response to selection Change, or response ( `\(R\)` ), is given by the basic equation: `\begin{align*} R = h^2S \end{align*}` Here S is the __selection differential__, which is equal to the mean value of the selected parents ( `\(\mu_S\)` ) minus the population mean ( `\(\mu\)` ). ------ -- <div align="center"> <img src="fig11.png" height=300> </div> --- # Response to selection Change, or response ( `\(R\)` ), is given by the basic equation: `\begin{align*} R = h^2S \end{align*}` Here S is the __selection differential__, which is equal to the mean value of the selected parents ( `\(\mu_S\)` ) minus the population mean ( `\(\mu\)` ). ------ The regression of offspring on mid-parent is equal to the heritability - no non-genetic cause of resemblance - no natural selection, i.e. same fertility and viability -- Therefore, `\begin{align*} R = b_{O\bar{P}}S \\ R = h^2S \end{align*}` --- # Response to selection Change, or response ( `\(R\)` ), is given by the basic equation: `\begin{align*} R = h^2S \end{align*}` Here S is the __selection differential__, which is equal to the mean value of the selected parents ( `\(\mu_S\)` ) minus the population mean ( `\(\mu\)` ). ------ .pull-left[ Improve the body weight <div align="center"> <img src="poultry.jpg" height=180> </div> ] .pull-right[ 1. Base population (N=100) mean = 1,000g 2. Selected 5 chicken as the parents, with mean =1,050g. What is `\(S\)`? ] --- # Response to selection Change, or response ( `\(R\)` ), is given by the basic equation: `\begin{align*} R = h^2S \end{align*}` Here S is the __selection differential__, which is equal to the mean value of the selected parents ( `\(\mu_S\)` ) minus the population mean ( `\(\mu\)` ). ------ .pull-left[ Improve the body weight <div align="center"> <img src="poultry.jpg" height=180> </div> ] .pull-right[ 1. Base population (N=100) mean = 1,000g 2. Selected 5 chicken as the parents, with mean =1,050g. If `\(b_{OP}\)` = 0.25. What is `\(R\)`? ] --- # Standardized selection differential With the assumption that the phenotypic distributions are normal. - Same `\(S\)` means different selection pressure if the variances of the normal distributions are different -- #### Selection differential in standard deviation units: `\begin{align*} \frac{S}{\sigma_P} \end{align*}` -- ### __Intensity of selection ( `\(i\)` )__ `\begin{align*} i = \frac{S}{\sigma_P} \end{align*}` - `\(i\)` is the "standarized selection differential" - it better reflects the selection effort --- # Intensity of selection ( `\(i\)` ) .pull-left[ - With the assumption that the phenotypic distributions are normal. - And standardize the selection differential: `\begin{align*} &i= S/\sigma_P \\ \end{align*}` ] .pull-right[ ```r curve(dnorm(x,0,1), xlim=c(-3,3), xaxt="n", xlab="Trait Value", main="", ylab="Density", lwd=3) fromd <- qnorm(.95); tod <- 3 S.x <- c(fromd, seq(fromd, tod, 0.01), tod) S.y <- c(0, dnorm(seq(fromd, tod, 0.01)), 0) polygon(S.x,S.y, col="grey") abline(v=mean(S.x**S.y)*2, col="blue", lwd=3); abline(v=0, col="red", lwd=3) ``` <img src="Ch11_c1_files/figure-html/unnamed-chunk-1-1.png" width="100%" style="display: block; margin: auto;" /> ] --- # Intensity of selection ( `\(i\)` ) .pull-left[ `\begin{align*} &i= S/\sigma_P \\ \end{align*}` - If `\(p\)` is __the proportion selected__, i.e. the proportion of the population falling beyond the point of truncation. - And `\(z\)` is the __height of the ordinate__ at the point of truncation. `\begin{align*} i = \frac{z}{p} \end{align*}` ] .pull-right[ ```r curve(dnorm(x,0,1), xlim=c(-3,3), xaxt="n", xlab="Trait Value", main="", ylab="Density", lwd=3) fromd <- qnorm(.95); tod <- 3 S.x <- c(fromd, seq(fromd, tod, 0.01), tod) S.y <- c(0, dnorm(seq(fromd, tod, 0.01)), 0) polygon(S.x,S.y, col="grey") abline(v=mean(S.x**S.y)*2, col="blue", lwd=3); abline(v=0, col="red", lwd=3) ``` <img src="Ch11_c1_files/figure-html/unnamed-chunk-2-1.png" width="100%" style="display: block; margin: auto;" /> ] --- # Intensity of selection ( `\(i\)` ) A table of these values is found in Appendix A of F&M. .pull-left[ ```r ifun <- function(p=0.5){ x=qnorm(p=(1-p)) # get the truncation point z=dnorm(x) # get z return(z/p) # get i } p <- seq(0.0001, 1, by=0.0001) i <- ifun(p) plot(p, i) ``` <img src="Ch11_c1_files/figure-html/unnamed-chunk-3-1.png" width="80%" style="display: block; margin: auto;" /> ] .pull-right[ ```r head(data.frame(p, i), 20) ``` ``` ## p i ## 1 0.0001 3.958480 ## 2 0.0002 3.789212 ## 3 0.0003 3.686955 ## 4 0.0004 3.612829 ## 5 0.0005 3.554381 ## 6 0.0006 3.505980 ## 7 0.0007 3.464589 ## 8 0.0008 3.428376 ## 9 0.0009 3.396149 ## 10 0.0010 3.367090 ## 11 0.0011 3.340611 ## 12 0.0012 3.316274 ## 13 0.0013 3.293746 ## 14 0.0014 3.272767 ## 15 0.0015 3.253129 ## 16 0.0016 3.234665 ## 17 0.0017 3.217235 ## 18 0.0018 3.200726 ## 19 0.0019 3.185040 ## 20 0.0020 3.170097 ``` ] --- # Breeder's equation Given: `\begin{align*} &R = h^2S\\ &S = i\sigma_P& \end{align*}` -- So now, we have, as a prediction: `\begin{align*} &R = h^2i\sigma_P\\ \end{align*}` where `\(i = \frac{z}{p}\)`. --- # Breeder's equation `\begin{align*} &R = h^2i\sigma_P\\ &i = \frac{z}{p}& \end{align*}` .pull-left[ ### Genders With differential selection opportunities between genders in sexual reproduction, we need to account for this in our predictions: `\begin{align*} &i = \frac{1}{2}(i_m + i_f)& \end{align*}` ] -- .pull-right[ ### Generation interval Generation interval (L) is the __"average age of the parents at the birth of their selected offspring"__. - it is when they effectively leave the next generation that is then sampled to repeat the selection process. `\begin{align*} &L = \frac{1}{2}(L_m + L_f)& \end{align*}` ] --- # Breeder's equation Now, on a time-constant basis to allow comparisons of alternatives: `\begin{align*} R & = \frac{i}{L}h^2\sigma_A\\ & = \frac{i_m + i_f}{L_m + L_f}h^2\sigma_A\\ \end{align*}` With the `\(h^2\)` and `\(\sigma_A\)` terms possibly constant, we can simply compare the i/L portions under different scenarios of selection and reproduction. --- # Example: selection for body weight in chicken A poultry breeder is selecting for 56-day body weight in chicken. - __Base population__: a random mating population of 154 males and 155 females. Mean value = 1,000g; sd = 50g. - __Selecion scheme__: 8 males and 48 females with the highest body weight to found the next generation. - __h2__: from previous parent-offspring regression and half-sib analysis, `\(h^2=0.45\)`. -- #### What is the predicted response to selection in next generation? `\begin{align*} & R = \frac{i_m + i_f}{2}h^2\sigma_P\\ \end{align*}` -- ```r i_m <- ifun(p= 8/154) #2.05 i_f <- ifun(p= 48/155) # 1.14 (i_m + i_f)/2*50*0.45 ``` ``` ## [1] 35.8358 ``` --- # Example: selection for body weight in chicken A chicken breeder is selecting for 56-day body weight in chicken. - __Base population__: a random mating population of 154 males and 155 females. Mean value = 1,000g; sd = 50g. - __Selecion scheme__: 8 males and 48 females with the highest body weight to found the next generation. - __h2__: from previous parent-offspring regression and half-sib analysis, h2=0.45. #### What is the selection differential for the male and female? `\begin{align*} &S = i\sigma_P& \end{align*}` -- ```r Sm <- i_m*50 #102 Sf <- i_f*50 # 57 ``` -- Why response 36g not equal to average value of the `\(S_m\)` and `\(S_f\)`? --- # Example: selection for body weight <div align="center"> <img src="fig11.png" height=300> </div> - `\(S_m =102\)` and `\(S_f=57\)` are the means of the selected parents - `\(R=36\)` is the mean of the offspring. -- Not equal because the heritability is less than one, which means that the phenotypes of the parents was less than a perfect indictor of their breeding values. --- # Improvement of response `\begin{align*} & R = \frac{i h^2\sigma_P}{L} \\ \end{align*}` The form of the breeder's equation allows us to clearly see how to maximize response to selection per unit of time. -- ### 1. Reduce the generation interval This factor is bound by the biology of the organisms, but ways to reduce the generation interval are possible. -- - selection during the off-season in plants - winter nursery - [speed breeding](https://youtu.be/Zwk1IGYPJiA) (Watson et al., 2018), up to 6 six generations per year -- - __genomic selection__ - Selection without phenotyping --- # Improvement of response `\begin{align*} & R = \frac{i h^2\sigma_P}{L} \\ \end{align*}` ### 2. Increase the heritability of the trait Depends on the genetic architecture, not always under the control of the breeder. -- - maximizing the repeatability of trait evaluation. - sound measurement methods and proper experimental design --- # Improvement of response `\begin{align*} & R = \frac{i h^2\sigma_P}{L} \\ \end{align*}` ### 3. Increase the selection intensity - reduce the number of individuals selected to serve as parents of the next generation -- - __Be cautious__: increase the inbreeding coefficient and hence loss of alleles through genetic drift. - __Increase `\(i\)` is likely not the most efficient path!__ i.e. p from 10% to 5% => i from 1.755 to 2.063. -- - increase the population size from which selections are made --- # Improvement of response `\begin{align*} & R = \frac{i h^2\sigma_P}{L} \\ \end{align*}` ### 4. Increase additive genetic variance `\begin{align*} & h^2 = \frac{\sigma_A^2}{\sigma^2_P}; & h = \frac{\sigma_A}{\sigma_P} \\ \end{align*}` -- `\begin{align*} & R = i h\sigma_A/L\\ \end{align*}` -- - The breeding population needs to contain adequate additive genetic variation for the trait of interest. - If no difference in breeding values exist between individuals within the population, genetic gain through selection is not possible.