Quant. Gen. IV: Response to selection

In this final lecture, we will address the second of the traditional concerns of quantitative genetics: Short term responses of population trait means to selection.

The QG equivalent of “if nothing happens, then nothing happens”

In our previous lecture on Correlation between relatives, we used linear regression primarily get estimates of trait heritabilities. These younger $\sim$ older generation regressions make predictions about expected phenotypes given relatives’ phenotypes, but nothing more. They do not tell us anything about how the distribution of phenotypes is expected to change in future generations.

This leads us to a quantitative genetics version of our “if nothing happens, then nothing happens” mantra. The slope of these regression lines is determined by the coefficient of relatedness, and the regression line, by definition, passes through the population mean. Under random mating, there will be no expected change in the trait means:

Figure 1: Without selection, nothing happens!

If we examine Figure 1 carefully, we can see that even if an offspring has an above-average phenotype, their mate will most likely have a smaller phenotype (remember, random mating!), and any resulting offspring in the next generation will regress towards the mean.

Insight

If there is no systematic difference in individuals’ fitness due to their phenotype, nothing happens: $\overline{z}_{off} = \overline{z}_{par}$.

What happens when fitness DOES covary with phenotype?

Fitness can covary with phenotype in a few different ways:

Figure 2: Three important forms of selection on quantitative traits

In this course, we are going to focus primarily on Directional selection. However, read through the brief explanations for stabilizing and disruptive selection to the right.

Under Stabilizing selection:: Given sufficient standing genetic variation, or if genetic variation is generated rapidly enough to counteract loss due to stabilizing selection, $V_G$ can be assumed to remain constant across generations, and nothing happens. i.e., $\Var(z_{off}) = \Var(z_{par})$

Under disruptive selection:: With disruptive selection, things get a bit tricky. If the concave fitness surface is perfectly symmetrical, then no change in the mean phenotype is expected (i.e., $\overline{z}_{off} = \overline{z}_{par}$). However, since individuals in the tail of the phenotype distribution have highest fitness, the variance of the distribution is expected to increase over time (i.e., i.e., $\Var(z_{off}) \neq \Var(z_{par})$). That is, $V_G$ cannnot be assumed to remain constant. However, the situation is complicated if the fitness surface is not symmetrical, as this can lead to net directional selection AND changing phenotypic variance. If strong enough, disruptive selection can sometimes lead to bifurcation of the trait distribution, leading to a bimodal distribution, or even evolutionary branching.

Response to directional selection

If fitness covaries with phenotype in a roughly linear fashion, we get directional selection on the trait in question. This will result in a change in mean phenoype in the next generation. If selection is not too strong, we can also make the simplifying assumption that the variance in the trait mean will remain the same.

Figure 3: Directional selection shifts the population mean phenotype.

Response to selection: The change in mean phenotype from $t = 0$ to $t = 1$ is the Response to selection, denoted $\mcal{R}$: \[ \mcal{R} = \overline{z}_R - \overline{z}_0 \]

The Breeder’s equation

We now have a simple expression for the response to selection, but how can we quantify the intensity of selection? and what is the relationship between the selection differential, $\mcal{S}$ and the response, $\mcal{R}$? Let’s return for a moment to a situation where we know the midparent value for each offspring:

Recall from last lecture that we know the mean value of the offspring with midparent value $P_M = z$ is equal to $h^2 z$. The expected phenotype of offspring from parents with midparent value $z_i$ is therefore equal to $h^2 z_i$. So, in a selection experiment with $n$ pairs of parents, the expected offspring phenotype is:

\[ \frac{1}{n} \sum_i^n \E \left[ P_O | P_M = z_i \right] = \frac{h^2}{n} \sum_i^n z_i \]

The term on the left is the average expected offspring value (expressed as a deviation from the midparent value), which is the selection response, $\mcal{R}$.
The sum on the right side is the mean phenotypic value of selected parents, or the selection differential, $\mcal{S}$.

Insight

In other words, the selection response is simply:

\[ \mcal{R} = h^2 \mcal{S} \]

This is the univariate (i.e., one trait) Breeder’s Equation, an extremely robust formula describing the single-generation response to selection for a quantitative trait.

Example: Truncation selection

Figure 4: Truncation Selection, a common strategy for breeding programs where breeding pairs are selected from the desired end of the phenotypic distribution.

A few important caveats

$\mcal{R} = h^2 \mcal{S}$ holds only if there is equal selection in both parents!. If there is selection on only one parent (e.g., milk production in cows), then $\mcal{R} = \frac{h^2}{2} \mcal{S}$.

If there is assortative mating by the selected trait, information from one parent also tells us about the other parent. Let’s denote the correlation between parent phenotypes as $\rho$, then¹:

¹ For other relatives, it gets more complicated…

\[ \text{Slope of offspring} \sim \text{single-parent regression is: } \frac{h^2}{2} (1 + \rho) \]

Supplement: Extending to multiple traits

Everything we have covered so far in quantitative genetics has focused on single traits. But organisms are complex entities with many traits that jointly determine an individual’s fitness. How do we deal with these realities? How do genetic constraints fit into our framework?

We defined the selection differential as $\mcal{S} = \bar{z}_s - \bar{z}_0$. To derive a more general expression requires defining relative fitness.

The frequency of individuals in the next generation that will be produced by individuals with phenotype $z$ in the current generation is:

\[ f^{\prime}(z) = f(z)\frac{W(z)}{\overline{W}(z)} = f(z) w(z) \]

We know the relation between offspring and parent phenotypes: $z_{\text{off}} = \bar{z} + h^2 (z - \bar{z}) + \epsilon$. With a little calculus, we can derive a more general expression for the change in mean phenotype over a single generation: $\Delta \bar{z} = h^2 \Cov[z,w]$.

Insight: Notice the many equivalences

\[ \begin{aligned} \mcal{S} &= \Cov[z,w] = \bar{z}_s - \bar{z} \\ \mcal{R} &= h^2 \mcal{S} \\ \Delta \bar{z} &= h^2 \Cov[z,w] \\ \beta &= \frac{\Cov[z,w]}{V_P}, \,\text{a regression coefficient}\\ h^2 &= \frac{V_A}{V_P} \\ \Delta \bar{z} &= V_A \beta \end{aligned} \]

The response to selection is proportional to the additive genetic variance times the selection coefficient!

To generalize to the evolution of multiple traits, we note that:

\[ \begin{aligned} \bar{z}^{\prime} &= \overline{z^o w^p} \\ &= \Cov[z^o,w^p] + \bar{z}^o \\ \Delta \bar{z} &= \Cov[z^o,w^p] + \bar{z}^o - \bar{z}^p \\ \Delta \bar{z} &= \Cov[z^o,w^p] \end{aligned} \]

Where the last step assumes that expected offspring is equal to the expected phenotype of their parents.

We then decompose total relative fitness of parents into the contributions by each trait:

\[ w^p = w_0 + \beta_1 z_1^p + \beta_2 z_2^p + \ldots + + \beta_n z_n^p \]

Substituting into the previous equation gives:

\[ \Delta \bar{z}_i = \sum_j \beta_j \Cov[z_i^o,z_j^p] \]

Or in a more compact matrix form as: $\Delta \bar{\mathbf{z}} = \mathbf{G} \beta$:

\[ \left[ \begin{array}{c} \Delta \bar{z}_1 \\ \Delta \bar{z}_2 \\ \vdots \\ \Delta \bar{z}_n \end{array} \right] = \left[ \begin{array}{cccc} G_{1,1} & G_{1,2} & \cdots & G_{1,n} \\ G_{2,1} & G_{2,2} & \cdots & G_{2,n}\\ \vdots & \vdots & \ddots & \vdots \\ G_{n,1} & G_{n,2} & \cdots & G_{n,n} \\ \end{array} \right] \left[ \begin{array}{c} \beta_1 \\ \beta_2 \\ \vdots \\ \beta_n \end{array} \right] \]

$\mathbf{G}$ is the Genetic variance-covariance matrix!

Diagonal terms of $\mathbf{G}$ are $V_A$’s.
Off-diagonal terms are genetic covariances between pairs of traits.

So… what does this tell us? Let’s expand it out a bit:

\[ \begin{aligned} \Delta \bar{z}_1 &= G_{11} \beta_1 + G_{12} \beta_2 + \ldots + G_{1n} \beta_n \\ \Delta \bar{z}_2 &= G_{21} \beta_1 + G_{22} \beta_2 + \ldots + G_{2n} \beta_n \\ \vdots &= \vdots \\ \Delta \bar{z}_n &= G_{n1} \beta_1 + G_{n2} \beta_2 + \ldots + G_{nn} \beta_n \\ \end{aligned} \]

Insight

Change in each trait depends on covariances with other traits!!!!!

If covariance terms in $\mathbf{G}$ are non-zero, traits will not evolve independently!
Terms including covariances represent indirect selection

Let’s illustrate how genetic covariances between traits can influence trait evolution in the short term:

Figure 5: Selection on two quantitative traits, illustrating effects of genetic covariances.

--- title: "Quant. Gen. IV: Response to selection" --- ::: {.hidden} $$ \usepackage{amsmath} \usepackage{amssymb} \def\mathbi#1{\textbf{\em #1}} \def\mbf#1{\mathbf{#1}} \def\mbb#1{\mathbb{#1}} \def\mcal#1{\mathcal{#1}} \newcommand{\bo}[1]{{\bf #1}} \newcommand{\tr}{{\mbox{\tiny \sf T}}} \newcommand{\bm}[1]{\mbox{\boldmath $#1$}} \newcommand{\norm}[1]{\left\lVert#1\right\rVert} \DeclareMathOperator{\E}{\mathbb{E}} \DeclareMathOperator{\Var}{\text{Var}} \DeclareMathOperator{\Cov}{\text{Cov}} \DeclareMathOperator{\Corr}{\text{Corr}} $$ ```{python} #| echo: FALSE from scipy import * import matplotlib.pyplot as plt import numpy as np import random import os from pylab import * exec(open('./code/Allele-A1-functions.py').read()) ``` ::: In this final lecture, we will address the second of the traditional concerns of quantitative genetics: _Short term responses of population trait means to selection_. ## The QG equivalent of "if nothing happens, then nothing happens" In our previous lecture on [Correlation between relatives](QuantGen-III.qmd), we used linear regression primarily get estimates of trait heritabilities. These younger $\sim$ older generation regressions make predictions about expected phenotypes given relatives' phenotypes, but nothing more. They do not tell us anything about how the distribution of phenotypes is expected to change in future generations. This leads us to a quantitative genetics version of our "if nothing happens, then nothing happens" mantra. The slope of these regression lines is determined by the coefficient of relatedness, and the regression line, by definition, passes through the population mean. Under random mating, there will be no expected change in the trait means: ![Without selection, nothing happens!](./img/QG-if-nothing-happens.jpeg){width=85% #fig-QG-nothing-happens} If we examine @fig-QG-nothing-happens carefully, we can see that even if an offspring has an above-average phenotype, their mate will most likely have a smaller phenotype (remember, random mating!), and any resulting offspring in the next generation will regress towards the mean. :::{.callout-tip} ## Insight If there is no systematic difference in individuals' fitness due to their phenotype, nothing happens: $\overline{z}_{off} = \overline{z}_{par}$. ::: ## What happens when fitness DOES covary with phenotype? Fitness can covary with phenotype in a few different ways: ![Three important forms of selection on quantitative traits](./img/QG-different-sel.jpeg){width=85% #fig-QG-diffSel} In this course, we are going to focus primarily on Directional selection. However, read through the brief explanations for stabilizing and disruptive selection to the right. :::{.column-margin} Under Stabilizing selection: : Given sufficient standing genetic variation, or if genetic variation is generated rapidly enough to counteract loss due to stabilizing selection, $V_G$ can be assumed to remain constant across generations, and nothing happens. i.e., $\Var(z_{off}) = \Var(z_{par})$ ::: :::{.column-margin} Under disruptive selection: : With disruptive selection, things get a bit tricky. If the concave fitness surface is perfectly symmetrical, then no change in the mean phenotype is expected (i.e., $\overline{z}_{off} = \overline{z}_{par}$). However, since individuals in the tail of the phenotype distribution have highest fitness, the variance of the distribution is expected to increase over time (i.e., i.e., $\Var(z_{off}) \neq \Var(z_{par})$). That is, $V_G$ **cannnot** be assumed to remain constant. However, the situation is complicated if the fitness surface is not symmetrical, as this can lead to net directional selection AND changing phenotypic variance. If strong enough, disruptive selection can sometimes lead to bifurcation of the trait distribution, leading to a bimodal distribution, or even evolutionary branching. ::: ## Response to directional selection If fitness covaries with phenotype in a roughly linear fashion, we get directional selection on the trait in question. This will result in a change in mean phenoype in the next generation. If selection is not too strong, we can also make the simplifying assumption that the variance in the trait mean will remain the same. ![Directional selection shifts the population mean phenotype.](./img/QG-Directional-R.jpeg){width=85% #fig-QG-directional} Response to selection : The change in mean phenotype from $t = 0$ to $t = 1$ is the **Response to selection**, denoted $\mcal{R}$: $$ \mcal{R} = \overline{z}_R - \overline{z}_0 $$ ## The Breeder's equation We now have a simple expression for the response to selection, but how can we quantify the _intensity_ of selection? and what is the relationship between the selection differential, $\mcal{S}$ and the response, $\mcal{R}$? Let's return for a moment to a situation where we know the midparent value for each offspring: Recall from last lecture that we know the mean value of the offspring with midparent value $P_M = z$ is equal to $h^2 z$. The expected phenotype of offspring from parents with midparent value $z_i$ is therefore equal to $h^2 z_i$. So, in a selection experiment with $n$ pairs of parents, the expected offspring phenotype is: $$ \frac{1}{n} \sum_i^n \E \left[ P_O | P_M = z_i \right] = \frac{h^2}{n} \sum_i^n z_i $$ - The term on the left is the average expected offspring value (expressed as a deviation from the midparent value), which is the selection response, $\mcal{R}$. - The sum on the right side is the mean phenotypic value of selected parents, or the selection differential, $\mcal{S}$. :::{.callout-tip} ## Insight In other words, the selection response is simply: $$ \mcal{R} = h^2 \mcal{S} $$ This is the univariate (i.e., one trait) Breeder's Equation, an extremely robust formula describing the single-generation response to selection for a quantitative trait. ::: ### Example: Truncation selection ![Truncation Selection, a common strategy for breeding programs where breeding pairs are selected from the desired end of the phenotypic distribution.](./img/QG-Trunc-Sel.jpeg){width=85% #fig-QG-truncation} ## A few important caveats $\mcal{R} = h^2 \mcal{S}$ **holds only if there is equal selection in both parents!**. If there is selection on only one parent (e.g., milk production in cows), then $\mcal{R} = \frac{h^2}{2} \mcal{S}$. If there is assortative mating by the selected trait, information from one parent also tells us about the other parent. Let's denote the correlation between parent phenotypes as $\rho$, then[^1]: $$ \text{Slope of offspring} \sim \text{single-parent regression is: } \frac{h^2}{2} (1 + \rho) $$ [^1]: For other relatives, it gets more complicated... ## Supplement: Extending to multiple traits Everything we have covered so far in quantitative genetics has focused on _single_ traits. But organisms are complex entities with _many traits_ that jointly determine an individual's fitness. How do we deal with these realities? How do genetic constraints fit into our framework? We defined the selection differential as $\mcal{S} = \bar{z}_s - \bar{z}_0$. To derive a more general expression requires defining relative fitness. The frequency of individuals in the next generation that will be produced by individuals with phenotype $z$ in the current generation is: $$ f^{\prime}(z) = f(z)\frac{W(z)}{\overline{W}(z)} = f(z) w(z) $$ We know the relation between offspring and parent phenotypes: $z_{\text{off}} = \bar{z} + h^2 (z - \bar{z}) + \epsilon$. With a little calculus, we can derive a more general expression for the change in mean phenotype over a single generation: $\Delta \bar{z} = h^2 \Cov[z,w]$. :::{.callout-tip} ## Insight: Notice the many equivalences $$ \begin{aligned} \mcal{S} &= \Cov[z,w] = \bar{z}_s - \bar{z} \\ \mcal{R} &= h^2 \mcal{S} \\ \Delta \bar{z} &= h^2 \Cov[z,w] \\ \beta &= \frac{\Cov[z,w]}{V_P}, \,\text{a regression coefficient}\\ h^2 &= \frac{V_A}{V_P} \\ \Delta \bar{z} &= V_A \beta \end{aligned} $$ **The response to selection is proportional to the additive genetic variance times the selection coefficient!** ::: To generalize to the evolution of multiple traits, we note that: $$ \begin{aligned} \bar{z}^{\prime} &= \overline{z^o w^p} \\ &= \Cov[z^o,w^p] + \bar{z}^o \\ \Delta \bar{z} &= \Cov[z^o,w^p] + \bar{z}^o - \bar{z}^p \\ \Delta \bar{z} &= \Cov[z^o,w^p] \end{aligned} $$ :::{.column-margin} Where the last step assumes that expected offspring is equal to the expected phenotype of their parents. ::: We then decompose total relative fitness of parents into the contributions by each trait: $$ w^p = w_0 + \beta_1 z_1^p + \beta_2 z_2^p + \ldots + + \beta_n z_n^p $$ Substituting into the previous equation gives: $$ \Delta \bar{z}_i = \sum_j \beta_j \Cov[z_i^o,z_j^p] $$ Or in a more compact matrix form as: $\Delta \bar{\mathbf{z}} = \mathbf{G} \beta$: $$ \left[ \begin{array}{c} \Delta \bar{z}_1 \\ \Delta \bar{z}_2 \\ \vdots \\ \Delta \bar{z}_n \end{array} \right] = \left[ \begin{array}{cccc} G_{1,1} & G_{1,2} & \cdots & G_{1,n} \\ G_{2,1} & G_{2,2} & \cdots & G_{2,n}\\ \vdots & \vdots & \ddots & \vdots \\ G_{n,1} & G_{n,2} & \cdots & G_{n,n} \\ \end{array} \right] \left[ \begin{array}{c} \beta_1 \\ \beta_2 \\ \vdots \\ \beta_n \end{array} \right] $$ $\mathbf{G}$ is the Genetic variance-covariance matrix! * Diagonal terms of $\mathbf{G}$ are $V_A$'s. * Off-diagonal terms are genetic covariances between pairs of traits. So... what does this tell us? Let's expand it out a bit: $$ \begin{aligned} \Delta \bar{z}_1 &= G_{11} \beta_1 + G_{12} \beta_2 + \ldots + G_{1n} \beta_n \\ \Delta \bar{z}_2 &= G_{21} \beta_1 + G_{22} \beta_2 + \ldots + G_{2n} \beta_n \\ \vdots &= \vdots \\ \Delta \bar{z}_n &= G_{n1} \beta_1 + G_{n2} \beta_2 + \ldots + G_{nn} \beta_n \\ \end{aligned} $$ :::{.callout-tip} ## Insight **Change in each trait depends on covariances with other traits!!!!!** * If covariance terms in $\mathbf{G}$ are non-zero, traits will not evolve independently! * Terms including covariances represent _indirect selection_ ::: Let's illustrate how genetic covariances between traits can influence trait evolution in the short term: ![Selection on two quantitative traits, illustrating effects of genetic covariances.](./img/QG-Genetic-Constraint.jpeg){width=85% #fig-QG-twoTrait}