Evolution of the mean phenotype

We're going to focus on how the mean phenotype in a population changes in response to natural selection, specifically in response to viability selection. Before we can do this, however, we need to think a bit more carefully about the relationship between genotype, phenotype, and fitness. Let $F_{ij}(x)$ be the probability that genotype $A_iA_j$ has a phenotype smaller than $x$.³ Then $x_{ij}$, the genotypic value of $A_iA_j$ is

$$x_{ij} = \int_{-\infty}^\infty x \mbox{\rm dF}_{ij}(x)$$

and the population mean phenotype is $p^2x_{11} + 2pqx_{12} + q^2x_{22}$. If an individual with phenotype $x$ has fitness $w(x)$, then the fitness of an individual with genotype $A_iA_j$ is

$$w_{ij} = \int_{-\infty}^\infty w(x) \mbox{\rm dF}_{ij}(x)$$

and the mean fitness in the population is $\bar w = p^2w_{11} + 2pqw_{12} + q^2w_{22}$.

Now, there's a well known theorem from calculus known as Taylor's theorem. It says that for any function⁴ $f(x)$

$$f(x) = f(a) + \sum_{k=1}^\infty \left({{(x-a)^k} \over k!}\right) f^{(k)}(a) \quad .$$

Using this theorem we can produce an approximate expression describing how the mean phenotype in a population will change in response to selection. Remember that the mean phenotype, $\bar x$, depends both on the underlying genotypic values and on the allele frequency. So I'm going to write the mean phenotype as $\bar x(p)$ to remind us of that dependency.

$$\begin{eqnarray*} {\bar x}(p') &=& {\bar x}(p) + (p' - p)\left({d{\bar x} \over... ...Delta{\bar x} &=& (\Delta p)\left(2(\alpha_1 - \alpha_2)\right) \end{eqnarray*}$$

Now you need to remember (from lo those many weeks ago) that

$$p' = {p^2w_{11} + pqw_{12} \over \bar w} \quad .$$

Thus,

$$\begin{eqnarray*} \Delta p &=& p' - p \\ &=& {p^2w_{11} + pqw_{12} \over \b... ... p\left(pw_{11} + qw_{12} - \bar w \over \bar w \right) \quad . \end{eqnarray*}$$

Now,⁵ let's do a linear regression of fitness on phenotype. After all, to make any further progress, we need to relate phenotype to fitness, so that we can use the relationship between phenotype and genotype to infer the change in allele frequencies, from which we will infer the change in mean phenotype.⁶ From our vast statistical knowledge, we know that the slope of this regression line is

$$\beta_1 = {\mbox{Cov}(w,x) \over \mbox{Var}(x)}$$

and its intercept is

$$\beta_0 = \bar w - \beta_1 \bar x \quad .$$

Let's use this regression equation to determine the fitness of each genotype. This is only an approximation to the true fitness,⁷ but it is adequate for many purposes.

$$\begin{eqnarray*} w_{ij} &=& \int_{-\infty}^\infty w(x) \mbox{\rm dF}_{ij}(x) \... ...+ \beta_1x_{ij} \\ \bar w &=& \beta_0 + \beta_1\bar x \quad . \end{eqnarray*}$$

If we substitute this into our expression for $\Delta p$ above, we get

$$\begin{eqnarray*} \Delta p &=& p\left(pw_{11} + qw_{12} - \bar w \over \bar w ... ...\\ &=& {pq\beta_1(\alpha_1 - \alpha_2) \over \bar w} \quad . \end{eqnarray*}$$

So where are we now?⁸ From the equation for $\Delta\bar x$ we get

$$\begin{eqnarray*} \Delta\bar x &=& (\Delta p)\left(2(\alpha_1 - \alpha_2)\right... ...\right) \\ &=& V_a \left(\beta_1 \over \bar w\right) \quad . \end{eqnarray*}$$

This is great if we've done the regression between fitness and phenotype, but what if we haven't?⁹ Let's look at $\mbox{Cov}(w,x)$ in a little more detail.

$$\begin{eqnarray*} \mbox{Cov}(w,x) &=& p^2\int_{-\infty}^\infty x w(x) \mbox{\rm... ... x_{ij} \right) \\ &=& \bar w (x_{ij}^* - x_{ij}) \quad, \end{eqnarray*}$$

where $x_{ij}^*$ refers to the mean phenotype of $A_iA_j$ after selection. So

$$\begin{eqnarray*} \mbox{Cov}(w,x) &=& p^2\bar w(x_{11}^* - x_{11}) + 2pq\bar w(... ...w(x_{22}^* - x_{22}) \\ &=& \bar w(\bar x^* - \bar x) \quad , \end{eqnarray*}$$

where $\bar x^*$ is the population mean phenotype after selection. In short,¹⁰ combining our equations for the change in mean phenotype and for the covariance of fitness and phenotype and remembering that $\beta_1 = \mbox{Cov}(w,x)/Var(x)$¹¹

$$\begin{eqnarray*} \Delta\bar x &=& V_a \left({\bar w(\bar x^* - \bar x) \over ... ...} \over \bar w \right) \cr &=& h^2_N (\bar x^* - \bar x) \cr \end{eqnarray*}$$

$\Delta\bar x = \bar x' - \bar x$ is referred to as the response to selection and is often given the symbol $R$. It is the change in population mean between the parental generation (before selection) and the offspring beneration (before selection). $\bar x^* - \bar x$ is referred to as the selection differential and is often given the symbol $S$. It is the difference between the mean phenotype in the parental generation before selection and the mean phenotype in the parental generation after selection. Thus, we can rewrite our final equation as

$$R = h^2_N S \quad .$$