Loss of beneficial alleles

We're going to confine our studies to our usual simple case: one locus, two alleles. We're also going to consider a very simple form of directional viability selection in which the heterozygous genotype is exactly intermediate in fitness.

$A_1A_1$	$A_1A_2$	$A_2A_2$
1 + s	$1 + \frac{1}{2}s$	1

After solving a reasonably complex partial differential equation, it can be shown that¹ the probability that allele $A_1$² is fixed, given that its current frequency is $p$ is

$$P_1(p) = \frac{1 - e^{-2N_esp}}{1 - e^{-2N_es}} \quad .$$ (1)

Now it won't be immediately evident to you, but this equation actually confirms our intuition that even selectively favored alleles may sometimes be lost as a result of genetic drift. How does it do that? Well, it's not too hard to verify that $P_1(p) < 1$.³ The probability that the beneficial allele is fixed is less than one meaning that the probability it is lost is greater than zero, i.e., there's some chance it will be lost.

How big is the chance that a favorable allele will be lost? Well, consider the case of a newly arisen allele with a beneficial effect. If it's newly arisen, there is only one copy by definition. In a diploid population of $N$ individuals that means that the frequency of this allele is $1/2N$. Plugging this into equation (1) above we find

$$\begin{eqnarray*} P_1(p) &=& \frac{1 - e^{-2N_es(1/2N)}}{1 - e^{-2N_es}} \\ ... ...ox& s\left(\frac{N_e}{N}\right) \hbox{ if $s$\ is \lq\lq small.''} \end{eqnarray*}$$

In other words, most beneficial mutations are lost from populations unless they are very beneficial. If $s=0.2$ in an ideal population, for example, a beneficial mutation will be lost about 80% of the time.⁴ Remember that in a strict harem breeding system with a single male $N_e \approx 4$ if the number of females with which the male breeds is large enough. Suppose that there are 99 females in the population. Then $N_e/N = 0.04$ and the probability that this beneficial mutation will be fixed is only 0.8%.

Notice that unlike what we saw with natural selection when we were ignoring genetic drift, the strength of selection⁵ affects the outcome of the interaction. The stronger selection is the more likely it is that the favored allele will be fixed. But it's also the case that the larger the population is, the more likely the favored allele will be fixed.⁶ Size does matter.