Random Drift of Quantitative Traits

We can write the value of a quantitative trait as the sum of effects of many genes, labeled i. For simplicity, suppose that there are two alleles of each gene (“+” and “–” at frequencies p_i, q_i). Each extra “+” allele at the ith gene increases the trait by α_i. In a diploid, the contribution of the ith gene to the trait is 2α_ip_i; the factor of 2 is needed because there are two copies of each gene. Overall, the trait mean is

where the constant is just the trait value for an individual carrying all “–” alleles.

We need to know the additive genetic variance, V_A, in terms of the allele frequencies p_i and allelic effects α_i. Each copy of the ith gene has a chance p_i of contributing α_i to the trait and a chance q_i of contributing zero. The variance of such a two-valued distribution is p_iq_i (see Box 15.1 and Table 28.1), so the total genetic variance is

Again, there is a factor of 2 because the two copies of each gene both contribute to the variance.

Now, we can use our knowledge of how random drift affects allele frequencies to find how it affects the mean and variance of a quantitative trait. On average, there is no change to the mean of the trait from one generation to the next (E[_{t + 1}] = E[_t]), because there is no change in the average allele frequency. (This would not be true with dominance or epistasis, because random drift can cause inbreeding depression; see pp. 515–518.) However, the variance in allele frequency causes random variation in the trait mean, such as that seen in Figure 15.4. The variance in mean after one generation of drift is

where p_1,i is the allele frequency at gene i after one generation of drift. Random drift generates a variance in the trait mean equal to the additive genetic variance divided by population size. This relationship does not depend on the number of genes or their effects, α_i. Thus, the effects of a drift on an additive trait can be predicted without knowing the detailed genetics.

Random drift reduces genetic diversity within populations and so, on average, the additive genetic variance should decrease. The additive variance is proportional to the product of allele frequencies, and we saw in Box 15.1 that this decreases as the variance of allele frequency increases. Thus, after one generation,

On average, the genetic variance decreases by a factor (1 – 1/2N) in every generation. Therefore, using (1 – x)^t ~ exp(xt) for x << 1 (Chapter 28), we see that V_A,t ~ V_A,0 exp(–t/2N). This is exactly the same pattern as for the decline in average heterozygosity, E[2pq] (Box 15.1). After 2N generations, random drift will have reduced additive genetic variance by a factor e^–1 ~ 37%.