Mutation/Selection Balance for a Quantitative Trait

Assume that fitness W depends on the trait z following a Gaussian curve. For simplicity, we set the optimum at 0:

Here, V_s is the variance of the Gaussian curve. The larger V_s is, the wider the range of phenotypes that can survive and the weaker is stabilizing selection (Fig. WN18.9).

Suppose that there is a rare allele that increases the trait by α_i. (The subscript i labels this particular allele.) We can assume that individuals homozygous for the other allele (or alleles) are at the optimum, and that individuals carrying one copy of the allele deviate from the optimum by α_i and have fitness exp(–/2V_s). If the deviation is small, this is approximately 1 – /2V_s (dashed line in Fig. WN18.9). So, the selection against these rare alleles, caused by the deviation of the trait from the optimum, is just s_i = /2V_s. In a mutation selection balance, the allele frequency is p_i = µ_i/s_i = 2 V_sµ_i/).

With two alleles at each locus, the additive genetic variance is V_A = 2Σ_ip_iq_i. Because the allele is assumed to be rare, q_i ~ 1. Substituting the allele frequency p_i we see that the allelic effect cancels, giving

V_A = 4V_sΣ_iµ_i = 2UV_s,

where U = 2Σ_iµ_i is the total rate of mutation to alleles affecting the trait per diploid genome.

Here, we have assumed that all the selection on the allele is due to stabilizing selection on the trait. If pleiotropy is extensive, we can just say that the selection against allele i is s_i and is determined by factors that have nothing to do with the effect on the trait α_i. Then, substituting the allele frequency p_i = µ_i/s_i into the formula for the additive genetic variance gives

But, we saw in Chapter 14 (p. 410) that the trait variance generated by mutation in each generation is V_M = 2Σ_iµ_i. Hence

where is the average selection coefficient weighted toward alleles with large effects on the trait and small effects on fitness.