The population genetic theory of hidden variation and genetic robustness

Abstract

One of the most solid generalizations of transmission genetics is that the phenotypic variance of populations carrying a major mutation is increased relative to the wild type. At least some part of this higher variance is genetic and due to release of previously hidden variation. Similarly, stressful environments also lead to the expression of hidden variation. These two observations have been considered as evidence that the wild type has evolved robustness against genetic variation, i.e., genetic canalization. In this article we present a general model for the interaction of a major mutation or a novel environment with the additive genetic basis of a quantitative character under stabilizing selection. We introduce an approximation to the genetic variance in mutation-selection-drift balance that includes the previously used stochastic Gaussian and house-of-cards approximations as limiting cases. We then show that the release of hidden genetic variation is a generic property of models with epistasis or genotype-environment interaction, regardless of whether the wild-type genotype is canalized or not. As a consequence, the additive genetic variance increases upon a change in the environment or the genetic background even if the mutant character state is as robust as the wild-type character. Estimates show that this predicted increase can be considerable, in particular in large populations and if there are conditionally neutral alleles at the loci underlying the trait. A brief review of the relevant literature suggests that the assumptions of this model are likely to be generic for polygenic traits. We conclude that the release of hidden genetic variance due to a major mutation or environmental stress does not demonstrate canalization of the wild-type genotype.

Figure 1.—

Different types of interactions in a schematic reaction-norm picture. Average (standard mean) allelic effects at various loci under “old” and “new” conditions (change of the environment or the genetic background) are shown. Left, α_i ≡ 0 and β ≠ 0 leading to a one-sided spread of lines (canalization scenario). Right, α_i > 0 and β = 0 leading to line crossing (variable interaction scenario).

Figure 2.—

The hidden variation coefficient Δ_G, in units of the epistasis parameter α, as a function of the effective population size. The three curves correspond to increasing leptokurcy of the distribution of the locus variances v_o, with shape parameters q = 0.2, q = 0.5, and q = 1. The parameters for mutation rates and the average selection strength are u = 10⁻⁵ and s〈v_o〉 = 1/800. For the strongly L-shaped distribution with q = 0.2, the values of 56.0α and 258.7α are reached for N_e = 10⁵ and N_e = 10⁶, respectively.

Figure 3.—

Reduction factor r_f for the contribution of conditionally neutral loci to the hidden variation if the system changes to conditions that are not altogether new, but just rare with frequency f. r_f is shown as a function of log₁₀(s_ef) = log₁₀(N_esvf) for three values of Θ = 4N_eu, 0.2, 2, and 20 (top to bottom).