Adaptation of a quantitative trait to a moving optimum

Abstract

We investigate adaptive evolution of a quantitative trait under stabilizing selection with a moving optimum. We characterize three regimes, depending on whether (1) the beneficial mutation rate, (2) the fixation time, or (3) the rate of environmental change is the limiting factor for adaptation. If the environment is rate limiting, mutations with a small phenotypic effect are preferred over large mutations, in contrast to standard theory.

Figure 1.—

The probability π_small (in percentage) of simulations where the mutant allele at the minor locus fixes before the mutant allele at the major locus, as a function of the population-wide mutation rate Θ/2 = Nμ (assuming μ = 10⁻⁵), the speed of the environmental change v, the selection strength σ, and the mutational effects of the two loci, α₁ and α₂. π_small is also indicated by shades of gray, with black corresponding to 0% and white to 100%. The results are based on 1000 simulation for each parameter combination. Numbers in parentheses show the prediction from the analytical approximation (Equation A7). Similar results were obtained when N was fixed at 10⁶ and μ varied between 10⁻⁸ to 10⁻⁵ (not shown).

Figure 2.—

Example simulation run, in which the mutant allele at the minor locus (locus 1) fixes first. In each plot, the thick solid line shows the frequency of the mutant allele at the respective locus, and the thin solid line shows the selection coefficient s (in the approximation A5) over time. The dotted line shows the selection coefficient for the other locus. Shading marks the time periods used in the analytical approximation. T_l is the lag time during which the mutant allele is deleterious, T_w is the waiting time for a successful mutation, and T_f is the fixation time (i.e., the time until the mutant allele reaches frequency 0.5). Note that T_l,1 < T _l,2 but T_f,1 > T_f,2. Parameters: α₁ = 0.25, α₂ = 0.75, μ = 10⁻⁵, N = 10⁴, σ = 0.025, v = 1/2000.

Figure 3.—

Schematic overview of the behavior of π_small as a function of the population-wide mutation rate, Θ, and the speed of the environmental change, v. As outlined in the main text, there are three main regimes, depending on the dominating time scale (i.e., on which of the three time periods in Figure 2 is longest). The dotted line shows how the boundaries of the regimes shift as the selection parameter σ becomes smaller.