Distributions of beneficial fitness effects in RNA

Abstract

Beneficial mutations are the driving force of evolution by natural selection. Yet, relatively little is known about the distribution of the fitness effects of beneficial mutations in populations. Recent work of Gillespie and Orr suggested some of the first generalizations for the distributions of beneficial fitness effects and, surprisingly, they depend only weakly on biological details. In particular, the theory suggests that beneficial mutations obey an exponential distribution of fitness effects, with the same exponential parameter across different regions of genotype space, provided only that few possible beneficial mutations are available to that genotype. Here we tested this hypothesis with a quasi-empirical model of RNA evolution in which fitness is based on the secondary structures of molecules and their thermodynamic stabilities. The fitnesses of randomly selected genotypes appeared to follow a Gumbel-type distribution and thus conform to a basic assumption of adaptation theory. However, the observed distributions of beneficial fitness effects conflict with specific predictions of the theory. In particular, the distributions of beneficial fitness effects appeared exponential only when the vast majority of small-effect beneficial mutations were ignored. Additionally, the distribution of beneficial fitness effects varied with the fitness of the parent genotype. We believe that correlation of the fitness values among similar genotypes is likely the cause of the departure from the predictions of recent adaptation theory. Although in conflict with the current theory, these results suggest that more complex statistical generalizations about beneficial mutations may be possible.

Figure 1.—

The distribution of absolute fitness of 3,636,520 random sequences. The data were divided into 10 equal-width bins and plotted so that the center of the column on the x-axis is at the upper bin bound. The y-axis is the fraction of sequences falling into a particular bin. Inset, the distribution of Δ₁, Δ₂, and Δ₃ (see text) for 15,880 sets of 229 absolute fitness values. The x-axis is the fitness effect and the y-axis is the fraction of fitnesses falling into a particular bin on a log scale. The bin width is 0.2 for Δ₁, 0.1 for Δ₂, and 0.67 for Δ₃.

Figure 2.—

The cumulative distribution of beneficial fitness effects of wild-type alleles from random walks. Data are from 5721 adaptive walks starting from random sequences—one wild-type genotype per rank per walk. The x-axis is the size of the beneficial fitness effect and the y-axis is the fraction of mutants with fitness greater than the x-axis value on a log scale. The dashed curve is i = 2 (n = 5004), the gray curve is i = 3 (n = 9908), and the black curve is i = 4 (n = 14871). Inset: exponential behavior when truncated at S = 0.20. Style and shading of curves match the main figure.

Figure 3.—

The cumulative distribution of all one-step beneficial fitness effects of wild-type alleles from the high-fitness walks. Data are from 6959 adaptive walks starting near fitness optima—one wild-type genotype per rank per walk. The x-axis is the size of the beneficial fitness effects and the y-axis is the fraction of mutants with fitness greater than the x-axis value on a log scale. The dashed curve is i = 2 (n = 6204), the gray curve is i = 3 (n = 12374), and the black curve is i = 4 (n = 18432). Inset: exponential behavior when truncated at S = 10.0. Style and shading of curves match the main figure.

Figure 4.—

The effect of truncation on the estimated mean beneficial effect. The distributions become approximately exponential when the curve shown here asymptotes. The top plot is the estimate of the mean using absolute fitness differences. The bottom plot is the estimate based on s-values.

Figure 5.—

Mean s for all beneficial mutations in the neighborhood of R ≤ 4 wild-type sequences across the length of an adaptive walk. Data are from 5721 random and 6959 high-fitness adaptive walks. The x-axis is the number of substitutions and the y-axis is the mean s of the one-step beneficial mutations from all low-rank wild-type alleles at that step. Bars indicate standard errors.