Selection Limits to Adaptive Walks on Correlated Landscapes

Abstract

Adaptation depends critically on the effects of new mutations and their dependency on the genetic background in which they occur. These two factors can be summarized by the fitness landscape. However, it would require testing all mutations in all backgrounds, making the definition and analysis of fitness landscapes mostly inaccessible. Instead of postulating a particular fitness landscape, we address this problem by considering general classes of landscapes and calculating an upper limit for the time it takes for a population to reach a fitness peak, circumventing the need to have full knowledge about the fitness landscape. We analyze populations in the weak-mutation regime and characterize the conditions that enable them to quickly reach the fitness peak as a function of the number of sites under selection. We show that for additive landscapes there is a critical selection strength enabling populations to reach high-fitness genotypes, regardless of the distribution of effects. This threshold scales with the number of sites under selection, effectively setting a limit to adaptation, and results from the inevitable increase in deleterious mutational pressure as the population adapts in a space of discrete genotypes. Furthermore, we show that for the class of all unimodal landscapes this condition is sufficient but not necessary for rapid adaptation, as in some highly epistatic landscapes the critical strength does not depend on the number of sites under selection; effectively removing this barrier to adaptation.

Keywords: correlated landscapes; cost of complexity; speed of adaptation; weak selection regime.

Figure 1

Used fitness functions applied to the same genotype. Values of contributing loci are highlighted in red.

Figure 2

(A) Time required to reach the fitness peak in function $f_{\mathrm{eq}}$ as a function of genome size. Solid black line represents the mean of 100 runs for given n and shaded area their SDs. Dashed line represents the theoretical upper bound on this expectation: $\left(1+\frac{1}{2\beta }\right)n\ln \left(n\right)+n.$ $N\beta$ was set to 100. (B) A sharp threshold on the strength of selection for the speed of adaptation. Black line represents the mean time to reach the fitness peak for a constant genome size $\left(n=500\right)$ and selection strength $\left(\beta =0.1\right),$ with increasing population size N, and shaded areas represent the SD. Dashed line represents the critical value of selection strength $\left[2\left(N-1\right)\beta =\ln n\right]$ separating the polynomial and exponential regimes for the time to reach the fitness peak. Simulations were stopped if they took longer than $6\times {10}^4$ iterations.

Figure 3

Time to reach different fractions of the total fitness for an exponential distribution of effects. For a fixed selection strength, there is a maximum fraction of the fitness that can be reached in $O\left(n\ln n\right)$ mutational trials. The time to reach lower fractions of this fitness scales linearly, while the time to adapt further scales exponentially. Data points correspond to means of 1000 runs, and lines correspond to the indicated scalings. N was set to 20, $\beta =0.1,$ and the effects were distributed as $w_i\sim \mathrm{Exp}\left(1\right).$

Figure 4

Time to reach the fitness peak of $f_{\mathrm{ridge}},$ a member of the unimodal class of functions. (A) A visualization of the landscape induced by this function for $n=8.$ z-coordinate represents trait values (bottom cluster z = 0, top genotype z = n). Links between genotypes (•) represent mutations with the only path of strictly increasing fitness from $0^n$ to the peak highlighted in black. (B) Symbols represent averages (of 100 runs) of the time to reach the peak (•) or to reach 50% of the maximum fitness (▪). Shaded areas represent their SDs. Dashed line represents the bound $\mathit{O}\left(n^2\right).$ Parameters were set to $N=100$ and $\beta =\mathrm{0.1.}$

Figure D1

f_eq function. Fitness increases linearly with increasing number of ones.

Figure F1

f_ridge function, various mutations, n = 8. For each genotype, only one mutation is positive, while many are either neutral or negative. Red color represents new mutations, green color represents free riders, which are loci adding to fitness that had no fitness effect before a suitable mutation occurred.

Figure F2

Fitness as a function of time for different genome sizes for f_ridge. Solid gray lines represent the mean of 100 simulations for n = 500, 1000, and 5000, and dashed black lines represent best-fit power laws of the form $a\cdot t^b.$ Fitness is scaled by the maximum fitness (n) and time scaled by $n^2.$ This shows that the time to reach the peak is well estimated by $\mathit{O}\left(n^2\right),$ and that the rate of approach is well approximated by a power law. Parameters were set to $N=100$ and $\beta =\mathrm{0.1.}$