Fig 1.
(A) Schematic illustration of how selection for an intermediate phenotypic value wopt can make a genotype-phenotype landscape incongruent with the resulting fitness landscape. (B) An additive three-locus, biallelic genotype-phenotype landscape with a single peak (gray filled circle; pgp = 1). One pairwise interaction is highlighted in blue. It exhibits no magnitude epistasis (ϵgp = 0) or sign epistasis. (C) The Gaussian phenotype-fitness map (Eq 1) is shown for three values of wopt (dashed line, wopt = 0; dotted line, wopt = 0.5; solid line, wopt = 1), with three values of σ shown for wopt = 0.5. (D) Applying the Gaussian phenotype-fitness map with wopt = 0 to the genotype-phenotype landscape results in a single-peaked fitness landscape (gray filled circle; pf = 1). The same pairwise interaction from (B) is highlighted in blue. It exhibits positive epistasis (ϵgp = 0.266), but no sign epistasis. (E) Applying the Gaussian phenotype-fitness map with wopt = 0.5 to the genotype-phenotype landscape results in a multi-peaked fitness landscape (gray filled circles; pf = 2). The same pairwise interaction from (B) is highlighted in red. It exhibits negative epistasis (ϵgp = −1.01), as well as reciprocal sign epistasis. (F) Applying the Gaussian phenotype-fitness map with wopt = 1 to the genotype-phenotype landscape results in a single-peaked fitness landscape (gray filled circle; pf = 1). The same pairwise interaction from (A) is highlighted in blue. It exhibits positive epistasis (ϵgp = 0.53), but no sign epistasis. In panels B, D-F, arrows point from genotypes with lower phenotypic or fitness values to genotypes with higher phenotypic or fitness values. The no sign epistasis motif is highlighted in blue and the reciprocal sign epistasis motif in red. Note the symmetry of landscapes in panels D and F.
Table 1.
List of symbols.
Fig 2.
Local incongruence: Magnitude epistasis.
(A) The probability of retaining the type of epistasis, shown in relation to the optimal phenotype wopt. The black line shows the theoretical prediction and the dots show the results from randomly generated genotype-phenotype landscapes for different values of σ. The theoretical approximation agrees well with the results from randomly generated genotype-phenotype landscapes for large σ (i.e., σ ≥ 1). (B) The probability of observing zero magnitude epistasis in the fitness landscape, shown in relation to wopt, for different values of σ, which we selected to show the range of variation in the probability of observing zero epistasis.
Fig 3.
Local incongruence: Sign epistasis.
The probability of retaining or changing the type of epistasis in the genotype-phenotype landscape, relative to the fitness landscape, shown in relation to the optimal phenotypic value wopt. Data are grouped based on whether the genotype-phenotype landscapes exhibits (A) no sign epistasis (blue), (B) simple sign epistasis (brown), or (C) reciprocal sign epistasis (red). The colours of the lines represent the type of epistasis in the resulting fitness landscape. These results are independent of σ, because they only depend on the rank ordering of fitness values. Notice the “U” shape of the probability of retaining the type of epistasis in each panel.
Fig 4.
Global incongruence: Mt. Fuji and House-of-Cards genotype-phenotype landscapes.
The absolute change in the number of peaks and the probability that the number of peaks changes in the fitness landscape, relative to (A,B) Mt. Fuji and (C,D) House-of-Cards genotype-phenotype landscapes, shown in relation to wopt for L ∈ {2, 3..8}. These results are independent of σ, because they only depend on the rank ordering of fitness values.
Fig 5.
Global incongruence: NK genotype-phenotype landscapes.
The absolute change in the number of peaks in the fitness landscape, relative to the genotype-phenotype landscape, is shown in relation to wopt for genotypes of length (A) L = 5 and (C) L = 8, as K increases from zero to L − 1. The corresponding probability of change in the number of peaks is shown in relation to wopt for genotypes of length (B) L = 5 and (D) L = 8, as K increases from zero to L − 1. These results are independent of σ, because they only depend on the rank ordering of fitness values.
Fig 6.
Global incongruence: Genotype-phenotype landscapes of transcription factor-DNA interactions.
Landscapes constructed using (A,B) the mismatch model and (C,D) experimental measurements from protein binding microarrays for 1,137 eukaryotic transcription factors. (A) The number of genotypes, shown in relation to mismatch class. (B) The absolute change in the number of peaks in the fitness landscape, relative to the genotype-phenotype landscape, shown in relation to the optimal mismatch class mopt. Labels indicate the number of genotypes per peak in the fitness landscape. Note the symmetry in the absolute change in the number of peaks around mismatch class mopt = 4, as well as the tripling of the number of genotypes per peak for each increment in mopt. The grey shaded circles are a schematic representation of the growing width of the peaks. (C) The number of genotypes per binding affinity class, where protein binding microarray E-scores are used as a proxy for relative binding affinity. Violin plots show the distribution, and box-and-whisker plots the 25–75% quartiles, across genotype-phenotype landscapes for the 1,137 transcription factors. Closed symbols and the dashed line denote the median of each distribution. (D) Violin plots of the distribution of the absolute change in the number of peaks in the fitness landscape, relative to the genotype-phenotype landscape, shown in relation to the optimal binding affinity wopt for σ = 0.15. The inset shows the mean absolute change in the number of peaks, in relation to wopt. The x-axes in (A,B) are arranged such that binding affinity increases when read from left to right, in qualitative agreement with the x-axes in (C,D). The results in panels A-C are independent of σ.
Fig 7.
Rugged fitness landscapes need not impede adaptation.
The average length of an adaptive walk 〈l〉 and the change in mean population fitness at equilibrium 〈f〉 is shown for landscapes of transcription factor-DNA interactions generated using (A,C) the mismatch model for e = 0.05 and (B,D) protein binding microarray data. In (B,D), violin plots show the distribution, and box-and-whisker plots the 25–75% quartiles, across the 1,137 empirical landscapes for each optimal binding affinity wopt. The large variability of 〈l〉 at intermediate and high wopt is a consequence of the random diffusion of the population on non-peak plateaus, which results in longer walks. In (C), μ = 0.1, σ = 1 while in (D) μ = 0.1, σ = 0.15.