Fig 1.
Different numbers of sequence-coding units K result in different mutational structures of the fitness landscape.
Examples of two fitness landscapes with K = 2 and K = 4 and the same number of genotypes N = 16. Grey arrows show the direction of increasing fitness. Two accessible pathways (black = with indirect mutations, green = without indirect mutations) have been highlighted. For K = 4, indirect mutations can be either backward (increasing the Hamming distance from the target genotype) or distance-neutral (the Hamming distance remains unchanged).
Fig 2.
Accessibility increases with increasing number of unit types K.
(a) Plots of accessibility A versus the number of genotypes N for different K (black, red, blue, and green triangles) and for pathways without backward mutations. A tends to zero as N → ∞ for K = 2. (b) Accessibility versus N for all pathways, including indirect ones (circles). In both panels the dashed lines correspond to asymptotic estimates of accessibility A∞ (cf. S1 Fig).
Fig 3.
Indirect mutations dominate evolutionary trajectories for fitness landscapes with K > 2.
(a) Average number of mutational steps (pathway length) along accessible pathway for a fixed length of genotype L. Fractions of forward, distance-neutral, and backward mutations are indicated by coloured areas between the curves, for example the fraction of forward mutations corresponds to the distance between the red and the gray curve. (b) Average length of accessible pathway for a fixed number of genotypes N. (c) Average length of accessible pathway for a fixed number of coding units K (circles) with straight lines fitted to the data (lines). Note that the fraction of shortest pathways (with only forward mutations) among all pathways quickly decreases (cf. S3 Fig).
Fig 4.
Accessibility strongly depends on the initial fitness.
(a) Average number of accessible pathways as a function of initial fitness f0 for different K and for N = 220. Shaded areas represent standard errors. (b) Accessibility A as a function of initial fitness f0 for different K and for N = 228. A steep decrease in accessibility at some critical f0 (dashed line) indicates that the probability of reaching the best-fit genotype can be very sensitive to the initial fitness.
Fig 5.
Accessible pathways cover a large part of FL for K > 2.
The total number of genotypes (grey) and the genotypes belonging to accessible pathways (colours black, red, and green) as a function of Hamming distance from the target genotype, for fitness landscapes with different number of coding units K = 2, 4, 16 (panels a, b, c) and the same number of genotypes N = 220. The shaded area under the curve corresponds to the total number of genotypes (for any distance) that belong to accessible pathways. Coverage (the fraction of genotypes in accessible pathways) is also shown.
Fig 6.
Accesibility of experimental FLs.
Panels (a-c): data from Rowe et al. [39], panels (d-f): data from Guenther et al. [42]. (a,d) Accessibility of the sub-FLs for K = 2, 4 (red), and for their randomized counterparts with the same fitness distribution (grey). Randomization decreases accessibility to that of a maximally-random FL. (b,e) Average genotype fitness in sub-landscapes with K = 2 as a function of the distance from the target genotype. Randomization removes correlations present in the original FL. (c,f) Histogram of the fitness of the initial genotype; the average fitness for each histograms is the same as the average fitness value of the antipodal genotype (the right-most points in Panels b, e).
Fig 7.
Fitness-to-distance correlation impacts accessibility and coverage of the experimental FLs.
Panels (a-c): data from Rowe et al. [39], panels (d-f): data from Guenther et al. [42]. (a,d) Accessibility of the sub-FLs for K = 2, 4 (red circles, red triangles), and for their randomized counterparts (grey, black) as a function of rescaled initial fitness . In the presence of fitness-to-distance correlations landscapes with high levels of
are mostly accessible compared to the case with no correlations. (b,e) Number of paths exhibits as a function of rescaled initial fitness
. Presence of fitness-to-distance correlations increased number of pathways by a few orders of magnitude compared with the randomized ensemble. (c,f) Coverage of experimental sub-FLs and their randomized counterparts as a function of
. The insets show the same data in semi-log scales. In the case of K = 2 correlations increase the fraction of accessible genotypes to levels observed for FLs with K = 4.