The question of accessibility of genotype with the highest fitness for different fitness landscapes has a long history. Dating back to works of Wright and Fisher, there are two contradicting intuitions about behaviour of global accessibility when the genotype space grows. On the one hand, as the dimensionality of fitness landscape is increased, the sheer increase in the number of pathways should allow for circumventing fitness barriers, rendering the global maximum accessible. On the other hand, the increasing number of genotypes results in a higher number of local fitness maxima that act as evolutionary traps, effectively making the global optimum inaccessible from a distant (in terms of mutations) genotype.

In the work by Franke et al. (PLoS Comput. Biol., 2011) authors revisited the question of evolutionary accessibility of mutational pathways for various fitness landscapes with underlying genotypes made of 0s and 1s. Here we extend this work by posting this question for fitness landscapes with higher number of possible states K_A per position in the genotype. Specifically the case with K_A = 4 might be seen as a proxy for the DNA fitness landscape where the four states represent the four DNA base pairs. Further the graph of possible mutations is a generalized hypercube such that any two genotypes (nodes) are connected only when they differ at a single position. We define accessibility as an average probability to reach the genotype with the highest fitness from the antipodal genotype by a series of mutations that increase fitness.

By performing extensive numerical simulations for system sizes up to 2²⁸ genotypes we are able to estimate the accessibility for the infinite genotype length. Interestingly, even for the fitness landscape with uncorrelated fitness values (e.g. drawn from uniform distribution), we find that accessibility is nonvanishing for infinite system sizes. Particularly, the increase of K_A from 2 to 4 results in an order of magnitude increase of the accessibility asymptotic value, i.e. from 0.034(2) to 0.426(4) for the former and the latter case respectively. Furthermore, the relatively high level of accessibility for K_A = 4 is almost not affected by introducing roughly 40% of mutant genotypes that are lethal (holes in the landscape), but it drops abruptly after exceeding this threshold.

fitness landscapes; evolutionary pathways; accessibility