Fig 1.
In the one-point crossover scheme, the parent genotypes are cut once between two randomly chosen loci and recombined to form the offspring. In the uniform crossover scheme, at each locus of the offspring, an allele present in one of the parents is chosen at random.
Fig 2.
Genotype (1,1) is lethal while the other three genotypes are viable with the same fitness. Here, genotype (0,0) is most robust since both its single mutants are viable.
Fig 3.
Equilibrium genotype frequencies in the two locus model.
Genotype frequencies in the stationary state are shown as a function of mutation rate for (A) strong recombination (ρ = 1) and (B) no recombination (ρ = 0).
Fig 4.
Mutational robustness as a function of mutation rate.
The figure shows the robustness in the two-locus model at ρ = 0 and ρ = 1. Recombination leads to a massive enhancement of robustness for small mutation rates.
Fig 5.
Mutational robustness as a function of recombination rate.
The figure shows the mutational robustness for one-point crossover (mopc) and uniform crossover (muc) and three different values of the mutation rate μ. When mutations are rare, a small amount of recombination is sufficient to significantly increase mutational robustness.
Fig 6.
Mutational robustness in a mesa landscape as a function of recombination rate.
Data points are obtained by numerically iterating the selection-mutation-recombination dynamics until the equilibrium state is reached. The parameters of the mesa landscape are L = 6, k = 2 and the mutation rate is μ = 0.001.
Fig 7.
Equilibrium genotype distributions in a mesa landscape for strongly and non-recombining populations.
Stationary states for populations with communal recombination and no recombination have been computed by assuming that only single point mutations occur with U = 0.01. Landscape parameters are L = 1000 and k = 100. The resulting mutational robustness is mnr ≈ 0.572 for the non-recombining population and mcr ≈ 1.000 for communal recombination. (A) Lumped mutation class frequencies on linear scales. In the absence of recombination the majority of the population is located at the critical Hamming distance d = k, whereas in the case of strong recombination the distribution is broader and shifted away from the brink of the mesa. (B) Genotype frequencies on semi-logarithmic scales. In both cases the genotype frequencies decrease exponentially with the Hamming distance to the wild type, but the distribution has much more weight at small distances in the case of recombination.
Fig 8.
Network representation of a percolation landscape.
The figure shows a percolation landscape with L = 8 loci and a fraction p = 0.2 of viable genotypes. Viable genotypes at Hamming distance d = 1 are connected by edges, and the node area of a genotype σ is proportional to , where the recombination weight λσ is defined in Eq (10). The recombination center is the genotype with the largest recombination weight.
Fig 9.
Stationary states in a percolation landscape.
The figure shows three different stationary population distributions in the percolation landscape depicted in Fig 8. Node areas are proportional to the stationary frequency of the respective genotype in the population, and the edge width eσ,τ between neighboring genotypes is proportional to the frequency of the more populated one, . (A) Unique stationary state of a non-recombining population. (B,C) Stationary states for recombining populations undergoing uniform crossover with r = 1. The recombination center (purple) is the most populated genotype in (A,B), but not in (C). In all cases the mutation rate is μ = 0.01.
Fig 10.
Average mutational robustness in the percolation landscape as a function of recombination rate.
Mutational robustness is computed for 250 randomly generated percolation landscapes with L = 6 and p = 0.4, and the results are averaged to obtain . The mutation rate is μ = 0.001.
Fig 11.
Mutational robustness in the percolation landscape as a function of the fraction of viable genotypes.
The robustness for recombining () and non-recombining (
) populations is obtained by averaging over 6800 randomly generated landscapes with L = 6 and μ = 0.001. In the same way the average maximal robustness
is estimated. The full line shows the analytic expression (39) for the robustness of a uniformly distributed population.
Fig 12.
Stationary states in two different sea-cliff landscapes with and without recombination.
(A,B) A single reference genotype with landscape parameters L = 8, d< = 1 and d> = 6. (C,D) Two reference genotypes which are antipodal to each other with landscape parameters L = 8, d< = 2 and d> = 4.2. (A,C) Stationary frequency distribution in the absence of recombination. (B,D) Stationary frequency distribution with uniform crossover and r = 1. In all cases node areas are proportional to genotype frequencies, and the recombination center is marked in blue. The edge width between neighboring genotypes is proportional to the frequency of the more populated one. The mutation rate is μ = 0.01.
Fig 13.
Mutational robustness correlates with recombination weight.
The recombination weight of genotypes is plotted against their mutational robustness for (A) a percolation landscape with parameters L = 8, p = 0.4 and (B) a sea-cliff landscape with parameters L = 8, d< = 2, d> = 6. For the evaluation of the recombination weight (10), uniform crossover at rate r = 1 is assumed.
Fig 14.
The empirical A. niger fitness landscape.
(A,B) Two-dimensional network representation of the fitness landscape with node sizes determined by the mutational robustness mσ and the recombination weight λσ, respectively. In order to make the differences between genotypes more conspicuous, the node area is chosen proportional to the sixth power of these quantities. The recombination weight is evaluated for uniform crossover with r = 1, and the recombination center is highlighted in purple. (C,D) Recombination weight plotted against mutational robustness and genotype fitness, respectively. Lethal genotypes with wσ = 0 appear only in panel D.
Fig 15.
Recombination weights and stationary states at different recombination rates.
(A-C) Two-dimensional network representation of the A. niger fitness landscape with node areas proportional to the sixth power of the recombination weight for recombination rates r = 0, r = 0.4 and r = 1, respectively. (D-F) Two-dimensional network representation of the A. niger fitness landscape with node areas proportional to the stationary genotype frequency at the same recombination rates and mutation rate μ = 0.005. The edge width between neighboring genotypes is proportional to the frequency of the more populated one.