Population dependent Fourier decomposition of fitness landscapes over recombination spaces: evolvability of complex characters.

The effect of recombination on genotypes can be represented in the form of P-structures, i.e., a map from the set of pairs of genotypes to the power set of genotypes. The interpretation is that the P-structure maps the pair of parental genotypes to the set of recombinant genotypes which result from the recombination of the parental genotypes. A recombination fitness landscape is then a function from the genotypes in a P-structure to the real numbers. In previous papers we have shown that the eigenfunctions of (a matrix associated with) the P-structure provide a basis for the Fourier decomposition of arbitrary recombination landscapes. Here we generalize this framework to include the effect of genotype frequencies, assuming linkage equilibrium. We find that the autocorrelation of the eigenfunctions of the population-weighted P-structure is independent of the population composition. As a consequence we can directly compare the performance of mutation and recombination operators by comparing the autocorrelations on the finite set of elementary landscapes. This comparison suggests that point mutation is a superior search strategy on landscapes with a low order and a moderate order of interaction p < n/3 (n is the number of loci). For more complex landscapes 1-point recombination is superior to both mutation and uniform recombination, but only if the distance among the interacting loci (defining length) is minimal. Furthermore we find that the autocorrelation on any landscape is increasing as the distribution of genotypes becomes more extreme, i.e., if the population occupies a location close to the boundary of the frequency simplex. Landscapes are smoother the more biased the distribution of genotype frequencies is. We suggest that this result explains the paradox that there is little epistatic interaction for quantitative traits detected in natural populations if one uses variance decomposition methods while there is evidence for strong interactions in molecular mapping studies for quantitative trait loci.