## 99-01-001

#
Abstract:

Evolvability of Complex Characters

### Population Dependent Fourier Decomposition of Fitness Landscapes over
Recombination Spaces

[PostScript]

##
Peter F. Stadler, Rudi Seitz, and Günter P. Wagner

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 base 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 on 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 auto-correlations on the
finite set of elementary landscapes. This comparison predicts that
point mutation is a superior search strategy on landscapes with low
and moderate order of interaction *p<n/3* (*n* is the
number of loci). For more complex landscapes one 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.

Submitted to *Bull.Math.Biol.*.

**Keywords:**
Fitness Landscapes, Recombination, P-Structures,
Autocorrelation, Epistatic Interactions

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