Polytopes, graphs and fitness landscapes

Darwinian evolution can be illustrated as an uphill walk in a landscape, where the surface consists of genotypes, the height coordinates represent fitness, and each step corresponds to a point mutation. Epistasis, roughly defined as the dependence between the fitness effects of mutations, is a key concept in the theory of adaptation. Important recent approaches depend on graphs and polytopes. Fitness graphs are useful for describing coarse properties of a landscape, such as mutational trajectories and the number of peaks. The graphs have been used for relating global and local properties of fitness landscapes. The geometric theory of gene interaction, or the shape theory, is the most fine-scaled approach to epistasis. Shapes, defined as triangulations of polytopes for any number of loci, replace the well established concepts of positive and negative epistasis for two mutations. From the shape one can identify the fittest populations, i.e., populations where allele shuffling (recombination) will not increase the mean fit- ness. Shapes and graphs provide complementary information. The approaches make no structural assumptions about the underlying fitness landscapes, which make them well suited for empirical work.