Higher order epistasis and fitness peaks

We show that higher order epistasis has a substantial impact on evolutionary dynamics by analyzing peaks in the fitness landscapes. There are 193,270,310 fitness graphs, or cube orientations, for 4-locus systems, distributed on 511, 863 isomorphism classes. We identify all fitness graphs with 6 or more peaks. 81 percent of them imply 4-way epistasis, whereas 9 percent of all 4-locus fitness graphs imply 4-way epistasis. Fitness graphs are useful in that they reflect the entire collection of fitness landscapes rather than focusing on a particular model. Our results depend on a characterization of fitness graphs that imply $n$-way epistasis. The characterization is expressed in terms of a partition property that can be derived from Hall's marriage theorem for bipartite graphs. A similar partition condition holds for any partial order. The result answers an open problem posed at a conference on interactions between algebra and the sciences at the Max Planck institute.