Relation between the number of peaks and the number of reciprocal sign epistatic interactions

Abstract

Empirical assays of fitness landscapes suggest that they may be rugged, that is having multiple fitness peaks. Such fitness landscapes, those that have multiple peaks, necessarily have special local structures, called reciprocal sign epistasis (Poelwijk et al. (2011)). Here, we investigate the quantitative relationship between the number of fitness peaks and the number of reciprocal sign epistatic interactions. Previously it has been shown (Poelwijk et al. (2011)) that pairwise reciprocal sign epistasis is the necessary but not sufficient condition for the existence of multiple peaks. Applying discrete Morse theory, which to our knowledge has never been used in this context, we extend this result by giving the minimal number of reciprocal sign epistatic interactions required to create a given number of peaks.