Background: Understanding interactions between mutations and how they affect fitness is a\udcentral problem in evolutionary biology that bears on such fundamental issues as the structure of\udfitness landscapes and the evolution of sex. To date, analyses of fitness landscapes have focused\udeither on the overall directional curvature of the fitness landscape or on the distribution of pairwise\udinteractions. In this paper, we propose and employ a new mathematical approach that allows a\udmore complete description of multi-way interactions and provides new insights into the structure\udof fitness landscapes.\udResults: We apply the mathematical theory of gene interactions developed by Beerenwinkel et al.\udto a fitness landscape for Escherichia coli obtained by Elena and Lenski. The genotypes were\udconstructed by introducing nine mutations into a wild-type strain and constructing a restricted set\udof 27 double mutants. Despite the absence of mutants higher than second order, our analysis of\udthis genotypic space points to previously unappreciated gene interactions, in addition to the\udstandard pairwise epistasis. Our analysis confirms Elena and Lenski's inference that the fitness\udlandscape is complex, so that an overall measure of curvature obscures a diversity of interaction\udtypes. We also demonstrate that some mutations contribute disproportionately to this complexity.\udIn particular, some mutations are systematically better than others at mixing with other mutations.\udWe also find a strong correlation between epistasis and the average fitness loss caused by\uddeleterious mutations. In particular, the epistatic deviations from multiplicative expectations tend\udtoward more positive values in the context of more deleterious mutations, emphasizing that\udpairwise epistasis is a local property of the fitness landscape. Finally, we determine the geometry\udof the fitness landscape, which reflects many of these biologically interesting features.\udConclusion: A full description of complex fitness landscapes requires more information than the\udaverage curvature or the distribution of independent pairwise interactions. We have proposed a\udmathematical approach that, in principle, allows a complete description and, in practice, can suggest\udnew insights into the structure of real fitness landscapes. Our analysis emphasizes the value of nonindependent\udgenotypes for these inferences.
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