The mapping from genotype to phenotype to fitness typically involves multiple nonlinearities that can transform the individual and combined effects of mutations. For example, mutations may contribute additively to a phenotype, but their effects on fitness may combine non-additively because selection favors a low or intermediate value of that phenotype. Here, we study transformability under a non-linear phenotype-to-fitness map. We do so by comparing the topographical properties of genotype-phenotype landscapes, constructed using a diversity of theoretical models and empirical data on transcription factor-DNA interactions, to the topographical properties of fitness landscapes when selection favors a low or intermediate phenotypic value. Using the theoretical models, we prove a number of fundamental results. For example, selection for low or intermediate phenotypic values cannot transform simple sign epistasis into reciprocal sign epistasis, yet it transforms reciprocal sign epistasis into simple sign epistasis and no sign epistasis with equal probability. More broadly, we show that such selection tends to create fitness landscapes that are more rugged than the underlying genotype-phenotype landscape, but surprisingly this increased ruggedness typically does not frustrate adaptive evolution, because the local adaptive peaks in the fitness landscape tend to be nearly as tall as the global peak. Many of these results carry forward to the empirical genotype-phenotype landscapes, which may help to explain why low- and intermediate-affinity transcription factor-DNA interactions are so prevalent in eukaryotic gene regulation.
