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The enormous size and complexity of genotypic sequence space frequentlyrequires consideration of coarse-grained sequences in empirical models. Wedevelop scaling relations to quantify the effect of this coarse-graining onproperties of fitness landscapes and evolutionary paths. We first considerevolution on a simple Mount Fuji fitness landscape, focusing on how the lengthand predictability of evolutionary paths scale with the coarse-grained sequencelength and alphabet. We obtain simple scaling relations for both the weak- andstrong-selection limits, with a non-trivial crossover regime at intermediateselection strengths. We apply these results to evolution on a biophysicalfitness landscape that describes how proteins evolve new binding interactionswhile maintaining their folding stability. We combine the scaling relationswith numerical calculations for coarse-grained protein sequences to obtainquantitative properties of the model for realistic binding interfaces and afull amino acid alphabet.

Scaling properties of evolutionary paths in a biophysical model of protein adaptation