  The enormous size and complexity of genotypic sequence space frequently
requires consideration of coarse-grained sequences in empirical models. We
develop scaling relations to quantify the effect of this coarse-graining on
properties of fitness landscapes and evolutionary paths. We first consider
evolution on a simple Mount Fuji fitness landscape, focusing on how the length
and predictability of evolutionary paths scale with the coarse-grained sequence
length and alphabet. We obtain simple scaling relations for both the weak- and
strong-selection limits, with a non-trivial crossover regime at intermediate
selection strengths. We apply these results to evolution on a biophysical
fitness landscape that describes how proteins evolve new binding interactions
while maintaining their folding stability. We combine the scaling relations
with numerical calculations for coarse-grained protein sequences to obtain
quantitative properties of the model for realistic binding interfaces and a
full amino acid alphabet.
