Scaling Laws of Robustness in Probabilistic Genotype-Phenotype Maps

Recent studies of genotype-phenotype (GP) maps have reported universally enhanced phenotypic robustness to genotype mutations, a feature essential to evolution representable as a graph-theoretic property of genotype networks. Virtually all of these studies make a simplifying assumption that each genotype maps deterministically to a single phenotype. Here, we introduce probabilistic GP (PrGP) maps, where each genotype maps to a vector of phenotype probabilities, as a more realistic and universal language for investigating robustness. In three model systems we show PrGP maps offer a generalized framework which can handle uncertainty emerging from various physical sources. We observe a novel biphasic scaling of robustness enhanced relative to random expectation for frequent phenotypes and approaching random expectation for less frequent phenotypes. We derive an analytical theory for this behavior and demonstrate the theory is highly predictive of empirical robustness.