Maximum Entropy Model for Genotype-Phenotype Map Organization Reveals Maximally Robust Neutral Networks*

Genotype-phenotype (GP) maps, including protein/RNA primary sequences mapping to folded structures, gene regulatory network interactions mapping to expression cycles, and even non-biological examples like spin glass bond configurations mapping to ground states, tend to universally display similar scaling laws for mutational robustness and other neutral network properties. We propose a maximum entropy model for GP map organization in which only global robustness is constrained, generating a mapping onto a Potts model on a Hamming graph with conserved phenotype frequencies. Our mean field theory and simulations show existence of two phases, robust and fragile. In the robust phase, the neutral networks organize into maximally robust "bricklayer's graphs" whose robustness is related to the sums-of-digits function. We argue that mutational robustness, base information error, and population neutrality cannot be simultaneously optimized in general. We find that bricklayer's graphs naturally occur as component networks in RNA/HP protein folding GP maps.