Average frustration and phase transition in large-scale biological networks: a statistical physics approach

For large-scale biological networks represented as signed graphs, the index of frustration measures how far a network is from a monotone system, i.e., how incoherently the system responds to perturbations. For the set of networks we analyze, we find that the frustration is systematically lower in transcriptional networks (modeled at functional level) than in signaling and metabolic networks (modeled at stoichiometric level). Interpreting this result in terms of energetic cost of an interaction, an erroneous or contradictory transcriptional action costs much more than a signaling/metabolic error, and therefore must be avoided as much as possible. Averaging over all possible perturbations, however, we also find that signaling/metabolic networks tend to undergo a phase transition to order in an energetic regime lower than for transcriptional networks, meaning that they may partially compensate for the higher frustration by lowering the interaction strenght needed to achieve a globally ordered response to perturbations.