Topological criterion for robust perfect adaptation of reaction fluxes in biological networks

Robust Perfect Adaptation (RPA) is a mechanism adopted by living systems to maintain homeostasis amidst fluctuating environments. Previous studies on RPA of concentrations and fluxes (e.g., the passage of metabolites) lacked generality due to complex theories and specific kinetic assumptions. Here, we present a simple and general topological criterion for networks to exhibit RPA of all reaction fluxes against perturbations of kinetic parameters and conserved quantities. Our criterion is solely based on network structure, making it valid for any kinetics (e.g., mass-action or Michaelis-Menten kinetics). Applying this to <i>Escherichia coli</i> metabolic pathways, we identify key genes determining metabolic fluxes, validated by single-gene knockout experiments. Our approach also solves the mystery of why plant growth is affected by perturbations in only aspartate transcarbamoylase among various enzymes involved in pyrimidine metabolism. Our work informs key components of complex networks governing fluxes and provides critical information regarding long-term dynamics solely from network topology.