Universal structural requirements for maximal robust perfect adaptation in biomolecular networks

Abstract: Consider a biomolecular reaction network that exhibits robust perfect adaptation to disturbances from several parallel sources. The well-known Internal Model Principle of control theory suggests that such systems must include a subsystem (called the “internal model”) that is able to recreate the dynamic structure of the disturbances. This requirement poses certain structural constraints on the network which we elaborate in this paper for the scenario where constant-in-time disturbances maximally affect network interactions and there is model uncertainty and possible stochasticity in the dynamics. We prove that these structural constraints are primarily characterized by a simple linear-algebraic stoichiometric condition which remains the same for both deterministic and stochastic descriptions of the dynamics. Our results reveal the essential requirements for maximal robust perfect adaptation in biology, with important implications for both systems and synthetic biology. We exemplify our results through many known examples of robustly adapting networks and we construct new examples of such networks with the aid of our linear-algebraic characterization.