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.

Noise in Biomolecular Systems: Modeling, Analysis, and Control Implications

“…A natural question is whether every reaction network exhibiting robust perfect adaptation involves an integral action. The answer to this question is affirmative, and Gupta & Khammash (111) proved that, in the case where the set point is defined in terms of two reaction rates, perfect adaptation implies the presence of an integral action in the network dynamics. In the deterministic setting, it was shown that the integral action can be implemented using at least one molecular species (in terms of constrained integral controllers), whereas in the stochastic setting, the integral action necessarily involves an antithetic pair.…”