Complete characterization of robust perfect adaptation in biochemical reaction networks

Perfect adaptation is a phenomenon whereby the output variables of a system can maintain certain values despite external disturbances. Robust perfect adaptation (RPA) refers to an adaptation property that does not require fine-tuning of system parameters. RPA plays a vital role for the survival of living systems in unpredictable environments. However, complex interaction patterns in biochemical systems pose a significant challenge in identifying RPA and associated regulatory mechanisms. The goal of this paper is to present a novel approach for finding all RPA properties that are realized for a generic choice of kinetics for general deterministic chemical reaction systems. This is accomplished by proving that an RPA property is represented by a subnetwork with certain topological features. This connection is exploited to show that these structures generate all kinetics-independent RPA properties, allowing us to systematically identify all RPA properties by enumerating these subnetworks. An efficient method is developed for this enumeration, and we provide a computational package for this purpose. We pinpoint the integral feedback controllers that work in concert to realize each RPA property, casting our results into the familiar control-theoretic paradigm of the Internal Model Principle. We further generalize the regulation problem to the multi-output scenario where the target values belong to a manifold of nonzero dimension, and provide a sufficient condition for this. The present work significantly advances our understanding of regulatory mechanisms that lead to RPA in endogenous biochemical systems, and it also provides rational design principles for synthetic controllers. The present results indicate that an RPA property is essentially equivalent to the existence of a "topological invariant", which is an instance of what we call the "Robust Adaptation is Topological"(RAT) principle.