Design principles for perfect adaptation in biological networks with nonlinear dynamics

Establishing a mapping between the emergent biological properties and network structure has always been of great relevance in systems and synthetic biology. Adaptation is one such biological property of paramount importance, which aids in regulation in the face of environmental disturbances. In this paper, we present a nonlinear systems theory-driven framework to identify the design principles for perfect adaptation in the presence of large disturbances. Based on the input-output configuration of the network, we use invariant manifold methods to deduce the condition for perfect adaptation to constant input disturbances. Subsequently, we translate these conditions to necessary structural requirements for adaptation in networks of small size. We then extend these results to argue that there exist only two classes of architectures for a network of any size that can provide local adaptation in the entire state space—namely, incoherent feed-forward structure and negative feedback loop with buffer node. The additional positiveness constraints further restrict the admissible set of network structures—this also aids in establishing the global asymptotic stability for the steady state given a constant input disturbance. The entire method does not assume any explicit knowledge of the underlying rate kinetics barring some minimal assumptions. Finally, we also discuss the infeasibility of the incoherent feed-forward structure to provide adaptation in the presence of downstream connections. Our theoretical findings are corroborated by detailed and extensive simulation studies. Overall, we propose a generic and novel algorithm based on a nonlinear systems theory to unravel the design principles for global adaptation.