Design principles for perfect adaptation in biological networks with nonlinear dynamics

Abstract

Establishing a mapping between the emergent biological properties and the repository of network structures has been of great relevance in systems and synthetic biology. Adaptation is one such biological property of paramount importance that promotes regulation in the presence of environmental disturbances. This paper presents a nonlinear systems theory-driven framework to identify the design principles for perfect adaptation. Based on the prior information about the network, we frame precise mathematical conditions for adaptation using nonlinear systems theory. We first deduce the mathematical conditions for perfect adaptation for constant input disturbances. Subsequently, we first translate these conditions to specific necessary structural requirements for adaptation in networks of small size and then extend 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 narrow 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 networks (IFFLP) to provide adaptation in the presence of downstream connections. Detailed and extensive simulation studies corroborate the theoretical findings. Moreover, we propose a generic and novel algorithm based on a nonlinear systems theory to unravel the design principles for global adaptation.