Theoretical investigations to understand the emergence of any biological property require unraveling the specific interconnections (design principles) among the constituents of the underlying biological network. Adaptation, the tendency of every living organism to regulate its essential activities in the presence of environmental fluctuations, is one such important property that has gained significant attention in systems and synthetic biology.
In this work, we present generic methodologies inspired by systems theory to discover the design principles for adaptation in four different contexts– i) Perfect infinitesimal adaptation for deterministic disturbance (PIAD), ii) Robust infinitesimal adaptation for deterministic disturbance (RIAD), iii) Robust infinitesimal adaptation for stochastic processes (RIAS), and iv) Perfect adaptation for a deterministic disturbance with arbitrary magnitude (PAD). In all the cases, firstly, we translate the necessary qualitative conditions for adaptation to mathematical constraints using the language of systems theory, which we then map back as design requirements for the underlying networks. Thus, contrary to the existing approaches, the proposed methodologies provide an exhaustive set of admissible network structures without resorting to computationally burdensome brute-force techniques. Further, the proposed frameworks do not assume any prior knowledge about the particular rate kinetics, thereby validating the conclusions for a large class of biological networks.