# Abstract

Designing biological networks that are capable of achieving specific functionality has been of sustained interest in the field of synthetic biology for nearly a decade. Adaptation is one such important functionality that is observed in bacterial chemotaxis, cell signalling and homoeostasis. It refers to the ability of a cell to cope with environmental perturbations. All of these adaptation networks, involve negative feedback loops or open loop control strategies. A typical enzymatic network is a circuit of enzymes whose connections are characterized by enzymatic reactions that exhibit non-linear dynamics. Previous approaches to design of enzymatic networks capable of perfect adaptation have used brute force searches encompassing the complete set of possibilities to identify suitable circuit designs. In contrast, this work presents a systematic algorithm for circuit design, using a linear systems-theoretic approach. The key idea is to set up a design-oriented problem formulation as against employing a brute force search in the space of possible circuits. To this effect, we first linearize the non-linear dynamical circuit, subsequently, we translate the requirements for adaptation to design specifications for a linear time-invariant system and imposing these design specifications on the linearized system, we obtain the minimal topologies or motifs that can perform perfect adaptation, with an optimal set of rate constants. The optimal set of rate constants is obtained by solving a structure-specific constrained optimisation problem. In effect, we demonstrate that the proposed approach identifies the key motifs of the biological network that were identified by the existing brute force approach, albeit in a systematic manner and with very little computational effort.