Designing biological network motif-based controllers by reverse engineering Hill function-type models from linear models.

Perfect adaptation, the ability to regulate and maintain gene expression to its desired value despite disturbances, is important in the development of organisms. Building biological controllers to endow engineered biological systems with such perfect adaptation capability is a key goal in synthetic biology. Model-guided exploration of such synthetic circuits has been effective in designing such systems. However, theoretical analysis to guarantee controller properties with nonlinear models, such as Hill functions, remains challenging, while use of linear models fails to capture the inherent nonlinear dynamics of gene expression systems. Here, we propose a reverse engineering approach to infer the kinetic parameters for nonlinear Hill function-type models from analysis of linear models and apply our method to design controllers, which achieve perfect adaptation. Focusing on three biological network motif-based controllers, we demonstrate via simulation the efficacy of the proposed approach in combining linear system theories with nonlinear modelling, to design multiple gene circuits that could deliver perfect adaptation. Given the ubiquitous use of Hill functions in describing the dynamics of biological regulatory networks, we anticipate the proposed reverse engineering approach to benefit a wide range of systems and synthetic biology applications.