A Singular Singular Perturbation Problem Arising from a Class of Biomolecular Feedback Controllers

Abstract

© 2017 IEEE. A long-standing challenge in synthetic biology is to engineer biomolecular systems that can perform robustly in highly uncertain cellular environments. Recently, there has been increasing interest to design biomolecular feedback controllers to address this challenge. Molecular sequestration is one of the proposed feedback mechanisms. For this type of design, when all reactions within the controller are sufficiently fast, the process output can reach a set-point regardless of parametric uncertainties. However, as we demonstrate in this letter, the way in which molecular sequestration affects the fast controller dynamics leads to a singular singularly perturbed (SSP) system. In an SSP system, the boundary layer Jacobian is singular and thus standard singular perturbation approaches cannot be applied, posing difficulties to analytically determine the performance of sequestration-based controllers. In this letter, we consider a class of linear systems that capture the key structure of sequestration-based controllers. We show that, under certain technical conditions, these SSP systems can still be approximated by reduced-order systems that are dependent on the small parameter. This result allows us to analytically evaluate the tracking performance of the linearized model of a sequestration-based controller.