A Hierarchy of Biomolecular Proportional-Integral-Derivative Feedback Controllers for Robust Perfect Adaptation and Dynamic Performance

Abstract

Proportional-Integral-Derivative (PID) feedback controllers have been the most widely used controllers in industry for almost a century due to their good performance, simplicity, and ease of tuning. Motivated by their success in various engineering disciplines, PID controllers recently found their way into synthetic biology, where the design of feedback molecular control systems has been identified as an important goal. In this paper, we consider the mathematical realization of PID controllers via biomolecular interactions. We propose an array of topologies that offer a compromise between simplicity and high performance. We first demonstrate that different Proportional-Integral (PI) controllers exhibit different capabilities for enhancing the dynamics and reducing variance (cell-to-cell variability). Next, we introduce several derivative controllers that are realized based on incoherent feedforward loops acting in a feedback configuration. Alternatively, we show that differentiators can be realized by placing molecular integrators in a negative feedback loop--an arrangement that can then be augmented by PI components to yield PID feedback controllers. We demonstrate that the derivative component can be exploited for enhancing system stability, dramatically increasing the molecular control systems dynamic performance, and for reducing the noise effect on the output of interest. The PID controller features are established through various deterministic and stochastic analyses as well as numerical simulations. Finally, we provide an experimental demonstration using a recently developed hybrid setup, the cyberloop, where the controller is implemented in silico to control a biological genetic circuit in vivo. The large array of novel biomolecular PID controllers introduced here forms a basis for the design and construction of advanced high-performance biomolecular control systems that robustly regulate the dynamics of living systems.