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A Class of Simple Biomolecular Antithetic Proportional-Integral-Derivative Controllers

Filo, Maurice; Khammash, Mustafa

Abstract

Proportional-Integral-Derivative (PID) feedback controllers have been the most widely used controllers in the industry for almost a century. This is mainly due to their simplicity and intuitive operation. Recently, motivated by their success in various engineering disciplines, PID controllers found their way into molecular biology. In this paper, we consider the mathematical realization of (nonlinear) PID controllers via biomolecular interactions in both the deterministic and stochastic settings. We propose several simple biomolecular PID control architectures that take into consideration the biological implementation aspect. We verify the underlying PID control structures by performing a linear perturbation analysis and examine their eects on the (deterministic and stochastic) performance and stability. In fact, we demonstrate that dierent proportional controllers exhibit dierent capabilities of enhancing the dynamics and reducing variance (cell-to-cell variability). Furthermore, we propose a simple derivative controller that is mathematically realized by cascading the antithetic integral controller with an incoherent feedforward loop without adding any additional species. We demonstrate that the derivative component is capable of enhancing the transient dynamics at the cost of boosting the variance, which agrees with the well known vulnerability of the derivative controller to noise. We also show that this can be mitigated by carefully designing the inhibition pathway of the incoherent feedforward loop. Throughout the paper, the stochastic analysis is carried out based on a tailored moment-closure technique and is also backed up by simulations.

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