Design and analysis of a Proportional-Integral-Derivative controller with biological molecules

Abstract

The ability of cells to regulate their function through feedback control is a fundamental underpinning of life. The capability to engineer de novo feedback control with biological molecules is ushering in an era of robust functionality for many applications in biotechnology and medicine. To fulfill their potential, feedback control strategies implemented with biological molecules need to be generalizable, modular and operationally predictable. Proportional-Integral-Derivative (PID) control fulfills this role for technological systems and is a commonly used strategy in engineering. Integral feedback control allows a system to return to an invariant steady-state value after step disturbances, hence enabling its robust operation. Proportional and derivative feedback control used with integral control help sculpt the dynamics of the return to steady-state following perturbation. Recently, a biomolecular implementation of integral control was proposed based on an antithetic motif in which two molecules interact stoichiometrically to annihilate each other's function. In this work, we report how proportional and derivative implementations can be layered on top of this integral architecture to achieve a biochemical PID control design. We illustrate through computational and analytical treatments that the addition of proportional and derivative control improves performance, and discuss practical biomolecular implementations of these control strategies.

In vitro implementation of robust gene regulation in a synthetic biomolecular integral controller

“…Even though the presented model is a coarse-grained mechanistic model, it enabled us to explain the measured response of the controller, and can also predict dynamic trajectories for a wide range of operating conditions accurately. Based on the extracted parameter values, we derived a simplified version of the 16 original model that is as effective as the original model and can be used for further theoretical analysis in order to gain deeper insight into the operation of the controller.…”

“…This strategy has been used previously to design closed-loop biomolecular controllers for reference tracking and disturbance rejection. [13][14][15][16][17][18] Although our design is a variation of those in, 13,26 it differs from them in substantial ways in its biological instantiation. Our results are unique in the sense that we have a well-characterized controller that is realized in all E. coli TXTL system, where biological noise is negligible compared to in vivo implementations.…”

“…Motivated by this analogy, various possible designs of such controllers have been discussed substantially in the context of biological systems. [13][14][15][16]26 The present work is inspired by our previous work, 26 in which we introduced a computational design based on RNA based controllers; no experimental validation was provided. Here, we start from that design, modifying it to allow for direct genetic regulation and provide an experimental validation.…”

“…The "internal model principle" in control theory states, in essence, that the existence of an integrator in the control loop is necessary for the tight regulation of the concentration of an output species in the presence of disturbances or stimuli. 12 Inspired by these ideas from engineering and control theory, there have been several recent designs [13][14][15][16] and implementations [17][18] of biomolecular integral feedback controllers. These implementations are not completely satisfactory in that it is very difficult to make quantitative predictive models of their behaviors, in large part due to the biological noise which is found in cellular systems, which makes it hard to achieve precise control over model parameters, thereby complicating the design and feasibility of the system.…”

Practical differentiation using ultrasensitive molecular circuits

“…The linearized dynamics of a rapid buffering process have been shown to be mathematically equivalent to derivative negative feedback and improve stability of a closed loop system with limited sensitivity to molecular noise [3]. An approximate derivative component for PID controllers was computationally considered in [4] by operating near saturation a two node network that achieves perfect adaptation [5], [6], [7], [8]. A model for genetic differentiation that combines two genetic elements tracking positive and negative slopes of an input, and computes their difference via molecular sequestration, was also proposed and numerically studied in [9].…”