# On the stability of nucleic acid feedback controllers

## TL;DR

This paper develops theoretical insights into the stability and dynamics of nucleic acid feedback controllers, revealing that nonlinear effects can destabilize systems even when linear designs suggest stability.

## Contribution

It introduces foundational theoretical results on equilibria and stability of nucleic acid controllers, highlighting the nonlinear dynamics introduced by DNA strand displacement reactions.

## Key findings

- Linear stability does not guarantee chemical network stability.
- Nonlinear dynamics can cause instability in nucleic acid feedback systems.
- Simulation confirms theoretical predictions of destabilization.

## Abstract

Recent work has shown how chemical reaction network theory may be used to design dynamical systems that can be implemented biologically in nucleic acid-based chemistry. While this has allowed the construction of advanced open-loop circuitry based on cascaded DNA strand displacement (DSD) reactions, little progress has so far been made in developing the requisite theoretical machinery to inform the systematic design of feedback controllers in this context. Here, we develop a number of foundational theoretical results on the equilibria, stability, and dynamics of nucleic acid controllers. In particular, we show that the implementation of feedback controllers using DSD reactions introduces additional nonlinear dynamics, even in the case of purely linear designs, e.g. PI controllers. By decomposing the effects of these non-observable nonlinear dynamics, we show that, in general, the stability of the linear system design does not necessarily imply the stability of the underlying chemical network, which can be lost under experimental variability when feedback interconnections are introduced. We provide an in-depth theoretical analysis of an example illustrating this phenomenon, whereby the linear design does not capture the instability of the full nonlinear system implemented as a DSD reaction network, and we further confirm these results using VisualDSD, a bespoke software tool for simulating nucleic acid-based circuits. Our analysis highlight the many interesting and unique characteristics of this important new class of feedback control systems.