  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.
