Uncertainty modelling and stability robustness analysis of nucleic acid-based feedback control systems

Recent advances in nucleic acid-based chemistry have highlighted its potential for the implementation of biomolecular feedback circuits. Here, we focus on a proposed design framework, which is able to approximate the inputoutput behaviour of key linear operators used in feedback control circuits by combining three elementary chemical reactions. The implementation of such circuits using DNA strand displacement introduces non-linear internal dynamics due to annihilation reactions among different molecular species. In addition, experimental implementation of in silico designs introduces significant levels of uncertainty and variability in reaction rate constants and equilibrium concentrations. Previous work using this framework has overlooked the practical
implications of these issues for the construction of nucleic acidbased feedback control circuits. Here, we analyse the impact of these nonlinearities and uncertainties on the stability of a biomolecular feedback loop.We show that a rigorous analysis of its nucleic acid-based implementation requires an investigation of the associated non-linear dynamics, to decide on realisable
parameters and acceptable equilibrium concentrations. We also show how the level of experimental uncertainty that is tolerated by the feedback circuit can be quantified using the structured singular value. Our results constitute a first step towards the development of a rigorous robustness analysis framework for nucleic acid-based feedback control circuits.