Describing Function-based Approximations of Biomolecular Systems

TL;DR

This paper adapts the describing function method to biomolecular systems, providing improved approximations of their responses and analytical error bounds, enhancing analysis of oscillations and system behavior.

Contribution

It introduces a novel application of describing functions to biomolecular systems, surpassing standard linearization in accuracy and enabling better limit cycle analysis.

Findings

Better approximation of biomolecular responses than linearization

Analytical upper bounds for computational errors

Enhanced limit cycle analysis with oscillation bounds

Abstract

Mathematical methods provide useful framework for the analysis and design of complex systems. In newer contexts such as biology, however, there is a need to both adapt existing methods as well as to develop new ones. Using a combination of analytical and computational approaches, we adapt and develop the method of describing functions to represent the input-output responses of biomolecular signalling systems. We approximate representative systems exhibiting various saturating and hysteretic dynamics in a way that is better than the standard linearization. Further, we develop analytical upper bounds for the computational error estimates. Finally, we use these error estimates to augment the limit cycle analysis with a simple and quick way to bound the predicted oscillation amplitude. These results provide system approximations that can add more insight into the local behaviour of these systems than standard linearization, compute responses to other periodic inputs, and to analyze limit cycles.