On Estimating Derivatives of Input Signals in Biochemistry

TL;DR

This paper provides a detailed mathematical analysis of a biochemical reaction network (CRN) pattern for estimating derivatives of input signals, clarifies its accuracy, and explores applications in synthetic and systems biology.

Contribution

It offers a rigorous analysis of a CRN derivative estimator, quantifies its error, and demonstrates its use in designing error correction and analyzing biological models.

Findings

The CRN derivative estimator's error depends on reaction parameters.

The analysis clarifies the computed quantity in the CRN.

Examples include models of circadian clock and bistable switch.

Abstract

The online estimation of the derivative of an input signal is widespread in control theory and engineering. In the realm of chemical reaction networks (CRN), this raises however a number of specific issues on the different ways to achieve it. A CRN pattern for implementing a derivative block has already been proposed for the PID control of biochemical processes, and proved correct using Tikhonov's limit theorem. In this paper, we give a detailed mathematical analysis of that CRN, thus clarifying the computed quantity and quantifying the error done as a function of the reaction kinetic parameters. In a synthetic biology perspective, we show how this can be used to design error correcting terms to compute online functions involving derivatives with CRNs. In the systems biology perspective, we give the list of models in BioModels containing (in the sense of subgraph epimorphisms) the core derivative CRN, most of which being models of oscillators and control systems in the cell, and discuss in detail two such examples: one model of the circadian clock and one model of a bistable switch.