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On Estimating Derivatives of Input Signals in Biochemistry

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Computational Methods in Systems Biology (CMSB 2023)

Part of the book series: Lecture Notes in Computer Science ((LNBI,volume 14137))

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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 compute online functions with CRNs augmented with an error correcting delay for derivatives. 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.

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Notes

  1. 1.

    We also explore in Figs. 2D and 3C what a non analyticity of \(X_\text {ext}\) imply for our model.

  2. 2.

    A species (resp. reaction) node can only be merged with another species (resp. reaction) node and the resulting node inherits of all the incoming and outcomming edges of the two nodes.

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Acknowledgment

This work benefited from ANR-20-CE48-0002 \(\delta \)ifference project grant.

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Authors and Affiliations

  1. Lifeware Project-Team, Inria Saclay, Palaiseau, France

    Mathieu Hemery & François Fages

Authors
  1. Mathieu Hemery
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  2. François Fages
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Corresponding author

Correspondence to François Fages .

Editor information

Editors and Affiliations

  1. University of Luxembourg, Esch-sur-Alzette, Luxembourg

    Jun Pang

  2. Inria Lille, Villeneuve d’Ascq, France

    Joachim Niehren

Appendix: Computation of Integration with a Delay

Appendix: Computation of Integration with a Delay

To prove that the drift of the output is a direct consequence of the delay, we first compute the input and the approximate derivative for our choice of input:

$$\begin{aligned} \begin{aligned} x(t)&= 1+\sin (t) \\ x'(t-\tau )&= \cos (t-\tau ) \\&= \cos (t) + \tau \sin (t) + o(\tau ^2) \end{aligned} \end{aligned}$$
(18)

Then we can compute the output up to the first order:

$$\begin{aligned} \begin{aligned} y(t)&= \int 2 x(s) x'(s-\tau ) ds \\&= \int 2 \left( 1+\sin (s)\right) \cos (s) ds + \int 2 \tau (\sin (s)+\sin ^2(s)) ds \\&= \left( 1+\sin (t)\right) ^2 + 2 \tau \int \sin (s)+\sin ^2(s) ds \\ y(t)&\simeq \left( 1+\sin (t)\right) ^2 + \tau t \end{aligned} \end{aligned}$$
(19)

Then, to correct the observed drift, we propose to introduce a delay signal and use it in the computation to produce the output species \(Y_+\) and \(Y_-\), with the following CRN:

$$\begin{aligned} \begin{aligned} X_\text {ext}&\xrightarrow {k_\text {diff}} X_\text {in},&\quad X_\text {in}&\xrightarrow {k_\text {diff}} X_\text {ext}, \\ X_\text {in}&\xrightarrow {k} X_\text {in}+ X_\text {delay},&\quad X_\text {delay}&\xrightarrow {k} \emptyset , \\ X_\text {ext}&\xrightarrow {k_\text {diff}k} X_\text {ext}+ D_+&\quad D_+&\xrightarrow {k} \emptyset \\ X_\text {in}&\xrightarrow {k_\text {diff}k} X_\text {in}+ D_-&\quad D_-&\xrightarrow {k} \emptyset \\ X_\text {delay}+ D_+&\xrightarrow {2} X_\text {delay}+ D_+ + Y_+&\quad X_\text {delay}+ D_-&\xrightarrow {2} X_\text {delay}+ D_- + Y_- \\ D_+ + D_-&\xrightarrow {\text {fast}} \emptyset&\quad Y_+ + Y_-&\xrightarrow {\text {fast}} \emptyset \\ \end{aligned} \end{aligned}$$
(20)

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Hemery, M., Fages, F. (2023). On Estimating Derivatives of Input Signals in Biochemistry. In: Pang, J., Niehren, J. (eds) Computational Methods in Systems Biology. CMSB 2023. Lecture Notes in Computer Science(), vol 14137. Springer, Cham. https://doi.org/10.1007/978-3-031-42697-1_6

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  • DOI: https://doi.org/10.1007/978-3-031-42697-1_6

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