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International Conference on Computational Methods in Systems Biology

CMSB 2017: Computational Methods in Systems Biology pp 108–127Cite as

Strong Turing Completeness of Continuous Chemical Reaction Networks and Compilation of Mixed Analog-Digital Programs

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Part of the book series: Lecture Notes in Computer Science ((LNBI,volume 10545))

Abstract

When seeking to understand how computation is carried out in the cell to maintain itself in its environment, process signals and make decisions, the continuous nature of protein interaction processes forces us to consider also analog computation models and mixed analog-digital computation programs. However, recent results in the theory of analog computability and complexity establish fundamental links with classical programming. In this paper, we derive from these results the strong (uniform computability) Turing completeness of chemical reaction networks over a finite set of molecular species under the differential semantics, solving a long standing open problem. Furthermore we derive from the proof a compiler of mathematical functions into elementary chemical reactions. We illustrate the reaction code generated by our compiler on trigonometric functions, and on various sigmoid functions which can serve as markers of presence or absence for implementing program control instructions in the cell and imperative programs. Then we start comparing our compiler-generated circuits to the natural circuit of the MAPK signaling network, which plays the role of an analog-digital converter in the cell with a Hill type sigmoid input/output functions.

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Notes

  1. 1.

    Restricting the definition to computable arguments might seem quite natural but is not the classical definition of computable analysis, see the Appendix of [48].

  2. 2.

    For the sake of reproducibility, all the examples described in this paper are directly executable online in Biocham v4 (http://lifeware.inria.fr/biocham4) notebooks available at http://lifeware.inria.fr/wiki/software/#CMSB17.

  3. 3.

    This definition can be generalized to functions of several variables over different domains [7].

  4. 4.

    The decreasing assumption is here to yield a simple way to decide when the result on the first component is correct with the required precision: given some precision \(\epsilon \), just wait until the second component is less than \(\epsilon \).

  5. 5.

    Note that we do not impose that concentration values are small values, less than 1 for instance. We consider arbitrary large concentration and molecule numbers [25].

  6. 6.

    Note also that the transformation to at most binary reactions is temporarily not included in our compiler.

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Acknowledgements

We are grateful to especially one reviewer for his expert proofreading which helped us to improve the presentation of our results, and to the editors for providing us with the necessary extra space. Part of this research is funded by the ANR-MOST Biopsy project. The first author acknowledges fruitful discussions with Jie-Hong Jiang (NTU, Taiwan) on the compilation of program control flows with reactions, and motivating discussions with Frank Molina (CNRS, Sys2Diag, Montpellier) on the biochemical implementation by enzymatic reactions in microfluidic vesicles.

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Authors and Affiliations

  1. Inria, Université Paris-Saclay, EP Lifeware, Palaiseau, France

    François Fages & Guillaume Le Guludec

  2. Sup Telecom, Paris, France

    Guillaume Le Guludec

  3. LIX, CNRS, Ecole Polytechnique, Palaiseau, France

    Olivier Bournez

  4. Max Planck Institute for Computer Science, Saarbrücken, Germany

    Amaury Pouly

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  1. François Fages
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  2. Guillaume Le Guludec
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Correspondence to François Fages .

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  1. Inria & École normale supérieure, Paris, France

    Jérôme Feret

  2. Technische Universität Darmstadt, Darmstadt, Germany

    Heinz Koeppl

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Fages, F., Le Guludec, G., Bournez, O., Pouly, A. (2017). Strong Turing Completeness of Continuous Chemical Reaction Networks and Compilation of Mixed Analog-Digital Programs. In: Feret, J., Koeppl, H. (eds) Computational Methods in Systems Biology. CMSB 2017. Lecture Notes in Computer Science(), vol 10545. Springer, Cham. https://doi.org/10.1007/978-3-319-67471-1_7

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