Abstract
The Turing completeness result for continuous chemical reaction networks (CRN) shows that any computable function over the real numbers can be computed by a CRN over a finite set of formal molecular species using at most bimolecular reactions with mass action law kinetics. The proof uses a previous result of Turing completeness for functions defined by polynomial ordinary differential equations (PODE), the dual-rail encoding of real variables by the difference of concentration between two molecular species, and a back-end quadratization transformation to restrict to elementary reactions with at most two reactants. In this paper, we present a polynomialization algorithm of quadratic time complexity to transform a system of elementary differential equations in PODE. This algorithm is used as a front-end transformation to compile any elementary mathematical function, either of time or of some input species, into a finite CRN. We illustrate the performance of our compiler on a benchmark of elementary functions relevant to CRN design problems in synthetic biology specified by mathematical functions. In particular, the abstract CRN obtained by compilation of the Hill function of order 5 is compared to the natural CRN structure of MAPK signalling networks.
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Notes
- 1.
For the sake of simplicity of the definition given here, we omit the error control mechanism that requires one extra CRN species z verifying:
\(\forall t>1\ |y(t)-f(x(0))|\le z(t),\ \forall t'>t\ z(t')<z(t)\) and \(\lim _{t\rightarrow \infty }z(t)=0\).
- 2.
http://lifeware.inria.fr/biocham/. All experiments described in this paper are available at https://lifeware.inria.fr/wiki/Main/Software#CMSB21.
- 3.
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Acknowledgment
We acknowledge fruitful discussions with Olivier Bournez, François Lemaire, Gleb Pogudin and Amaury Pouly. This work was supported by ANR-DFG SYMBIONT “Symbolic Methods for Biological Networks” project grant ANR-17-CE40-0036, and ANR DIFFERENCE “Complexity theory with discrete ODEs” project grant ANR-20-CE48-0002.
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Hemery, M., Fages, F., Soliman, S. (2021). Compiling Elementary Mathematical Functions into Finite Chemical Reaction Networks via a Polynomialization Algorithm for ODEs. In: Cinquemani, E., Paulevé, L. (eds) Computational Methods in Systems Biology. CMSB 2021. Lecture Notes in Computer Science(), vol 12881. Springer, Cham. https://doi.org/10.1007/978-3-030-85633-5_5
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