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The Turing completeness result for continuous chemical reaction networks(CRN) shows that any computable function over the real numbers can be computedby a CRN over a finite set of formal molecular species using at mostbimolecular reactions with mass action law kinetics. The proof uses a previousresult of Turing completeness for functions defined by polynomial ordinarydifferential equations (PODE), the dualrail encoding of real variables by thedifference of concentration between two molecular species, and a back-endquadratization transformation to restrict to elementary reactions with at mosttwo reactants. In this paper, we present a polynomialization algorithm ofquadratic time complexity to transform a system of elementary differentialequations in PODE. This algorithm is used as a front-end transformation tocompile any elementary mathematical function, either of time or of some inputspecies, into a finite CRN. We illustrate the performance of our compiler on abenchmark of elementary functions relevant to CRN design problems in syntheticbiology specified by mathematical functions. In particular, the abstract CRNobtained by compilation of the Hill function of order 5 is compared to thenatural CRN structure of MAPK signalling networks.

Compiling Elementary Mathematical Functions into Finite Chemical Reaction Networks via a Polynomialization Algorithm for ODEs