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

TL;DR

This paper introduces a polynomialization algorithm that efficiently compiles elementary mathematical functions into finite chemical reaction networks, enabling their implementation in synthetic biology and biochemical systems.

Contribution

The paper presents a quadratic-time polynomialization algorithm that transforms differential equations into CRNs, facilitating the synthesis of complex functions in biochemical systems.

Findings

The algorithm successfully compiles elementary functions into CRNs.

Performance demonstrated on relevant synthetic biology functions.

Comparison shows the abstract CRN aligns with natural signaling networks.

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 dualrail 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.