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

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.

Algebraic Biochemistry: a Framework for Analog Online Computation in Cells

“…It is worth remarking that any ODE system made of elementary mathematical functions can be transformed in a polynomial ODE system [17], hence one can wonder why we restrict here to polynomial expressions. This comes from the condition that asks that the domain D be of plain dimension in Def.…”

“…Similarly to our previous pipeline for compiling any elementary function in an abstract CRN that computes it [17,16,12], our compilation pipeline for generating stabilizing CRNs follows the same sequence of symbolic transformations:…”

“…Our previous MAXSAT algorithm [16] and heuristics [17] can again be used here with the slight modification mentioned above concerning the introduction of new variables.…”

“…With this provision to omit error control, the Turing-completeness result of continuous CRNs was used in [12] to design a compilation pipeline to implement any mathematical elementary function in abstract chemistry. This compiler, implemented in our CRN modeling, analysis and synthesis software Biocham [2] as the one presented here 1 , generates a CRN over a finite set of abstract molecular species, through several symbolic computation steps, mainly composed of polynomialization [17], quadratization [16] and lazy dual-rail encoding of negative variables. A similar approach is undertaken in the CRN++ system [22], also related to [3].…”

“…This marks a fundamental difference with many natural CRNs which perform a kind of online computation by adapting the response to the evolution of the input. This is the case for instance of the ubiquitous MAPK signaling network which computes an input/output function well approximated by a Hill function of order 4.9 [18], while our synthesized CRNs to compute the same function consume their input and do not correctly adapt to change of the input value during computation [17].…”