On the Complexity of Quadratization for Polynomial Differential Equations

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

“…That PIVP of degree 3 can now be used as input of our previous compilation pipeline [11]. It is first transformed in quadratic form [19], in this case by introducing one variable, BT = B.T , and removing the time T , giving the following quadratic PIVP:…”

“…The quadratization algorithm [19] now introduces new variables BX and T BX for the corresponding monomials, and removes variables T and B. This generates the following quadratic PIVP:…”

“…For example α = 1 3 may be solved with β = −2 3 and i = 3, j = 4, k = 0, l = 2. But this uses a polynomial of order 7 which may impact negatively the final quadratization phase [19]. For these reasons, we chose to treat exponentiation by systematically introducing two variables.…”

“…That definition for input/output functions computed by a CRN is used in [11] to show the Turing completeness of continuous CRNs in the sense 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 equation initial value problems (PIVP) [2], the dual-rail encoding of real variables by the difference of concentration between two molecular species [21,18], and a back-end quadratization transformation to restrict to elementary reactions with at most two reactants [5,19,3]. This proof gives rise to a pipeline, implemented in BIOCHAM-4 2 , to compile any computable real function presented by a PIVP into a finite CRN.…”

“…In Sec. 5 we evaluate this approach on a benchmark of elementary functions relevant to CRN design problems in synthetic biology used in [19]. In particular, we compare the CRN synthesized for the Hill function of order 5 to the structure of the natural MAPK signalling CRNs that have been shown to implement a similar input/output function [20].…”