Mapping dynamical systems into chemical reactions

Polynomial dynamical systems (DSs) can model a wide range of physical processes. A special subset of these DSs that can model chemical reactions under mass-action kinetics is called chemical dynamical systems (CDSs). A fundamental problem, central to synthetic biology, is to map polynomial DSs into dynamically similar CDSs. In this paper, we introduce the quasi-chemical map (QCM) that can systematically solve this problem. The QCM introduces suitable state-dependent perturbations into any given polynomial DS which then becomes a CDS under sufficiently large translations of variables. This map preserves robust features, such as generic equilibria and limit cycles, and generic bifurcations, as well as some temporal properties, such as periods of oscillations. Furthermore, the resulting CDSs are at most one degree higher than the original DSs. We showcase the QCM by designing relatively simple CDSs with oscillations, chaos and bifurcations, and addressing Hilbert's 16th problem in chemistry.