Real-Time Equivalence of Chemical Reaction Networks and Analog Computers

Abstract

This paper investigates the class $\mathbb {R}_{RTCRN}$ of real numbers that are computable in real time by chemical reaction networks (Huang, Klinge, Lathrop, Li, Lutz, 2019), and its relationship to general purpose analog computers. Roughly, $\alpha \in \mathbb {R}_{RTCRN}$ if there is a chemical reaction network (CRN) with integral rate constants and a designated species $X$ such that, when all species concentrations are initialized to zero, $X$ converges to $\alpha$ exponentially quickly. In this paper, we define a similar class $\mathbb {R}_{RTGPAC}$ of real numbers that are computable in real time by general purpose analog computers, and show that $\mathbb {R}_{RTGPAC}= \mathbb {R}_{RTCRN}$ using a construction similar to that of the difference representation introduced by Fages, Le Guludec, Bournez, and Pouly. We prove this equivalence by showing that $\mathbb {R}_{RTCRN}$ is a subfield of $\mathbb {R}$ which solves a previously open problem. We also prove that a CRN with integer initial concentrations can be simulated by a CRN with all zero initial concentrations. Using these results, we give simple and natural constructions showing $e$ and $\pi$ are members of $\mathbb {R}_{RTCRN}$, which was not previously known.

This research was supported in part by National Science Foundation Grant 1545028.