This paper provides novel CRN designs for the construction of an asynchronous computing device based on a bi-molecular reaction motif inspired by the Approximate Majority network (Angluin et al. 2008; Cardelli and Csikász-Nagy 2012). The motif employs catalytic reactions to achieve bistable switching of molecular concentrations, which emulates high and low voltage signals in digital electronics. All components are produced with simple reactions and uniform reaction rates, where we assume a well-mixed solution under mass action kinetics, and are independent of a universal clock. Moreover, any design provided in this paper could in principle be realised as a DNA strand displacement device (Cardelli 2010). We work with the dual-rail design methodology and employ a variant of the diagrammatic language of Cardelli (2014) to represent the designs at the high level. Starting from the Muller C-element, we design the main components of a complete asynchronous computing device in terms of CRNs in a principled way, including logic gates, control flow and basic arithmetic, as well as more complex structures such as queues. We validate the designs by exploring their time evolution for all possible combinations of inputs using Microsoft’s Visual GEC tool,1 with the latter also approximated using an experimental implementation of the Linear Noise Approximation (LNA) of Cardelli et al. (2015) provided by Visual GEC that offers better scalability. We use the LNA to highlight a flaw with a key design component. Further, we demonstrate the correct behaviour of the circuits against temporal logic specifications with the probabilistic model checker PRISM2 (Kwiatkowska et al. 2011). Our designs constitute the first feasible implementation of asynchronous computational components as CRNs, and are relevant for a multitude of applications in synthetic biology and biosensing.