The C-element design is based on the AM CRN, which computes the majority of two finite populations by converting the minority population into the majority population, so that a single population emerges as output. It uses a third ‘undecided’ state of the population, from where catalysis can drive the individuals into either of the final states. Interestingly, since approximate majority cannot be exactly computed by a bi-molecular CRN with less than 4 reactions (Mertzios et al. 2014), below we present the bi-molecular AM CRN with exactly four reactions:\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} X + Y \rightarrow X + \lambda \\ Y + X \rightarrow Y + \lambda \\ X + \lambda \rightarrow X + X \\ Y + \lambda \rightarrow Y + Y \end{aligned}$$\end{document}X+Y→X+λY+X→Y+λX+λ→X+XY+λ→Y+Ywhere X, Y are both the input and output species and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda$$\end{document}λ is the aforementioned catalytic driver. The intuition behind this reaction network is that we have two competing initial populations of X and Y, both of which try to eliminate the other by transforming their counterpart into the intermediary \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda$$\end{document}λ. If \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda$$\end{document}λ then interacts with X, it transforms itself into X, else into Y. Presented below in Fig. 7a is the same AM CRN in our diagrammatic notation.Fig. 7Two diagrammatic CRNs which are capable of computing Approximate Majority Angluin et al. (2008). In a we present the original in which the inputs X, Y, dependent on which species has the majority, influence outputs X, Y. b Shows a similar AM circuit, but now the input species are catalysed to arbitrary output species W, Z