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Chemical-induced dimerization (CID) modules enable users to implement ligand-controlled cellular and biochemical functions for a number of problems in basic and applied biology. A special class of CID modules occur naturally by plants involving a hormone receptor which binds hormone, triggering a conformational change in the receptor which enables recognition by a second binding protein. Two recent reports show that such hormone receptors can be engineered to sense dozens of structurally diverse compounds. As a closed form model for molecular ratchets would be of immense utility in forward engineering of biological systems, here we have developed a closed form model for these distinct CID modules. These modules, which we call molecular ratchets, are distinct from more common CID modules called molecular glues in that they engage in saturable binding kinetics and are well characterized by a Hill equation. A defining characteristic of molecular ratchets is that the sensitivity of the response can be tuned by increasing the molar ratio between the hormone receptor and binding protein. Thus, the same molecular ratchet can have a picomolar or micromolar EC50 depending on the concentration of the different receptor and binding proteins. Closed form models are derived for a base elementary reaction rate model, for ligand-independent complexation of receptor and binding protein, and for homodimerization of the hormone receptor. Useful governing equations for a variety of in vitro and in vivo applications are derived, including ELISA-like microplate assays, transcriptional activation in prokaryotes and eukaryotes, and ligand-induced split protein complementation.