A catalytic system and method of catalyzing reactions that uses a novel toehold exchange mechanism that allows a specified input to catalyze the release of a specified output, which in turn can serve as a catalyst for other reactions is provided. This toehold exchange catalyst system, which can be driven forward by the configurational entropy of the released molecule, provides an amplifying circuit element that is simple, fast, modular, composable, and robust. Using this toehold exchange catalyst system it has been possible to construct and characterize several circuits that amplify nucleic acid signals, including a feed-forward cascade with quadratic kinetics and a positive feedback circuit with exponential growth kinetics.