Cell-free biosensors are powerful platforms for monitoring human and environmental health. Here, we expand their capabilities by interfacing them with toehold-mediated strand displacement circuits, a dynamic DNA nanotechnology that enables molecular computation through programmable interactions between nucleic acid strands. We develop design rules for interfacing a small molecule sensing platform called ROSALIND with toehold-mediated strand displacement to construct hybrid RNA–DNA circuits that allow fine-tuning of reaction kinetics. We use these design rules to build 12 different circuits that implement a range of logic functions (NOT, OR, AND, IMPLY, NOR, NIMPLY, NAND). Finally, we demonstrate a circuit that acts like an analog-to-digital converter to create a series of binary outputs that encode the concentration range of the molecule being detected. We believe this work establishes a pathway to create ‘smart’ diagnostics that use molecular computations to enhance the speed and utility of biosensors.