Stimulus counters switch between discrete states in response to specific triggers22,35, whereas distributed counters are induced and then an observable element (for example, a bright particle) autonomously segregates as cells divide (a). The number of elapsed generations is encoded as the ratio of the number of particles to the number of cells. DCDC was implemented using a self-assembling protein (SAP) fused to a red fluorescent protein (RFP) under control of an arabinose-inducible promoter (PAra), producing monomers that self-assemble into a bright, fluorescent particle (b). After the cells are washed, particle production ceases but existing particles remain. We started with 10 designs for DCDC, and determined the best design as outlined in the schematic (c). To verify expression, cells were imaged after 3 h of induction with 1 mM arabinose using confocal microscopy (d, scale bar, 1 μm) and analysed using flow cytometry (e). The flow cytometry data from (e) were analysed using a range of thresholds between bright and dark cells (f) to create a receiver-operating characteristic curve36, which plots the true positive rate versus the false-positive rate for varying thresholds (g). The two best-performing designs were analysed by flow cytometry after an induction time course (h). To determine the optimal induction time, we calculated the area under the receiver operating characteristic curve for the P12/P9 design (i).