Tuning a Genetic Circuit with Double Negative Feedforward Loops to Approximate Square Waves

Precise temporal control of gene expression is important for programming cellular functions. Existing circuits, however, often couple amplitude and duration. This prevents independent tuning of key dynamic features. We therefore set out to design a genetic system capable of generating square-wave outputs with externally programmable geometry. To achieve this, we constructed a layered regulatory design that separates production and removal into two independently controlled modules. An activating input turns expression on and builds the reporter. A repressing input stops synthesis and removes existing protein. Together, these actions confine the active window and produce square-wave outputs or pulses on demand. Waveform geometry depends on input dose and timing. Analysis of limiting cases reveals that strong repression compresses dynamic range, and finite degradation capacity restricts decay rate. Enhancing removal capacity alleviates this bottleneck and sharpens the off-transition. Together, this work establishes a general framework for programming gene-expression dynamics with independently tunable amplitude and duration.