Bistability in Gene Regulation: Simulating Positive Feedback and Toggle Circuits Using Python and Hill Functions

Cellular decision-making relies heavily on bistable gene regulatory networks, which allow systems to respond to internal or external stimuli by switching between several stable expression states. Processes like cell differentiation, epigenetic memory, and the creation of artificial biological switches all depend on these dynamics. In this work, we introduce a simple and reproducible Python framework for modeling bistability in genetic feedback systems by means of ordinary differential equations (ODEs) driven by Hill functions. We employ two fundamental motifs, both of which are recognized for their ability to generate bistable behavior: a two-gene mutual inhibition toggle switch and a single-gene positive feedback loop. We investigate the effects of different Hill coefficients, production rates, and initial expression levels on system dynamics by numerical integration using SciPy. Our simulations show phase-plane convergence to several attractors, map expression outcomes over a grid of beginning circumstances, and illustrate the onset of bistability above a key Hill threshold. The delicate reliance of final states on cooperativity and beginning values is further demonstrated by heatmaps and bifurcation-like graphs. For accessibility, all code is hosted at Google Colab and is written in open-source Python. This study promotes research and teaching in synthetic biology, systems biology, and computational modeling while providing a simple yet effective computational framework for investigating the fundamentals of gene circuit bistability.