Simulating Gene Regulatory Feedback Loops with Ordinary Differential Equations: A Reproducible Python-Based Framework

In order to preserve cellular homeostasis, regulate the levels of gene expression, and permit intricate dynamic behaviors like bistability, oscillations, and ultrasensitivity, gene regulatory networks (GRNs) mostly depend on feedback mechanisms. In this study, we introduce a computational framework that is simple and reproducible for simulating the fundamental dynamics of both positive and negative feedback loops in gene regulation through the use of ordinary differential equations (ODEs). With no need for specific software installations and a reliance only on Python and publicly available libraries like NumPy, SciPy, and Matplotlib within Google Colab, our method is intended to be both approachable and instructive. We put into practice canonical models of autoregulatory circuits, in which a gene product either suppresses or increases its own production, and examine how important system characteristics, such as the Hill coefficient, degradation constant, and production rate, influence the system's temporal behavior. By means of numerical integration of the governing ODEs, we demonstrate how positive feedback produces switch-like dynamics and possible bistability under specific parameter regimes, while negative feedback stabilizes gene expression and buffers noise. Additionally, we use comparison simulations to depict the evolution of gene expression over time, analyze phase space trajectories, and compare the response profiles of the two feedback types. We simulate the system under various initial conditions and parameter perturbations to increase the pedagogical value and demonstrate the robustness and sensitivity of feedback-regulated gene expression. Crucially, we show that straightforward, quantitative modeling of regulatory motifs can yield biologically significant insights without claiming any new biological discoveries. We present a simple model that can be used as a teaching template for systems biology courses and for computational studies of feedback in artificial or natural gene circuits. Because the entire code and visualizations are publicly accessible and cloud-executable, this system is ideal for incorporation into instructional modules, reproducible research, and the development of initial hypotheses in computational biology.