DAF-16/FOXO Shuttling: Stochastic Kinetics Model

This is the 2D counterpart to the rotatable 3D cell: instead of animating particles toward a threshold-based target, every DAF-16/FOXO molecule here runs its own independent three-state continuous-time Markov chain — cytoplasmic-phosphorylated, cytoplasmic-dephosphorylated, and nuclear — with transition probabilities computed each frame from real rate constants using the standard 1 − e^−rate·dt kinetic Monte Carlo formula, and diffuses by true Brownian motion inside a reflecting compartment diagram. A live strip chart plots the simulated ensemble-average nuclear fraction against the closed-form mean-field steady state derived from the same rate constants, and a Hill-function readout converts nuclear DAF-16 into stress-response gene activity. Dial signaling down to mimic a daf-2 mutant, knock out daf-16 to reproduce the classic epistasis experiment, or fire an insulin pulse to watch the whole population re-equilibrate in real time.