Lambda Phage Toggle Switch: CI/Cro Phase Portrait

When bacteriophage lambda infects an E. coli cell, a bistable mutual-repression circuit between the CI repressor and the Cro anti-repressor decides whether the cell goes lysogenic or lytic. This 2D companion to the 3D cell-rendering simulation draws that decision the way a systems biologist actually studies it: as a phase portrait. The two population densities are plotted directly as (Cro, CI) coordinates, the real nullclines and vector field are computed from the governing ODEs and drawn on the plane, and a shaded background shows the true basin of attraction for each fate — obtained by numerically integrating the noiseless dynamics from a grid of starting concentrations, not guessed. A population of infected cells races across this plane in real time (RK4-integrated with molecular noise), straddling the unstable saddle point until noise tips each one into the lysogeny or lysis basin, exactly reproducing the coin-flip logic of the real circuit. Multiplicity of infection and nutrient stress visibly shift the nullcline crossing and the basin boundary, and the SOS button drives an established lysogen's cell back across the saddle for real prophage induction.