Programmed cell death is executed by a proteolytic amplifier with real turnover: initiator caspase-9 (activated at the apoptosome, proportional to the death signal) cleaves procaspase-3 into active executioner caspase-3 via cooperative, Hill-type zymogen-activation kinetics — the same ultrasensitive activation seen in real caspase cascades. Active caspase-3 loops back and activates more caspase-9 through a second Hill-type feedback term, closing a positive-feedback amplifier. The inhibitor XIAP continuously binds and neutralizes active caspase-3, and SMAC/DIABLO competitively blunts XIAP. Procaspase-9/-3 are continuously synthesized and degraded (basal protein turnover), and active caspase-9/-3 are cleared at their own basal rates — this synthesis/degradation balance, combined with cooperative positive feedback, is exactly the mechanism (Legewie et al. 2006; Bagci et al. 2006) that turns the cascade into a genuine bistable switch: below threshold the cascade decays back to the healthy steady state even with the trigger held on for a while, but once amplification outruns clearance the caspase-3 level keeps climbing on its own even after the trigger is pulled back to zero.
Integrated each frame with 4th-order Runge–Kutta (concentrations, not fractions — nothing here is scripted):
dP9/dt = ksyn9 - kdeg9·P9 - k1·S·P9 - k3·H(C3)·P9
dC9/dt = k1·S·P9 + k3·H(C3)·P9 - kdegC9·C9
dP3/dt = ksyn3 - kdeg3·P3 - k2·H(C9)·P3
dC3/dt = k2·H(C9)·P3 - k4·(1-SMAC)·X·C3 - kdegC3·C3
dX/dt = kxsyn·(Xtarget - X) - k4·(1-SMAC)·X·C3
H(v) = v^2 / (Km^2 + v^2) (Hill, cooperativity n=2)
- Initial signal — sets S, the apoptosome-driven procaspase-9 activation rate; can be held constant or fired as a short pulse.
- XIAP level — sets the free-XIAP target the inhibitor relaxes toward; higher XIAP raises the commitment threshold.
- SMAC/DIABLO — competitively cuts XIAP's effective binding rate, lowering the threshold.
- Fire a pulse, then drag the signal back to 0% — if the pulse pushed caspase-3 above the threshold, the C3→C9 feedback term keeps the cascade running even with S=0, exactly the irreversible commitment step real cells show.
- The bifurcation sweep below finds this threshold directly: it integrates the same ODE to steady state at many fixed signal strengths, once increasing and once decreasing, and reports where the two curves jump — a genuine hysteresis loop computed from the live equations, not a lookup table.