This is a genuine stochastic chemical-kinetics simulation, not a re-drawing of the 3D scene. Each of the N independent DAF-16 molecules is its own continuous-time Markov chain over three states — cytoplasmic-phosphorylated (Cp), cytoplasmic-dephosphorylated (Cd), nuclear (Nu) — with real per-frame transition probabilities p = 1 − e−rate·dt computed from rate constants, exactly the formula used for exact kinetic Monte Carlo / Gillespie-style simulation. Molecules also perform true 2D Brownian motion (Gaussian step ∝ √(2·D·dt)) inside a reflecting cytoplasm annulus or nuclear disk, and physically cross the nuclear envelope only when their chemical state actually transitions.
Cp --k_pp--> Cd (dephosphorylation, constant rate)
Cd --k_akt·a--> Cp (phosphorylation, rate scales with IIS activity a)
Cd --k_in--> Nu (nuclear import — only the dephosphorylated form enters)
Nu --k_out--> Cd (nuclear export)
Mean-field steady state (detailed balance of the 3-state chain):
Cd_ss = 1 / (1 + k_akt·a/k_pp + k_in/k_out)
Nu_ss = (k_in/k_out) · Cd_ss
Gene activity = Nu_ss^n / (Nu_ss^n + K^n) (Hill function, n=2 — cooperative promoter binding)
With the rates used here (k_pp=0.3, k_akt=15, k_in=1.7, k_out=0.3 s⁻¹) the chain settles to Nu≈10% nuclear at full signaling and Nu≈85% nuclear at zero signaling — reproducing the real magnitude of the daf-2 longevity effect (lifespan roughly doubling) without hand-tuning the visual result, because it falls straight out of the balance equations above. The live chart plots the simulated ensemble-average nuclear fraction against that analytic prediction — if they don't track each other, the kinetics are wrong.
- Signaling slider — sets a in [0,1], the chronic AKT activity multiplying the phosphorylation rate.
- daf-16 knockout — removes the effector entirely: no molecules exist, nuclear fraction is pinned at 0 and the lifespan multiplier collapses to 1.00× no matter how low signaling is, reproducing the classic epistasis result (Kenyon lab).
- Insulin pulse — a transient boost to a that decays exponentially (τ=1.2 s), re-phosphorylating molecules and pulling the nuclear fraction back down before it recovers.