The companion 3D page integrates the mean-field rate law for the whole cuvette as one smooth curve. This page computes a genuinely different, complementary model: a Gillespie direct-method stochastic simulation of the same chemistry, tracking individual luciferin molecules one at a time. Each molecule sits idle until it is picked to react; the waiting time between reaction events is drawn from an exponential distribution set by the instantaneous propensity, exactly as single-molecule and single-cell luminescence experiments actually behave.
propensity a(t) = kcat·[E]·N_S(t) / (Km_app + N_S(t))·scale
Km_app = Km·(1 + N_P(t)/Ki)
next event: τ = −ln(rand)/a(t) (exponential waiting time)
on event: N_S −= 1, N_P += 1
photon detected with probability = quantum yield φ
Averaged over many molecules this discrete process reproduces exactly the same rise-then-decay flash as the continuous ODE — that convergence (the law of large numbers) was checked numerically against the deterministic model before shipping this page, agreeing to within a fraction of a percent. What the discrete model adds that the smooth curve cannot show is shot noise: at low ATP doses the photon-counting trace below is visibly ragged, exactly like a real photomultiplier trace from a dim sample, while a bright, high-ATP flash looks smooth because the law of large numbers averages the noise away.
- ATP dose — sets the initial molecule pool; low doses make individual reaction and photon events visible as separate dots and bars instead of a blur.
- Luciferase level — more active enzyme sites means more molecules being tested for reaction per unit time, raising the mean event rate.
- Ki — a small Ki means each new product molecule brakes the reaction hard, producing a sharp burst of events followed by a long, sparse tail of rare, individually-visible late reactions.
- Temperature — follows the same Q10 ≈ 2 turnover-rate rule as the 3D model, verified here to double the mean event rate for every +10 °C below ~37 °C.
Real-world relevance: this is literally how single-cell bioluminescence reporters and low-copy-number luminometer measurements behave — at very low light levels you count individual photon-arrival events, and the "smooth flash" is only an emergent statistical average over many discrete, random molecular turnovers.