A + B →k₁ C | C →k₂ A + B | A₀ = B₀ = Ω·c₀, C₀ = 0
Gillespie Algorithm (1977)
The exact stochastic simulation algorithm (SSA) treats reactions as discrete events.
At each step:
1. Compute propensities: a₁ = k₁·A·B, a₂ = k₂·C
2. Sample time to next event: τ ~ Exp(a₀ = a₁+a₂)
3. Select reaction μ with probability aμ/a₀
4. Update molecule counts and advance time by τ
Deterministic ODE (dashed white): d[C]/dt = k₁[A][B] − k₂[C], valid for Ω→∞.
Stochastic trajectories (colored): fluctuate around ODE with amplitude ∝ 1/√Ω.
Equilibrium: K_eq = k₁/k₂. For small Ω, discrete fluctuations and rare events dominate.