GillespieAlgorithmus

The Gillespie algorithm, named after Daniel T. Gillespie, is a stochastic simulation algorithm for generating exact trajectories of coupled chemical reactions in well-mixed systems. It provides statistically correct realizations of the chemical master equation, capturing intrinsic noise arising from the discrete nature of molecular interactions.

In each step, the system state x comprises molecule counts. For each reaction μ with stoichiometry sμ and rate constant cμ, a propensity aμ(x) is computed, reflecting the probability that reaction μ fires in the next infinitesimal time interval. The sum a0 = sumμ aμ(x) is the overall hazard. Two independent random numbers are drawn: tau = (1/a0) ln(1/r1) gives the time to the next reaction, and a second number selects which reaction occurs with probability aμ/a0. The state is updated by x := x + sμ, and time advances by tau. The process repeats until a chosen end time or state condition is met.

Variants of the method include the Direct Method (the standard SSA as originally described by Gillespie), the First Reaction Method, and the Next Reaction Method, as well as tau-leaping approaches that approximate multiple reactions over a larger time step to gain speed on larger systems. The algorithm assumes a well-mixed environment and discrete reactant counts; it is exact for the chemical master equation in that context, but can be computationally intensive for large networks.

Applications span chemical kinetics and systems biology, including gene expression modeling, enzymatic networks, and intracellular signaling where stochastic fluctuations influence behavior. The method is a foundational tool for studying noise, rare events, and emergent dynamics in small systems.