Efficient Exact Stochastic Simulation of Chemical Systems with Many Species and Many Channels

There are two fundamental ways to view coupled systems of chemical equations: as continuous, represented by differential equations whose variables are concentrations, or as discrete, represented by stochastic processes whose variables are numbers of molecules. Although the former is by far more common, systems with very small numbers of molecules are important in some applications, e.g., in small biological cells or in surface processes. In both views, most complicated systems with multiple reaction channels and multiple chemical species cannot be solved analytically. There are exact numerical simulation methods to simulate trajectories of discrete, stochastic systems, methods that are rigorously equivalent to the Master Equation approach, but they do not scale well to systems with many reaction pathways. This paper presents the Next Reaction Method, an exact algorithm to simulate coupled chemical reactions that is also efficient: it (a) uses only a single random number per simulation event, and (b) takes time proportional to the logarithm of the number of reactions, not to the number of reactions itself. The Next Reaction Method is extended to include time-dependent rate constants and non-Markov processes and it is applied to a sample application in biology: the lysis/lysogeny decision circuit of lambda phage. When run on lambda the Next Reaction Method requires approximately 1/15th as many operations as a standard implementation of the existing methods.