A modified Next Reaction Method for simulating systems with time varying rate constants and systems with delays

Biochemical reaction systems with a low to moderate number of molecules are typically modeled as discrete jump Markov processes. These systems are oftentimes simulated using the Gillespie Algorithm or the Next Reaction Method, which are exact simulation methods. In this paper we make explicit use of the fact that the initiation times of each reaction are given by the firing times of an independent, unit Poisson process with integrated propensity function. We use this representation to develop a modified version of the Next Reaction Method. We then demonstrate how this modified Next Reaction Method is the appropriate method to use to simulate systems in which the rate constants are functions of time (that is, systems that are time dependent Markov Processes). Finally, we extend our modified Next Reaction Method to systems with delays and compare its efficiency with those of the recent algorithms designed for systems with delays.