A stochastic simulation algorithm for biochemical reactions with delays

Biochemical systems can be described by biochemical reactions. Biochemical reactions can be investigated through mathematical modeling and stochastic simulations. Deterministic and stochastic models are two basic categories of models for biochemical reactions. Due to transmembrane transportation of biochemical species and delayed degradation, time delays are ubiquitous in coupled biochemical systems. Therefore, models for biochemical reactions can be further classified into delayed and un-delayed ones. For biochemical systems without delays, researchers have established the connections between deterministic models and stochastic models directly from the deterministic ones. For delayed biochemical systems, researchers have proposed some stochastic simulation methods to cope with biochemical reactions with time delays. However, the existing delayed stochastic simulation algorithms (SSA) are all incapable of realizing the comparison between highly nonlinear deterministic delayed models and stochastic models directly from the the deterministic ones. In this paper, we proposed a delayed SSA, which can realize the comparison between deterministic models and its stochastic counterparts. Furthermore, one can also use the algorithm to investigate intrinsic noise-induced behaviors, and the effect of system volumes. Several numerical examples show the effectiveness and correctness of our algorithm.