A common approach for studying stochastic biochemical behavior utilizes the stochastic simulation algorithm (SSA). For rare events, characterization with the SSA is not feasible, as the number of realizations needed to witness even a single rare event is very large. To address this limitation, Kuwahara and Mura developed the weighted SSA (wSSA), which uses importance sampling (IS) to bias the system of interest toward the rare event. Although the wSSA can be more efficient than the SSA, its performance is highly sensitive to the choice of biasing parameters, which are problem-dependent constants that are chosen by insight and numerical experiment. Also, the wSSA lacks a measure of accuracy for its estimates.

We first describe procedural extensions to the wSSA, which provide an unambiguous measure of the optimality of a given set of IS parameters. We then develop the state-dependent wSSA (swSSA), which employs state-dependent IS parameters to efficiently characterize rare events in systems with highly variable states.

Next, we introduce the doubly weighted SSA (dwSSA). Unlike the wSSA and the swSSA, the dwSSA provides an automated mechanism for learning constant IS parameters that yield a low-variance estimate. Lastly, we describe an extension to the dwSSA---the state-dependent dwSSA (sdwSSA)---that employs state-dependent IS parameters. The sdwSSA efficiently and automatically computes these parameters by dynamically discretizing the relative propensity ranges during a multilevel cross-entropy simulation.

All algorithms presented are applied to systems of varying size and complexity, to gauge their efficiency as well as accuracy. In addition to the results described here, we provide the dwSSA and the sdwSSA as freely available software packages that are applicable to any systems that can be written in the mass action formulation of chemical kinetics.