The discrete Chemical Master Equation (dCME) provides a fundamental framework to describe the underlying stochastic processes of biological reaction networks. However, the direct solution of dCME, although advantageous, has not yet been widely used due to the enormous computational requirement. We have developed the finite buffer method that can optimally enumerate the state space and directly solve the dCME for both steady state and time evolution [*Cao and Liang, BMC Systems Biology 2008*, *Cao ang Liang, JCP 2013*]. Our finite buffer dCME method has been adapted by the NanoHUB.org. **Start your Finite Buffer dCME simulation** * here*.

Phage lambda is a fully parameterized model system for studying cellular fate determination. We have successfully applied the finite buffer dCME method in revealing the mechanisms controlling the efficiency and stability of the phage lambda epigenetic lysogeny-lysis switch. By solving the steady state probability landscapes, we were able to predict experimental outcomes in wild-type phage lambda and four different mutants. Our work also led to identification of the critical cooperativity interaction largely responsible for the emergence of stability of lysogenic maintenance in phage lambda. [*Cao, Lu and Liang PNAS 2010*]

Critical events that occur rarely in biological processes are of great importance, but are challenging to study using Monte Carlo simulation. We develop a practical algorithm called adaptively biased sequential importance sampling (ABSIS) method for efficiently estimating probabilities of rare events. By adopting a look-ahead strategy and by enumerating short paths from the current state, we estimate the reaction-specific and state-specific forward and backward moving probabilities of the system, which are then used to bias reaction selections. The ABSIS algorithm can automatically detect barrier-crossing regions, and can adjust bias adaptively at different steps of the sampling process, with bias determined by the outcome of exhaustively generated short paths. Regardless of the number of the reactions and the complexity of the network, ABSIS has only two bias parameters to be determined. ABSIS can accurately and efficiently estimate rare event probabilities in four testing examples, often with smaller variance than other importance sampling algorithms. The ABSIS method is general and can be applied to study rare events of other stochastic networks with complex probability landscape. [*Cao and Liang, JCP 2013*]

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