Multi-Finite Buffer Method for Direct Solution of Discrete Chemical Master Equation

Abstract

Many biological reaction networks are intrinsically stochastic due to random thermal fluctuations. Stochasticity is significant when the copy number of participating molecular species are small. The discrete Chemical Master Equation (dCME) provides a general framework to study the underlying stochastic processes of biological networks. Although the direct solution of dCME is advantageous over approximation methods such as the Langevin and the Fokker-Planck equations, it is challenging to obtain exact solution to the dCME. The Finite Buffer dCME Method provides an optimal algorithm to enumerate the underlying state space, and has been used to compute the exact solutions of dCME for several problems. In this study, we extend the finite buffer method by introducing multiple buffer queues for more effective construction of the state space and for quantitative control of errors, when buffer sizes are limited. By introducing the concept of open Independent Birth-Death (IBD) units, which are non-intersecting sets of reactions grouped by common synthesis and degradation reactions, we can enumerate the state space optimally and assess errors for each open IBD from the probability of buffer depletion, when the buffer size is limited. We also describe theoretical estimation of the error bound for any given buffer size of an IBD, so its buffer size can be optimized. We demonstrate the effectiveness of our approach in computing time-evolving and steady state probability landscapes, as well as first passage time distribution using the birth-death process, the bistable Schlogl model, the bistable toggle switch model, and the phage lambda lysogenic-lytic switching model as examples. We also compare our results with those using other methods.