Truncating the Time-evolving Discrete Chemical Master Equation for Stochastic Biological Networks

Abstract

The discrete Chemical Master Equation (dCME) provides a rigorous framework for modeling stochastic biochemical reaction networks but suffers from an exponentially growing state space, making its direct solution intractable. Traditional truncation methods rely on trial-and-error approaches or finite state projections, often introducing inaccuracies by arbitrarily selecting state space boundaries. In this study, we present a truncation strategy for the exact solution of the time-evolving dCME. We provide a systematic and mathematically grounded framework to truncate the time-evolving dCME by leveraging transient solutions of independent birth-death processes. The theoretically estimated truncation bounds provide a conservative limit on computed errors, eliminating the need for heuristic state space selection. The framework also accounts for both upper and lower truncation boundaries, ensuring a conservative yet computationally efficient truncation strategy. These together enable the a priori determination of truncation points at each time step while maintaining a predefined probability error tolerance level. We apply this approach to diverse biochemical networks, including a single-gene expression model, an activation-inhibition feedback system, a genetic toggle switch, a feedforward genetic loop, and an enzymatic cascade network. The results demonstrate an efficient computation of time-evolving probability landscapes while preserving key stochastic properties such as multimodality. The present approach thus offers a significant advancement in solving the dCME by systematically balancing accuracy and computational efficiency. While previous truncation strategies were developed for the steady state, we introduce a theoretically grounded, time-dependent truncation framework that avoids relying on steady-state limits and instead computes probabilities only for states relevant at each time point.