Analytical time-dependent distributions for two common signaling systems

The chemical master equation (CME) in principle provides a method for quantifying the probabilistic behavior of any reaction network including signaling networks with time-varying reaction rates, but solving this equation is a great challenge. Here, we apply the Doi-Peliti formalism combined with both the Wei-Norman method and Lie algebra to analytically solve the CME. When this method is applied to two common signaling modules with time-varying reaction rates: a reversible binding motif and an irreversible modification motif, we successfully derive the analytical transient distribution for each motif, including the time-dependent joint and marginal distributions for all the reactive species. The stochastic simulations confirm the correctness of our analytical results. The method proposed here has broad applications in finding the time-evolutional distributions of biochemical networks with time-varying reaction rates.