Exact first-passage time distributions from time-dependent solutions of the chemical master equation. I. Nonlinear networks with bimolecular reactions and Poisson-product initial conditions

Abstract

The first passage time (FPT) is a generic measure that quantifies when a random quantity reaches a specific state. We consider the FTP distribution in nonlinear stochastic biochemical networks, where obtaining exact solutions of the distribution is challenging because it requires time-dependent solutions of the chemical master equation (CME). Even simple two-particle collisions lead to strong nonlinearities that hinder obtaining such solutions and, hence, the full FPT distribution. Previous research has either focused on analyzing the mean FPT, which provides limited information about a system, or has considered time-consuming stochastic simulations that do not clearly expose causal relationships between parameters and dynamics. This paper presents the first exact solution of the full FPT distribution in a broad class of chemical reaction networks involving A + B → C type of second-order reactions. We obtained this distribution by deriving a closed-form, time-dependent solution of the system’s underlying CME. Approximate analytical solutions are unable to match the exact results of our method. Examples indicate that the approximations can deviate from the exact FPT distribution by more than 100% in both the mean and the variance. Furthermore, our exact method outperforms stochastic simulations in terms of computational efficiency. Given the prevalence of bimolecular reactions in biochemical systems, our approach has the potential to enhance the understanding of real-world biochemical processes.