Changqian Rao

In biochemical reaction networks, the first passage time (FPT) of a reaction quantifies the time, from the initial state, that it takes for the reaction to first occur. While the mean FPT historically served as a summary metric, a far more comprehensive characterization of the dynamics of the network is contained within the complete FPT distribution. The relatively uncommon theoretical treatments of the FPT distribution that have been given in the past have been confined to linear systems with zero- and first-order processes. Recently, we presented mathematically exact solutions for the FPT distribution within nonlinear systems involving two-particle collisions, such as A + B → C. Although this research yielded invaluable results, it was based upon the assumption of initial conditions in the form of a Poisson distribution. This somewhat restricts its relevance to real-world biochemical systems, which frequently display intricate behavior and initial conditions that are non-Poisson in nature. Our current study extends prior analyses to accommodate arbitrary initial conditions, thereby expanding the applicability of our theoretical framework and providing a more adaptable tool for capturing the dynamics of biochemical reaction networks.

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.

Refractory period (RP), the waiting time between signals, can induce complex signaling dynamics, such as acceleration, adaptation, and oscillation, within many cellular biochemical networks. However, its underlying molecular mechanisms are still unclear. Rigorously estimating the RP distribution may be essential to identify its causal regulatory mechanisms. Traditional methods of estimating the RP distribution depend on solving the underlying Chemical Master Equations (CMEs), the dominant modeling formalism of biochemical systems. However, exact solutions of the CME are only known for simple reaction systems with zero- and first-order reactions or specific systems with second-order reactions. General solutions still need to be derived for systems with bimolecular reactions. It is even more challenging if large state-space and nonconstant reaction rates are involved. Here, we developed a direct method to gain the analytical RP distribution for a class of second-order reaction systems with nonconstant reaction rates and large state space. Instead of using the CME, we used an equivalent path-wise representation, which is the solution to a transformed martingale problem of the CME. This allowed us to bypass solving a CME. We then applied the method to derive the analytical RP distribution of a real complex biochemical network with second-order reactions, the Drosophila phototransduction cascade. Our approach provides an alternative to the CMEs in deriving the analytical RP distributions of a class of second-order reaction systems. Since the bimolecular reactions are common in biological systems, our approach could enhance understanding real-world biochemical processes.