Exact first passage time distribution for nonlinear chemical reaction networks II: monomolecular reactions and a A + B - C type of second-order reaction with arbitrary initial conditions

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Abstract
In biochemical reaction networks, the first passage time (FPT) of a reaction quantifies the time it takes for the reaction to first occur, from the initial state. 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 theoretically 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 behaviour 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.
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