Analytical refractory period distribution for a class of time-variant biochemical systems with second-order reactions

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.