We study a class of Stochastic Differential Equations (SDEs) with jumps modeling multistage Michaelis--Menten enzyme kinetics, in which a substrate is sequentially transformed into a product via a cascade of intermediate complexes. These networks are typically high dimensional and exhibit multiscale behavior with strong coupling between different components, posing substantial analytical and computational challenges. In particular, the problem of statistical inference of reaction rates is significantly difficult, and becomes even more intricate when direct observations of system states are unavailable and only a random sample of product formation times is observed. We address this in two stages. First, in a suitable scaling regime consistent with the Quasi-Steady State Approximation (QSSA), we rigorously establish two asymptotic results: (i) a stochastic averaging principle yielding a reduced model for the product--substrate dynamics; and (ii) a Functional Central Limit Theorem (FCLT) characterizing the associated fluctuations. Guided by the reduced-order dynamics, we next construct a novel Interacting Particle System (IPS) that approximates the product-substrate process at the particle level. This IPS plays a pivotal role in the inference methodology; in particular, we establish a propagation of chaos result that mathematically justifies an approximate product-form likelihood based solely on a random sample of product formation times, without requiring access to the system states. Numerical examples are presented to demonstrate the accuracy and applicability of the theoretical results.

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