Particle-based simulation of non-elementary bimolecular kinetics

Particle-based simulations are an essential tool for the study of biochemical systems for scales between molecular/Brownian dynamics and the reaction-diffusion master equation. These simulations utilise proximity-based reaction conditions and are typically limited to elementary (mass-action) kinetics. We present a novel framework for directly simulating non-elementary bimolecular kinetics in a particle-based framework. By mimicking the behaviour of a third implicit reactant, we adapt non-elementary reaction conditions, previously restricted to trimolecular chemical interactions, to biomolecular reactions for the first time. We implement our approach in an event-driven simulation, which we validate by reproducing Michaelis-Menten kinetics. We then demonstrate its utility by simulating the classical Goldbeter model of circadian oscillations completely at the level of individual molecules. This model features multiple non-elementary reactions and requires the incorporation of several existing simulation techniques. Our method accurately reproduces the target non-elementary kinetics, without simulating the implied underlying fast elementary reactions, thereby significantly reducing the computational cost. This work expands the class of reaction networks accessible to particle-based simulations and provides a practical alternative to explicitly simulating all elementary steps in systems where quasi-steady-state approximations are applicable.