Enzyme kinetics simulation at the scale of individual particles

Enzyme-catalyzed reactions involve two distinct timescales: a short timescale on which enzymes bind to substrate molecules to produce bound complexes and a comparatively long timescale on which the molecules of the complex are transformed into products. The uptake of the substrate in these reactions is the rate at which the product is made on the long timescale. Models often only consider the uptake to reduce the number of chemical species that need to be modeled and to avoid explicitly treating multiple timescales. Typically, the uptake rates cannot be described by mass action kinetics and are traditionally derived by applying singular perturbation theory to the system's governing differential equations. This analysis ignores short timescales by assuming that a pseudo-equilibrium between the enzyme and the enzyme-bound complex is maintained at all times. This assumption cannot be incorporated into current particle-based simulations of reaction-diffusion systems because they utilize proximity-based conditions to govern the instances of reactions that cannot maintain this pseudo-equilibrium for infinitely fast reactions. Instead, these methods must directly simulate the dynamics on the short timescale to accurately model the system. Due to the disparate timescales, such simulations require excessive amounts of computational time before the behavior on the long timescale can be observed. To resolve this problem, we use singular perturbation theory to develop a proximity-based reaction condition that enables us to ignore all fast reactions and directly reproduce non-mass action kinetics at long timescales. To demonstrate our approach, we implement simulations of a specific third order reaction with kinetics reminiscent of the prototypical Michaelis-Menten system.