Monte Carlo simulations of enzymatic reactions in crowded media. Effect of the enzyme-obstacle relative size

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We perform Monte Carlo simulations in three-dimensional (3D) lattice in order to study diffusion-controlled and mixed activation-diffusion reactions following an irreversible Michaelis–Menten scheme in crowded media. The simulation data reveal the rate coefficient dependence on time for diffusion-controlled bimolecular reactions developing in three-dimensional media with obstacles, as predicted by fractal kinetics approach. For the cases of mixed activation-diffusion reactions, the fractality of the reaction decreases as the activation control increases. We propose a modified form of the Zipf–Mandelbrot equation to describe the time dependence of the rate coefficient, k(t)=k0(1+t/τ)−h. This equation provides a good description of the fractal regime and it may be split into two terms: one that corresponds to the initial rate constant (k0) and the other one correlated with the kinetics fractality. Additionally, the proposed equation contains and links two limit expressions corresponding to short and large periods of time: k1=k0 (for t≪τ) that relates to classical kinetics and the well-known Kopelman’s equation k∼t−h (for t≫τ) associated to fractal kinetics. The τ parameter has the meaning of a crossover time between these two limiting behaviours. The value of k0 is mainly dependent on the excluded volume and the enzyme-obstacle relative size. This dependence can be explained in terms of the radius of an average confined volume that every enzyme molecule feels, and correlates very well with the crossover length obtained in previous studies of enzyme diffusion in crowding media.