Diffusion-Controlled Reactions in Crowded Environments
1. Introduction
Diffusion-limited reactions are commonly found in biochemical processes, such as enzyme catalysis, protein aggregation or complexation in cells. The simplest model of diffusion-limited encounter has been introduced by Smoluchowski a century ago under the hypothesis of infinite dilution and chemically isotropic spherical reactants. The Smoluchowski framework has been widely used to describe reactions occurring in vivo. However, biological environments are crowded: the cells cytoplasm, for instance, contains a large number of proteins, nucleic acids, and other smaller molecules that can occupy up to 30÷40 % of the available volume. Thus, crowding effects are expected to impact considerably on the thermodynamics and kinetics of biological processes occurring within the cell.
2. Results
We propose a first extension of the Smoluchowski framework that incorporates excluded-volume effects, adapting Event-Driven Brownian Dynamics to the particular configuration of an absorbing sink located at the center of a spherical bounding box and substrate spherical particles that diffuse around and get absorbed. The simulation scheme is designed to enforce constant flux toward the sink through particle injection at the spherical boundary. For large absorbers, the density dependence of the encounter rate obtained from the simulations can be explained in terms of a finite-pressure correction to the Smoluchowski value. However, reducing the sink-to-particle size ratio, the encounter is substantially depressed at intermediate packing until it becomes non-monotonic. Concomitantly, a peculiar ordering of the diffusing particles occurs in the vicinity of the absorbing center, with the appearance of stationary density waves. A mean-field analysis confirms that these pehenomena are the result of the increasing competition among the diffusing particles when approaching a sink of smaller size. Finally, we show how the addition of an infinitesimal amount of non absorbing impurities can also slow down dramatically the reaction.