BB25: Stochastic modelling of reaction-diffusion processes with size-exclusion

Project completed November 1, 2012

Background

Many important cellular processes can be described in terms of diffusing and reacting particles. When these particles are present in high concentrations, effects due to molecular crowding can have a significant impact on the thermodynamics and kinetics of biological processes, such as association reactions at membranes or in chemotaxis. These effects are called excluded-volume or steric interactions, and they arise because the particles have a fixed volume and cannot be condensed.

A common approach to modelling such reaction–diffusion processes is to use stochastic discrete models at the particle level. These describe the dynamics and interactions of each particle explicitly and are especially relevant in systems with low numbers of particles, but become computationally intractable with large numbers of particles. These models have been formulated as both off-lattice and on-lattice models. The latter assumes the motion of particles is restricted to taking place on a lattice (see Figure 1). A common issue with on-lattice models is that often they are unable to capture correctly important quantities as a result of non-physical motion and interaction rules on the lattice. This is in contrast with the more realistic off-lattice models. 

An alternative approach is to consider a continuous population-level model based on partial differential equations  (PDEs) for the population density of cells or molecules. This can be a very useful tool for systems with large numbers of particles, but the challenge is to predict the correct macroscopic description of the key attributes at the particle or microscopic level.

Excluded-volume effects can change the macroscopic diffusion coefficient and alter reaction kinetics. An important question is how the chosen microscopic model and its corresponding macroscopic counterpart describe key quantities, such as the collective and self-diffusion coefficients, for a given physical system. 

In this project, researchers at the Oxford Centre for Collaborative Applied Mathematics (OCCAM) studied the differences between on- and off-lattice models. In particular, they developed improved on-lattice models, which preserve conceptual simplicity whilst also accurately capturing the key system attributes.

Techniques and Challenges

The starting point was a model for a system of diffusive particles with size-exclusion interactions on a regular two- or three-dimensional square lattice. While the most common approach is the so-called asymmetric exclusion process (in which only one particle can fit in one lattice cell at a time),   this work investigated developing a more efficient model that does not require such a fine discretisation of space. The chosen approach was a hybrid discrete and continuous model that allows more than one particle per lattice. As the lattice site becomes more occupied, the diffusion coefficient increases. This means that a particle at the same site as other particles is more likely to diffuse away from it than if it was the only particle on the site. 

Results

The resulting master equation describes the time-evolution of the joint probability of particles on the lattice. Two strategies were considered in order to compare this lattice-based model against its continuous counterpart. Firstly, a closure condition for the stochastic variance was derived from the continuum space limit of the equation for the mean value of particles per lattice site or stochastic mean. Secondly, the master equation can be interpreted as the finite difference discretisation of a PDE. Using these ideas, the researchers showed that the discrete model is the second order discretisation of the corresponding continuum model derived in another OCCAM project (M6). This implies that a lattice can be made coarser by allowing more than one particle per lattice site. This makes the model more computationally efficient, while still capturing the key excluded-volume interactions. Moreover, this particular result is proof of the viability of integrating a population-level property concept, such as the collective diffusion coefficient, into a one-particle (or discrete) random walk on a lattice.

The Future

We will examine several on-lattice models and derive their associated macroscopic models for the population density, and test their performance by comparison with the corresponding off-lattice Brownian model. We will perform stochastic simulations of the various particle-level models and compare these with the solution of the population-level partial differential equation models. In particular, we will use Spatiocyte (a lattice-based Monte Carlo method), developed by researchers in RIKEN, Japan, as part of the E-Cell System platform.

Related Publications

[10/65] Bruna M., Chapman S.J.: Excluded-volume effects in the diffusion of hard spheres, Phys. Rev. E, vol. 85, p. 011103, 2012