Reconciling transport models across scales: The role of volume exclusion

Diffusive transport is a universal phenomenon, throughout both biological and physical sciences, and models of diffusion are routinely used to interrogate diffusion-driven processes. However, most models neglect to take into account the role of volume exclusion, which can significantly alter diffusive transport, particularly within biological systems where the diffusing particles might occupy a significant fraction of the available space. In this work we use a random walk approach to provide a means to reconcile models that incorporate crowding effects on different spatial scales. Our work demonstrates that coarse-grained models incorporating simplified descriptions of excluded volume can be used in many circumstances, but that care must be taken in pushing the coarse-graining process too far.

Figure 1

Representation of particle positions within compartments of different capacity. Shaded cells represent particles, and white cells represent unused capacity. As $m$ increases, the spatial resolution coarsens, and this work reconciles the descriptions at these different scales.

Figure 2

Schematic of the interpolation process for $m=4$.

Figure 3

Comparing the (a) mean and (b) variance of particle numbers for the $m=1$ and $m=8$ cases at $t=1$, with $K=128/m$. Dark gray bars: Results from simulation of the random walk model with $m=1$. Light gray bars: Results from the numerical solution of Eq. (6)/Eq. (7) with $m=8$. Black dashed line in (a): Solution of the limiting diffusion equation.

Figure 4

HDEs for the (a) means and (b) variances in particle numbers as $m$ is varied [22].