Percolation Thresholds for Diffusing Particles of Nonzero Radius: Circular Obstacles in the Two-dimensional Continuum

Abstract

Lateral diffusion in the plasma membrane is obstructed by proteins bound to the cytoskeleton. The most important parameter describing diffusion in the presence of immobile obstacles is the percolation threshold, where long-range connected paths disappear and the long-range diffusion coefficient D goes to zero. To describe obstructed diffusion, it is more accurate to find the threshold directly than to extrapolate a low-density expansion in the obstacle concentration to find the concentration at which D = 0. The thresholds are well-known for point diffusing particles on various lattices or the continuum. But for particles of nonzero radius, the threshold depends on the excluded area, not just the obstacle concentration. Earlier results [Saxton, Biophys J 64 (1993) 1053] for the triangular lattice showed a very rapid decrease in the threshold as the radius of the diffusing particle increases, but a lattice model gives very low resolution. The current work finds the percolation threshold for circular obstacles in the two-dimensional continuum as a function of the radius of the diffusing species. The thresholds are obtained by a Monte Carlo method based on the Voronoi diagram for the obstacles. Each Voronoi bond is by definition the path equidistant from the nearest pair of obstacles, so the separation of that pair determines whether a diffusing particle of a given diameter can traverse that bond. For point obstacles, then, one can choose a threshold corresponding to the diameter of the diffusing particle, set the conductivity of all bonds narrower than that diameter to zero and all wider bonds to one, and test for bond percolation on the resulting Voronoi diagram. The results are used to find the thresholds for lipids and for proteins of different diameters. (Supported by NIH grant GM038133)