Two-Dimensional Continuum Percolation Threshold as a Function of the Radius of the Diffusing Particles

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 conducting paths disappear and the long-range diffusion coefficient therefore goes to zero. The thresholds are well-known for point diffusing particles on various lattices or the continuum. But for diffusing particles of nonzero radius, the threshold depends on the excluded area, not just the obstacle concentration. For the triangular lattice, the threshold is known to be highly sensitive to the size of the diffusing particle [Saxton, Biophys J 64 (1993) 1053], but lattice calculations give very low resolution. Here high-resolution results are obtained for circular obstacles on the continuum. Random obstacle configurations are generated by Brownian dynamics or Monte Carlo methods, and tested for percolation by examining bond percolation on the Voronoi diagram of the obstacles. The percolation threshold is expressed as the diameter of the largest diffusing particle that can cross a set of obstacles at a prescribed number density. For the simplest case, random overlapping obstacles, the analytical solution is known and the Monte Carlo results agree with it quantitatively. When the obstacles are disks with a 1/r⊥12 repulsion, the percolating diameter is around 10% lower than for overlapping obstacles. Disks with a 1/r⊥6 or 1/r⊥18 repulsion behave similarly. The results are used to find the thresholds for lipids, and for proteins of various diameters. (Supported by NIH grant GM038133)