Lateral Diffusion and Percolation in Membranes

An algorithm based on Voronoi tessellation and percolation theory is presented to study the diffusion of model membrane components (solutes) in the plasma membrane. The membrane is modeled as a two-dimensional space with integral membrane proteins as static obstacles. The Voronoi diagram consists of vertices, which are equidistant from three matrix obstacles, joined by edges. An edge between two vertices is said to be connected if solute particles can pass directly between the two regions. The percolation threshold, $p_c$, determined using this passage criterion is $p_c\approx 0.53$. This is smaller than if the connectivity of edges were assigned randomly, in which case the percolation threshold $p_r=2/3$, where $p$ is the fraction of connected edges. Molecular dynamics simulations show that diffusion is determined by percolation of clusters of edges.

Figure 1

Simulation results for time exponents of the mean-square displacement, $W(t)$, as a function of ${\varphi }_m$. The inset shows $W(t)$ for ${\varphi }_m=0.1$, 0.2, and 0.25.

Figure 2

Voronoi diagram for a matrix realization for ${\varphi }_m=0.2$. Circles represent matrix particles. Solid lines and dashed lines are connected and disconnected edges, respectively, determined using the passage criterion.

Figure 3

Fraction of connected edges as a function of ${\varphi }_m$ for ${\sigma }_f={\sigma }_m$ (open symbols), and as a function of ${\sigma }_f$ for ${\varphi }_m=0.2$ (solid symbols). The inset shows the corresponding value of the exponent $\alpha$.

Figure 4

Mean cluster size, $S(p,L)$, as a function of $p$ for several system sizes. Solid and open symbols represent $S(p,L)$ for edge connectivities determined by the passage criterion ($p$ is varied by changing ${\varphi }_m$) and for randomly connected edges, respectively.