Anomalous tracer diffusion in the presence of extended obstacles on a triangular lattice

Proteins diffuse to their sites of action within cells in a crowded, strongly interacting environment of nucleic acids and other macromolecules. An interesting question is how the highly crowded environment of biological cells affects the dynamic properties of passively diffusing particles. The Lorentz model is a generic model covering many of the aspects of transport in a heterogeneous environment. We investigate biologically relevant situations of immobile obstacles of various shapes and sizes. The Monte Carlo simulations for the diffusion of a tracer particle are carried out on a two-dimensional triangular lattice. Obstacles are represented by non-overlapping lattice shapes that are randomly placed on the lattice. Our simulation results indicate that the mean-square displacement displays anomalous transport for all obstacle shapes, which extends to infinite times at the corresponding percolation thresholds. In the vicinity of this critical density the diffusion coefficient vanishes according to a power law, with the same conductivity exponent for all obstacle shapes. At the fixed density of obstacles, we observe that the diffusion coefficient is higher for the smaller obstacles if the object size is defined as the highest projection of the object on one of the six directions on the triangular lattice. The dynamic exponent, which describes the anomalous transport at the critical density, is the same for all the obstacle shapes. Here we show that the values of critical exponents estimated for all disordered environments do not depend on the microscopic details of the present model, such as obstacle shape, and agree with the predicted values for the underlying percolation problem. We also provide the evidence for a divergent non-Gaussian parameter close to the percolation transition for all obstacle shapes.