Probing microscopic origins of confined subdiffusion by first-passage observables

Abstract

Subdiffusive motion of tracer particles in complex crowded environments, such as biological cells, has been shown to be widespread. This deviation from Brownian motion is usually characterized by a sublinear time dependence of the mean square displacement (MSD). However, subdiffusive behavior can stem from different microscopic scenarios that cannot be identified solely by the MSD data. In this article we present a theoretical framework that permits the analytical calculation of first-passage observables (mean first-passage times, splitting probabilities, and occupation times distributions) in disordered media in any dimensions. This analysis is applied to two representative microscopic models of subdiffusion: continuous-time random walks with heavy tailed waiting times and diffusion on fractals. Our results show that first-passage observables provide tools to unambiguously discriminate between the two possible microscopic scenarios of subdiffusion. Moreover, we suggest experiments based on first-passage observables that could help in determining the origin of subdiffusion in complex media, such as living cells, and discuss the implications of anomalous transport to reaction kinetics in cells.

Fig. 1.

Two scenarios of subdiffusion for a tracer particle in crowded environments. (a) Random walk in a dynamic crowded environment. The tracer particle evolves in a cage whose typical life time diverges with density. This situation can be modeled by a CTRW with power-law distributed waiting times. (b) Random walk with static obstacles. This situation can be modeled by a random walk on a percolation cluster.

Fig. 2.

Numerical simulation of first-passage observables for random walks on three-dimensional percolation clusters. All of the embedding domains have reflecting boundary conditions. (a) MFPT for random walks on 3-dimensional critical percolation clusters. For each size of the confining domain, the MFPT, normalized by the number of sites N, is averaged both over the different target and starting points separated by the corresponding chemical distance, and over percolation clusters. The black plain curve corresponds to the prediction of Eq. 15 with d_w^c − d_c^f ≃ 1. (b) Splitting probability for random walks on 3-dimensional critical percolation clusters. The splitting probability P₁ to reach the target T₁ before the target T₂ is averaged both over the different target points T₂ and over the percolation clusters. The chemical distance ST₁ = 10 is fixed whereas the chemical distance ST₂ = T₁T₂ is varied. The black plain curve corresponds to the explicit theoretical expression 14 with d_w^c − d_f^c ≃ 1. (c) Occupation time for random walks on critical percolation clusters. For each size of confining domain, the occupation time of site T₁ before the target T₂ is reached for the first time is averaged over the different target points T₂ and over the percolation clusters. The chemical distance ST₁ = 10 is fixed whereas the chemical distance ST₂ = T₁T₂ is varied. The black plain curve corresponds to the prediction of Eq. 15 with d_w^c − d_f^c ≃ 1. (d) The MFPT for random walks on percolation clusters above criticality for a 25 × 25 × 25 confining domain. The MFPT, normalized by the number of sites N, is averaged both over the different target and starting points separated by the corresponding chemical distance, and over the percolation clusters.

Fig. 3.

Schematic proposed set-up to measure first-passage observables.