Characterizing transport through a crowded environment with different obstacle sizes

Abstract

Transport through crowded environments is often classified as anomalous, rather than classical, Fickian diffusion. Several studies have sought to describe such transport processes using either a continuous time random walk or fractional order differential equation. For both these models the transport is characterized by a parameter α, where α = 1 is associated with Fickian diffusion and α < 1 is associated with anomalous subdiffusion. Here, we simulate a single agent migrating through a crowded environment populated by impenetrable, immobile obstacles and estimate α from mean squared displacement data. We also simulate the transport of a population of such agents through a similar crowded environment and match averaged agent density profiles to the solution of a related fractional order differential equation to obtain an alternative estimate of α. We examine the relationship between our estimate of α and the properties of the obstacle field for both a single agent and a population of agents; we show that in both cases, α decreases as the obstacle density increases, and that the rate of decrease is greater for smaller obstacles. Our work suggests that it may be inappropriate to model transport through a crowded environment using widely reported approaches including power laws to describe the mean squared displacement and fractional order differential equations to represent the averaged agent density profiles.

Modeling transport through an environment crowded by a mixture of obstacles of different shapes and sizes

“…To minimize the computational expense, we regenerate the obstacle field every R realizations [11,14]. Provided that R is sufficiently small, this has no observable effect on the averaged results [11,14]. For all results presented we always checked that the results were insensitive to the size of the lattice.…”

“…The initial location of the motile agent is randomly chosen in each realization. To minimize the computational expense, we regenerate the obstacle field every R realizations [11,14]. Provided that R is sufficiently small, this has no observable effect on the averaged results [11,14].…”

“…Many previous studies have analyzed this kind of data by assuming that the mean squared displacement (MSD) follows a power law [11][12][13][14] where D [L 2 T −α ] is a generalized diffusion coefficient, 0 < α < 2 indicates the type of transport process taking place, with α = 1 corresponding to Fickian diffusion, α < 1 corresponding to subdiffusion, α > 1 corresponding to superdiffusion [26,28], and ⟨·⟩ denotes an average over a large ensemble of simulations. In this work we focus on 0 < α ≤ 1 because transport through a crowded environment is thought to be subdiffusive [26,28], whereas transport through an uncrowded environment is known to be Fickian.…”

“…Therefore, the development of reliable mathematical models of this transport process is very important. Several previous studies have sought to describe crowded transport processes using either random walk simulation models [8][9][10][11][12][13][14][15][16][17][18] or fractional order diffusion equation (FDE) models [19][20][21][22][23][24][25][26][27][28][29]. a b Fig.…”

Spatial moment description of birth-death-movement processes incorporating the effects of crowding and obstacles

“…Cell migration in living tissues involves complicated heterogeneous environments that are occupied by various biological structures and scaffolds, including macromolecules and cells of varying size, shape and adhesive properties (Ellery et al 2014; Ellery et al 2016). Such obstacles and scaffolds can have a significant impact on the migration of cells due to the interplay between crowding and cell-to-substrate adhesion (Welch 2015; Sun and Zaman 2017; Simpson and Plank 2017).…”