Particle-environment interactions in arbitrary dimensions: A unifying analytic framework to model diffusion with inert spatial heterogeneities

Seeralan Sarvaharman*, Luca Giuggioli*

*Corresponding author for this work

Research output: Contribution to journalArticle (Academic Journal)peer-review

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Inert interactions between randomly moving entities and spatial disorder play a crucial role in quantifying the diffusive properties of a system, with examples ranging from molecules advancing along dendritic spines to antipredator displacements of animals due to sparse vegetation. Despite the ubiquity of such phenomena, a general framework to model the movement explicitly in the presence of spatial heterogeneities is missing. Here, we tackle this challenge and develop an analytic theory to model inert particle-environment interactions in domains of arbitrary shape and dimensions. We use a discrete space formulation, which allows us to model the interactions between an agent and the environment as perturbed dynamics between lattice sites. Interactions from spatial disorder, such as impenetrable and permeable obstacles or regions of increased or decreased diffusivity, as well as many others, can be modelled using our framework. We provide exact expressions for the generating function of the occupation probability of the diffusing particle and related transport quantities such as first-passage, return, and exit probabilities and their respective means. We uncover a surprising property, the disorder indifference phenomenon of the mean first-passage time in the presence of a permeable barrier in quasi-1D systems. We demonstrate the widespread applicability of our formalism by considering three examples that span across scales and disciplines. (1) We explore an enhancement strategy of transdermal drug delivery. (2) We represent the movement decisions of an animal undergoing thigomotaxis, the tendency to remain at the peripheries of its enclosure, using a spatially disordered environment. (3) We illustrate the use of spatial heterogeneities to model inert interactions between particles by modeling the search for a promoter region on the DNA by transcription factors during gene transcription.
Original languageEnglish
Article number043281
Number of pages28
JournalPhysical Review Research
Publication statusPublished - 22 Dec 2023

Bibliographical note

Funding Information: S.S. and L.G. acknowledge funding from, respectively the Biotechnology and Biological Sciences Research Council (BBSRC) Grant No. BB/T012196 and the Engineering and Physical Sciences Research Council (EPSRC) Grant No. S108151-11. S.S. and L.G. also thank the Isaac Newton Institute for Mathematical Sciences for support and hospitality during the programme Mathematics of Movement an interdisciplinary approach to mutual challenges in animal ecology and cell biology, when part of the work on this paper was undertaken, supported by the EPSRC Grant No. EP/R014604/1.

Structured keywords

  • Engineering Mathematics Research Group


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