{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/discrete-and-continuous-mathematical-models","title":"Discrete and continuous mathematical models of sharp-fronted collective cell migration and invasion","arxiv_id":"2310.07938","date":"2023-10-11","proceeding":null,"authors":["Matthew J Simpson","Keeley M Murphy","Scott W McCue","Pascal R Buenzli"],"abstract":"Mathematical models describing the spatial spreading and invasion of populations of biological cells are often developed in a continuum modelling framework using reaction-diffusion equations. While continuum models based on linear diffusion are routinely employed and known to capture key experimental observations, linear diffusion fails to predict well-defined sharp fronts that are often observed experimentally. This observation has motivated the use of nonlinear degenerate diffusion, however these nonlinear models and the associated parameters lack a clear biological motivation and interpretation. Here we take a different approach by developing a stochastic discrete lattice-based model incorporating biologically-inspired mechanisms and then deriving the reaction-diffusion continuum limit. Inspired by experimental observations, agents in the simulation deposit extracellular material, that we call a substrate, locally onto the lattice, and the motility of agents is taken to be proportional to the substrate density. Discrete simulations that mimic a two--dimensional circular barrier assay illustrate how the discrete model supports both smooth and sharp-fronted density profiles depending on the rate of substrate deposition. Coarse-graining the discrete model leads to a novel partial differential equation (PDE) model whose solution accurately approximates averaged data from the discrete model. The new discrete model and PDE approximation provides a simple, biologically motivated framework for modelling the spreading, growth and invasion of cell populations with well-defined sharp fronts","url_abs":"https://arxiv.org/abs/2310.07938v3","url_pdf":"https://arxiv.org/pdf/2310.07938v3.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"discrete-and-continuous-mathematical-models","repo_url":"https://github.com/profmjsimpson/discretesubstrate","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}