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 frontsComment: 47 Pages, 8 Figure