New mathematical modelling tools for co-culture experiments: when do we need to explicitly account for signalling molecules?

Mathematical models are often applied to describe cell migration regulated by diffusible signalling molecules. A typical feature of these models is that the spatial and temporal distribution of the signalling molecule density is reported by solving a reaction–diffusion equation. However, the spatial and temporal distributions of such signalling molecules are not often reported or observed experimentally. This leads to a mismatch between the amount of experimental data available and the complexity of the mathematical model used to simulate the experiment. To address this mismatch, we develop a discrete model of cell migration that can be used to describe a new suite of co–culture cell migration assays involving two interacting subpopulations of cells. In this model, the migration of cells from one subpopulation is regulated by the presence of signalling molecules that are secreted by the other subpopulation of cells. The spatial and temporal distribution of the signalling molecules is governed by a discrete conservation statement that is related to a reaction–diffusion equation. We simplify the model by invoking a steady state assumption for the diffusible molecules, leading to a reduced discrete model allowing us to describe how one subpopulation of cells stimulates the migration of the other subpopulation of cells without explicitly dealing with the diffusible molecules. We provide additional mathematical insight into these two stochastic models by deriving continuum limit partial differential equation descriptions of both models. To understand the conditions under which the reduced model is a good approximation of the full model, we apply both models to mimic a set of novel co–culture assays and we systematically explore how well the reduced model approximates the full model as a function of the model parameters.