Modelling adhesion in stochastic and mean-field models of cell migration

Adhesion between cells plays an important role in many biological processes such as tissue morphogenesis and homeostasis, wound healing and cancer cell metastasis. From a mathematical perspective, adhesion between multiple cell types has been previously analysed using discrete and continuum models including the Cellular Potts models and partial differential equations (PDEs). While these models can represent certain biological situations well, Cellular Potts models can be computationally expensive and continuum models only capture the macroscopic behaviour of a population of cells, ignoring stochasticity and the discrete nature of cell dynamics. Cellular automaton models allow us to address these problems and can be used for a wide variety of biological systems. In this paper, we consider a cellular automaton approach and develop an on-lattice agent-based model (ABM) for cell migration and adhesion in a population composed of two cell types. By deriving and comparing the corresponding PDEs to the ABM, we demonstrate that cell aggregation and cell sorting are not possible in the PDE model. Therefore, we propose a set of stochastic mean equations (SMEs) which better capture the behaviour of the ABM in one and two dimensions.