  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.
