Coarse graining in simulated cell populations

The main mechanisms that control the organization of multicellular tissues are still largely open. A commonly used tool to study basic control mechanisms are in vitro experiments in which the growth conditions can be widely varied. However, even in vitro experiments are not free from unknown or uncontrolled influences. One reason why mathematical models become more and more a popular complementary tool to experiments is that they permit the study of hypotheses free from unknown or uncontrolled influences that occur in experiments. Many model types have been considered so far to model multicellular organization ranging from detailed individual-cell based models with explicit representations of the cell shape to cellular automata models with no representation of cell shape, and continuum models, which consider a local density averaged over many individual cells. However, how the different model description may be linked, and, how a description on a coarser level may be constructed based on the knowledge of the finer, microscopic level, is still largely unknown. Here, we consider the example of monolayer growth in vitro to illustrate how, in a multi-step process starting from a single-cell based off-lattice-model that subsumes the information on the sub-cellular scale by characteristic cell-biophysical and cell-kinetic properties, a cellular automaton may be constructed whose rules have been chosen based on the findings in the off-lattice model. Finally, we use the cellular automaton model as a starting point to construct a multivariate master equation from a compartment approach from which a continuum model can be derived by a systematic coarse-graining procedure. We find that the resulting continuum equation largely captures the growth behavior of the CA model. The development of our models is guided by experimental observations on growing monolayers.