Multi-Scale Analysis of Contact-Dependent Interaction in Tissue Aggregation and Invasion

Understanding the mechanisms that drive tissue generation and maintenance has remained a major challenge. The interplay and feedback between different spatial and temporal scales—ranging from predominantly fast molecular via intermediate cellular to comparatively slow tissue scales, necessitates the use of mathematical modeling and multi-scale analysis to disentangle the principles of the complex interdependence between the different factors. Here we review the mathematical models, methods, algorithms and tools that can be used to systems-biologically analyze problems of tissue aggregation and invasion. To frame ideas within a specific application, we focus exemplarily on methods to study and describe the role of adhesion, migration and cell sorting for the emergence and maintenance of tissue boundaries. We present major cell-based model classes, concentrating on discrete on-lattice cellular automaton models and shortly discussing continuous off-lattice models, which can be used to describe contact-dependent interactions between cells in a straightforward manner and to study their potential effects at the tissue scale by numerical simulations or theoretical analysis. We also present methods to derive continuum equations for the cell density evolution on the tissue scale in processes of tissue aggregation and invasion. Such equations are valuable since they provide the basis for applying powerful mathematical tools which facilitate comprehensive understanding.

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