Individual-based models to growth and folding in one-layered tissues: Intestinal crypts and early development

In this paper we present individual-based models of the spat io-temporal dynamics in one-layered multicellular biological systems and outline how they can be adapted to experimental systems (individuum here: single cell). Mathematically these systems present a now and interesting class of problems which allow to study how to link individual-based models with continuum theory in problems with internally growing manifolds. As model systems we mainly focus on intestinal crypts since they axe paradigmatic in several regards: (i) they represent a simple, epithelial tissue, (ii) they contain only a small number of cells (N approximate to 300), (iii) much is known about them and they are fairly well accessible to experiments, (iv) perturbations in their spatio-temporal dynamics have. been found to result in folded patterns (crypt fission, polyps and adenoma) which are pro-patterns of intestinal cancer. We show how the formation of folded patterns can be explained within a single-cell based, lattice-free model. We, further illustrate, that the, folding principles are expected to carry over to other tissue layers as well as to a developing blastula or the oral mucosa, supported by computer simulations and a simple field theory.