This thesis extends an existing cell population modelling framework to investigate two different hypotheses for what drives the phenomenon of cell sorting, which is the spontaneous self-reorganization of cell populations. This behaviour cause cells to find their way back into their original configuration after they have been scrambled. Original tissue function may also be regained. The modelling framework is called discrete Laplacian cell mechanics (DLCM), and models cell movement on a lattice as a result of pressure differences. The first hypothesis suggests that cells exhibit type-specific adhesion properties which cause cells of the same type to adhere more to each other than to cells of a different kind. The other, more recent, hypothesis explains cell sorting behaviour as a consequence of interfacial tension, where cells of different types exhibit larger tension between them compared to cells of the same type. Adhesion is implemented as a passive force between cells of the same type, which counteract the pressure-driven events, while interfacial tension is implemented as pressure sources arising due to contact with cells of a different type. This thesis investigates whether these additions on the scale of individual cells can be sufficient to induce sorting behaviour on the cell population scale. Subsequently the suitability of implementing these effects in the DLCM framework can be evaluated. Starting from a scrambled cell configuration of two types, the results show that differential adhesion can result in the cell population sorting into smaller clusters, with the addition of Brownian motion improving the sorting ability significantly. Differential interfacial tension as it is implemented here demonstrates the effect of dissociation between cells of different type, but this is not sufficient to achieve sorting. The behaviour can be likened to a form of localized Brownian motion where more unsorted areas are prone to more movement events. Therefore, differential tension is not deemed suitable within the DLCM framework on its own. The cohesive effect of differential adhesion together with the dissociative effect of differential interfacial tension proved to work well together, acheiving a high degree of sorting both overall and compared to the case of only differential adhesion with some Brownian motion. Full separation into one distinct cell mass for each cell type present could not be achieved.

Dans cette thèse les théories spectrale et de la diffusion sur des graphes périodiques sont investigué. Le chapitre 1 présente des résultats de préservation de la nature fine du spectre pour des opérateurs de Schrödinger perturbés dans le cadre de cristaux topologiques perturbés. Le chapitre 2 étend ses résultats à des opérateurs du première ordre connu sous le nom de opérateurs de Gauss-Bonnet discrets. Finalement, le chapitre 3 présente des résultats de continuité de composantes spectrales pour des familles de opérateurs de Schrödinger magnétiques sur Z^d / In this thesis we investigate the spectral and scattering theories for crystal lattices. In chapter one we present results concerning the preservation of the nature of the spectrum for perturbed Schrödinger operators acting con perturbed topological crystals. In Chapter 2 we extend this results to some first order operators knowns as discrete Gauss-Bonnet operators. Finally, in chapter 3 we give some results dealing with the continuity of the spectrum for a family of magnetic Schrödinger operators acting on Z^d