The purpose of this thesis is to develop a vertex model framework that can be used to perform computational experiments related to the dynamics of epithelial tissues in developmental systems. We focus on three example systems: the Drosophila wing imaginal disc, the Drosophila epidermis and the visceral endoderm of the mouse embryo. Within these systems, key questions pertaining to size-control mechanisms and coordination of cell migration remain unanswered and are amenable to computational testing. The vertex model presented here builds upon existing frameworks in three key ways. Firstly, we include novel force terms, representing, for example, the reaction of a cell to being compressed and its shape becoming distorted during a highly dynamic process such as cell migration. Secondly, we incorporate a model of diffusing morphogenetic growth factors within the vertex framework, using an arbitrary Lagrangian-Eulerian formulation of the diffusion equation and solving with the finite-element method (FEM). Finally, we implement the vertex model on the surface of an ellipsoid, in order to simulate cell migration in the mouse embryo. Throughout this thesis, we validate our model by running simple simulations. We demonstrate convergence properties of the FEM scheme and discuss how the time taken to solve the system scales with tissue size. The model is applied to biological systems and its utility demonstrated in several contexts. We show that when growth is dependent on morphogen concentration in the Drosophila wing disc, proliferation occurs preferentially in regions of high concentration. In the Drosophila epidermis, we show that a recently proposed mechanism of compartment size-control, in which a growth-factor is released in limited amounts, is viable. Finally, we examine the phenomenon of rosettes in the mouse embryo, which occur when five or more cells meet at a common vertex. We show, by running simulations both with and without rosettes, that they are crucial facilitators of ordered migration, and are thus critical in the patterning of the early embryo.

The centrosome is a dynamic organelle found in all animal cells that serves as a microtubule organizing center during cell division. Most of the centrosome components have been identified by genetic screens over the last decade, but little is known about how these components interact with each other to form a functional centrosome. Towards a better understanding of the molecular organization of the centrosome, we investigated the mechanism that regulates the size of the centrosome in the early C. elegans embryo. For this, we monitored fluorescently labeled centrosomes in living embryos and developed a suite of image analysis algorithms to quantify the centrosomes in the resulting 3D time-lapse images. In particular, we developed a novel algorithm involving a two-stage linking process for tracking entrosomes, which is a multi-object tracking task. This fully automated analysis pipeline enabled us to acquire time-resolved data of centrosome growth in a large number of embryos and could detect subtle phenotypes that were missed by previous assays based on manual image analysis. In a first set of experiments, we quantified centrosome size over development in wild-type embryos and made three essential observations. First, centrosome volume scales proportionately with cell volume. Second, beginning at the 4-cell stage, when cells are small, centrosome size plateaus during the cell cycle. Third, the total centrosome volume the embryo gives rise to in any one cell stage is approximately constant. Based on our observations, we propose a ‘limiting component’ model in which centrosome size is limited by the amounts of maternally derived centrosome components. In a second set of experiments, we tested our hypothesis by varying cell size, centrosome number and microtubule-mediated pulling forces. We then manipulated the amounts of several centrosomal proteins and found that the conserved centriolar and pericentriolar material protein SPD-2 is one such component that determines centrosome size.

The centrosome is a dynamic organelle found in all animal cells that serves as a microtubule organizing center during cell division. Most of the centrosome components have been identified by genetic screens over the last decade, but little is known about how these components interact with each other to form a functional centrosome. Towards a better understanding of the molecular organization of the centrosome, we investigated the mechanism that regulates the size of the centrosome in the early C. elegans embryo. For this, we monitored fluorescently labeled centrosomes in living embryos and developed a suite of image analysis algorithms to quantify the centrosomes in the resulting 3D time-lapse images. In particular, we developed a novel algorithm involving a two-stage linking process for tracking entrosomes, which is a multi-object tracking task. This fully automated analysis pipeline enabled us to acquire time-resolved data of centrosome growth in a large number of embryos and could detect subtle phenotypes that were missed by previous assays based on manual image analysis. In a first set of experiments, we quantified centrosome size over development in wild-type embryos and made three essential observations. First, centrosome volume scales proportionately with cell volume. Second, beginning at the 4-cell stage, when cells are small, centrosome size plateaus during the cell cycle. Third, the total centrosome volume the embryo gives rise to in any one cell stage is approximately constant. Based on our observations, we propose a ‘limiting component’ model in which centrosome size is limited by the amounts of maternally derived centrosome components. In a second set of experiments, we tested our hypothesis by varying cell size, centrosome number and microtubule-mediated pulling forces. We then manipulated the amounts of several centrosomal proteins and found that the conserved centriolar and pericentriolar material protein SPD-2 is one such component that determines centrosome size.

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