On Computational Modeling of Biological Development

This thesis considers the mathematical modeling of patterning and growth in biological systems. The fundamental objective is to build realistic, yet efficient and well-described mathematical models of morphogenesis to help understand not just the specific systems under study, but ultimately the evolutionary processes that sculpt the biological forms, from tissues to complete organs. Developmental mechanisms both guide and restrict the evolution of biological forms, and mathematical models provide a powerful tool to understand the role and significance of those restrictions. The first study considers the developmental mechanisms of evolutionary transitions in mammalian dentition. In the study, the development of mouse molars is experimentally controlled, and a smooth transition in tooth morphology is demonstrated in response to controlling a single signaling morphogen. The experimental results are plausibly recapitulated in a computational model of tooth development, which is implemented in a user-friendly interface with the aim of bridging the traditional gap between biologists doing the experiments and mathematicians building the models. The second study investigates the relation between size and spacing of the taste papillae in the mouse tongue. It is observed that experimentally controlling the FGF signaling changes the size but not the spacing of the taste papillae. Based on the results it is suggested that FGF signaling affects the extend of canonical Wnt signaling by changing the local diffusion of Wnt. Support for the hypothesis is obtained from simulations of a reaction-diffusion (Turing) model, where the activator morphogen diffusion constant is replaced with a non-linear function. In the third study, a mathematical model of tooth enamel formation is proposed. The model assumes that the process of laying down the enamel in teeth is limited by the diffusion of nutrients required by the secreting cells. Using computer simulations it is demonstrated how the model is capable of plausibly reproducing the enamel surface patterns on the top of thick enamel, resulting from the uneven surface of the underlying tooth structures. The fourth study considers the application of the Stokes equation for the simulation of tissue growth. The system is coupled with a reaction-diffusion system augmented with a differentiation mechanism to mimic the state-changes in cells, such as the establishment of signaling centers. An efficient numerical framework for solving the system in 3D is developed, and the implementation is validated using numerical tests with known solutions. To demonstrate the model dynamics, idealized tissue growth is simulated in scenarios with varying amounts of surface tension and incorporating non-homogeneous viscosity profiles.