Mathematical models of cell migration and self-organization in embryogenesis

In this thesis we deal with mathematical models and numerical simulations for cell migration and self-organization in embryogenesis. The part of biology which studies the formation and development of the embryo from fertilization until birth is called embryology. Morphogenesis is then the part of embryology which is concerned with the development of patterns and forms. It is well known that although morphogenesis processes are controlled at the genetic scale, genes themselves cannot create the pattern. In general a series of biological mechanisms of self-organization intervene during the early development and the formation of particular biological structures can not be anticipated solely by genetic information. This needs to be taken into account in the choice of a suitable mathematical formulation of such phenomena. Two main main topics will be investigated: we will analyze and mathematically model the self-organizing cell migration in the morphogenesis of the lateral line in the zebrafish (Danio rerio); in a second part, starting from this model, we will propose, and will study both from the analytical and the numerical point of view, a mathematical model of collective motion under only alignment and chemotaxis effects. The present thesis is organized in four chapters. In Chapter 1 we will introduce biological elements about the morphogenetic process occurring in the development of the lateral line in a zebrafish. After a first discussion on the lateral line system and on its fundamental relevance in the current scientific research, we will focus on the main mechanisms of chemical signaling and collective cell migration that will be taken into account later in our mathematical formulation of the phenomenon. In Chapter 2 we will provide a mathematical-modelling background that, starting from the morphogenesis on the chemical scale, will gradually lead us to discuss the existing mathematical models, proposed in the last years to describe collective motion in living system and in particular in the biological field. Example of numerical simulations, and their comparison with experimental evidences will be briefly shown, taken from the recent modelling literature. In Chapter 3 we will introduce a mathematical model describing the self-organizing cell migration in the zebrafish lateral line primordium. We will discuss the derivation of the model, justifying our modelling choices and comparing them with the existing literature. The proposed model will adopt a hybrid “discrete in continuous” description, where cells are treated as discrete entities moving in a continuous space, and chemical signals at molecular level are described by continuous variables. On the chemical scale we will employ diffusion and chemotaxis equations, while on the cellular scale a Newtonian second order equation for each cell will take into account typical effects arising from collective dynamics models. Cell dimension will be recovered introducing suitable detection radii and nonlocal effects. Particular steady states, corresponding to emerging structures, said neuromasts, will then be investigated and their stability will be numerically assessed. Moreover, after a description of the designed numerical approximation scheme, some dynamical simulations will be proposed to show the powerful and the limit of our approach. Finally, we will discuss the estimate of the parameters of the model, derived in part by the biological and the modelling literature, in part by the stationary model or by a numerical data fitting. In Chapter 4 we will propose a Cucker and Smale-like mathematical model of collective motion. Our hybrid model will describe a system of interacting particles under an alignment and chemotaxis effect. From an analytical point of view local and global existence and uniqueness of the solution will be proved. Furthermore, the asymptotic behaviour of the model will be investigated on a linearized form of the system. From a numerical point of view, through an approximation scheme based on finite differences, the full nonlinear system will be simulated and some significant dynamical tests will be shown. Numerical results will be compared with those analytical, and new perspectives will be proposed.