Mathematical Modelling of Stem Cell Dynamics during Post-embryonic Organ Growth

Abstract

The current work is devoted to the mathematical modelling of the development of fish respiratory organs, called gills or branchiae. The model organism chosen for the task is the Japanese rice fish (Oryzias latipes), more colloquially known as medaka. Their gills are analysed in the attempt to answer three main developmental questions via mathematical modelling, with possible applications beyond the scope of this thesis. Firstly, how many stem cells are needed to build the organ? What kind of heterogeneities exist among these stem cells? And, finally, what properties and relations with each-other do these stem cells have, that give the organ its shape? Relying on experimental data from our collaborators in the group of Prof. Lazaro Centanin, Centre for Organismal Studies, Heidelberg University, we use a variety of methods to study the aforementioned aspects. These methods were selected, adapted and developed based on the goal of each project and on the available data. Thus, a combination of stochastic and deterministic techniques are employed throughout the thesis, including Gillespie-type simulations, Markov chains theory and compartmental models. The study of stem cell numbers and heterogeneities is approached via stochastic simulations extended from the algorithm of Gillespie, and further improved by Markov chains methods. Results suggest that not only very few stem cells are sufficient to build and maintain the organ but, more importantly, these stem cells are heterogeneous in their division behaviour. In particular, they rely on alternating activation and quiescence phases, such that once a stem cell has divided, it becomes activated and divides multiple times before allowing another one to take the lead. For the study of growth and shape of gills, multiple deterministic models based on different assumptions and investigating various hypotheses have been developed. All these models have a compartmental structure, with increasing number of compartments governed by indicator functions which, in turn, depend on explicit or implicit algebraic equations. For each model, the existence, uniqueness and non-negativity of solutions are proved, the analytical solutions are found and their regularity is discussed. The models are compared based on their ability to reproduce part of the data, and the best one is selected. The chosen model is then applied to further data and speculations on hypotheses supporting the model are made. Results suggest that the main stem cell types, responsible for growing the organ, slow down their proliferation in time, either due to ageing or to the lack of sufficient nutrients. The main results and strengths of this thesis consist of the high variety of models developed and methods employed, their capability to answer important biological questions and, even more, to uncover new insights on mechanisms previously unknown.