Analisi delle biforcazioni di oscillatori genetici

Many biological processes can be described through mathematical models; bistability and oscillation of chemical reactions are fundamental characteristics of what are called biological switches and biological clocks. Some examples of these dynamics are the circadian rhythm, the cell cycle and the gene regulatory network. The thesis studies thoroughly two systems of biological interest in which two proteins are present, one of which behaves as an activator of its own repressor. The difference between the two designs is the level at which operates the repressor: transcriptional level in the first and post-transcriptional in the second. The objective is to analyze the differences between these two cases with a study of the bifurcations and trajectories of the systems and then also to see how these different characteristics are reflected on the oscillatory behavior of the two models. These types of systems can often be divided, depending on the characteristics of the solutions, into integrator or resonator: in the first case they are strictly sensitive to variations in the amount of data input, in the second case they depend on the frequency of the inputs. Neurons, for example, can exhibit both these behaviours according to the different bifurcations in the models. In the first chapter of the paper we will introduce some biological facts and see how they can be modelled through the Hill function, then we will present the concepts of the biological switch and clock using some examples. In the second chapter, mainly of a mathematical nature, we will deal with the theory that we will need later to study the two models focusing on stability, bifurcations and limit cycles. The topics just mentioned will be useful in the third chapter that develops around the two particular systems mentioned above. After showing how both models were found we proceed with the study of their behavior, also using programs like Maple and Mathematica. What is found is that the systems have a very different dynamics both in terms of bifurcations and solutions and, as in the case of neurons, the first system will behave as an integrator near a saddle-node bifurcation with the creation of limit cycle, while the second will have the characteristics of a resonator near a Hopf bifurcation.