In this thesis, I discuss two main subjects coming from biology and I propose two models that mimic the behaviours of the biological networks studied. The first part of the thesis deals with intracellular transport of molecules. Proteins, RNA and, generally, any kind of cargo molecules move freely in the cytoplasm: intracellular transport as a consequence of Brownian motion is classically modelled as a diffusion process. Some specific proteins, like the tumour suppressor p53, use microtubules to facilitate their way towards the nucleus. Microtubules are a dense network of filaments that point towards the cell centre. Motor proteins bind to these filaments and move along, bearing a cargo bound to them. I propose a simplified bi-dimensional model of nucleocytoplasmic transport taking into account the kinetic processes linked to microtubule transport. Unlike in other models we know, I represented the position of a single MT filament. This model is given by a system of partial differential equations which are cast in different dimensions and connected by suitable exchange rules. A numerical scheme is introduced and several scenarios are presented and discussed to answer to the question of which proteins benefit from microtubule transport, depending on their diffusion coefficients. In the second part of the thesis, I design and analyse a physiologically based model representing the accumulation of protein p53 in the nucleus after triggering of the sentinel protein ATM by DNA damage. The p53 protein plays an essential role in the physiological maintenance of healthy tissue integrity in multicellular organisms (regulation of cell cycle arrest, repair pathways and apoptosis). Firstly, I developed a compartmental ODE model to represent the temporal dynamics of the protein. Since the p53 protein is known for its oscillatory behaviour, I performed a numerical bifurcation study to verify the existence, in the model, of stable periodic solutions. Next, I have expanded the model by the addition of a spatial variable and analysed the spatio-temporal dynamics of p53. After checking the existence of oscillations in the spatial setting, I have analysed the robustness of the system under spatial variations (diffusion and permeability coefficients, cell shape and size).
