DYNAMICAL FEEDBACK MODELS IN CELLULAR BIOPHYSICS

In the present PhD thesis we collected two distinct contributions to the field of mathematical modelling of complex biological phenomena (at the cellular and subcellular level), with the tools of differential equations and spatial stochastic simulations. In the first part we develop a method to determine whether oscillatory temporal signals are a common feature of genetic regulatory networks or they require a fine-tuning of the coupling parameters between nodes of the networks. Modelling a two-nodes genetic network with a system of coupled delay differential equations, we performed a Monte Carlo sampling of the space of parameters of the system, biasing the search in favor of highly oscillating solutions. Estimating with thermodynamic techniques the fraction of the parameters' space associated with oscillations, we conclude that oscillations are indeed a rare feature of these biological control systems. Their dependence on the values of the parameters has been analysed, revealing some simple patterns. In the second part we propose a simple theoretical model for the dynamics of cancer cells subpopulations as observed in experiments in vitro, where a populations of melanoma cells reacts to the depletion of most of its cancer stem cells (a small subpopulation with supposed stem-cells properties and tumorigenic potential) with a large overshoot in the fraction of CSC before returning to homeostatic values. Our model, consisting in a system of delay differential equations, is able to reproduce quite well experimental data and to provide a clear picture of the cellular and molecular details of the control mechanisms, based on experimental evidence and on the emerging paradigm of phenotypic plasticity of cancer cells. A multiscale hybrid-continuum stochastic spatial model on a 2D lattice is also developed to investigate the spatial distribution of cells within the growing tumour, to be verified in immunohistochemistry experiments, with particular regard to CSCs clusterization.