Mathematical models for cellular aggregation: the chemotactic instability and clustering formation

In this thesis we present a mathematical formulation of the interaction between microorganisms such as bacteria or amoebae and chemicals, often produced by the organisms themselves. This interaction is called chemotaxis and leads to cellular aggregation. We derive some models to describe chemotaxis. The first is the pioneristic Keller-Segel parabolic-parabolic model and it is derived by two different frameworks: a macroscopic perspective and a microscopic perspective, in which we start with a stochastic differential equation and we perform a mean-field approximation. This parabolic model may be generalized by the introduction of a degenerate diffusion parameter, which depends on the density itself via a power law. Then we derive a model for chemotaxis based on Cattaneo's law of heat propagation with finite speed, which is a hyperbolic model. The last model proposed here is a hydrodynamic model, which takes into account the inertia of the system by a friction force. In the limit of strong friction, the model reduces to the parabolic model, whereas in the limit of weak friction, we recover a hyperbolic model. Finally, we analyze the instability condition, which is the condition that leads to aggregation, and we describe the different kinds of aggregates we may obtain: the parabolic models lead to clusters or peaks whereas the hyperbolic models lead to the formation of network patterns or filaments. Moreover, we discuss the analogy between bacterial colonies and self gravitating systems by comparing the chemotactic collapse and the gravitational collapse (Jeans instability).

Mathematical modelling of integrin-like receptors systems

Nel presente lavoro, ho studiato e trovato le soluzioni esatte di un modello matematico applicato ai recettori cellulari della famiglia delle integrine. Nel modello le integrine sono considerate come un sistema a due livelli, attivo e non attivo. Quando le integrine si trovano nello stato inattivo possono diffondere nella membrana, mentre quando si trovano nello stato attivo risultano cristallizzate nella membrana, incapaci di diffondere. La variazione di concentrazione nella superficie cellulare di una sostanza chiamata attivatore dà luogo all’attivazione delle integrine. Inoltre, questi eterodimeri possono legare una molecola inibitrice con funzioni di controllo e regolazione, che chiameremo v, la quale, legandosi al recettore, fa aumentare la produzione della sostanza attizzatrice, che chiameremo u. In questo modo si innesca un meccanismo di retroazione positiva. L’inibitore v regola il meccanismo di produzione di u, ed assume, pertanto, il ruolo di modulatore. Infatti, grazie a questo sistema di fine regolazione il meccanismo di feedback positivo è in grado di autolimitarsi. Si costruisce poi un modello di equazioni differenziali partendo dalle semplici reazioni chimiche coinvolte. Una volta che il sistema di equazioni è impostato, si possono desumere le soluzioni per le concentrazioni dell’inibitore e dell’attivatore per un caso particolare dei parametri. Infine, si può eseguire un test per vedere cosa predice il modello in termini di integrine. Per farlo, ho utilizzato un’attivazione del tipo funzione gradino e l’ho inserita nel sistema, valutando la dinamica dei recettori. Si ottiene in questo modo un risultato in accordo con le previsioni: le integrine legate si trovano soprattutto ai limiti della zona attivata, mentre le integrine libere vengono a mancare nella zona attivata.

Structure learning of graphical models for task-oriented robot grasping

In the collective imaginaries a robot is a human like machine as any androids in science fiction. However the type of robots that you will encounter most frequently are machinery that do work that is too dangerous, boring or onerous. Most of the robots in the world are of this type. They can be found in auto, medical, manufacturing and space industries. Therefore a robot is a system that contains sensors, control systems, manipulators, power supplies and software all working together to perform a task. The development and use of such a system is an active area of research and one of the main problems is the development of interaction skills with the surrounding environment, which include the ability to grasp objects. To perform this task the robot needs to sense the environment and acquire the object informations, physical attributes that may influence a grasp. Humans can solve this grasping problem easily due to their past experiences, that is why many researchers are approaching it from a machine learning perspective finding grasp of an object using information of already known objects. But humans can select the best grasp amongst a vast repertoire not only considering the physical attributes of the object to grasp but even to obtain a certain effect. This is why in our case the study in the area of robot manipulation is focused on grasping and integrating symbolic tasks with data gained through sensors. The learning model is based on Bayesian Network to encode the statistical dependencies between the data collected by the sensors and the symbolic task. This data representation has several advantages. It allows to take into account the uncertainty of the real world, allowing to deal with sensor noise, encodes notion of causality and provides an unified network for learning. Since the network is actually implemented and based on the human expert knowledge, it is very interesting to implement an automated method to learn the structure as in the future more tasks and object features can be introduced and a complex network design based only on human expert knowledge can become unreliable. Since structure learning algorithms presents some weaknesses, the goal of this thesis is to analyze real data used in the network modeled by the human expert, implement a feasible structure learning approach and compare the results with the network designed by the expert in order to possibly enhance it.