Bistable elliptic equations with fractional diffusion

This work concerns the study of bounded solutions to elliptic nonlinear equations with fractional diffusion. More precisely, the aim of this thesis is to investigate some open questions related to a conjecture of De Giorgi about the one-dimensional symmetry of bounded monotone solutions in all space, at least up to dimension 8. This property on 1-D symmetry of monotone solutions for fractional equations was known in dimension n=2. The question remained open for n>2. In this work we establish new sharp energy estimates and one-dimensional symmetry property in dimension 3 for certain solutions of fractional equations. Moreover we study a particular type of solutions, called saddle-shaped solutions, which are the candidates to be global minimizers not one-dimensional in dimensions bigger or equal than 8. This is an open problem and it is expected to be true from the classical theory of minimal surfaces.

Stochastic models and dynamic measures for the characterization of bistable circuits

During the last few years, a great deal of interest has risen concerning the applications of stochastic methods to several biochemical and biological phenomena. Phenomena like gene expression, cellular memory, bet-hedging strategy in bacterial growth and many others, cannot be described by continuous stochastic models due to their intrinsic discreteness and randomness. In this thesis I have used the Chemical Master Equation (CME) technique to modelize some feedback cycles and analyzing their properties, including experimental data. In the first part of this work, the effect of stochastic stability is discussed on a toy model of the genetic switch that triggers the cellular division, which malfunctioning is known to be one of the hallmarks of cancer. The second system I have worked on is the so-called futile cycle, a closed cycle of two enzymatic reactions that adds and removes a chemical compound, called phosphate group, to a specific substrate. I have thus investigated how adding noise to the enzyme (that is usually in the order of few hundred molecules) modifies the probability of observing a specific number of phosphorylated substrate molecules, and confirmed theoretical predictions with numerical simulations. In the third part the results of the study of a chain of multiple phosphorylation-dephosphorylation cycles will be presented. We will discuss an approximation method for the exact solution in the bidimensional case and the relationship that this method has with the thermodynamic properties of the system, which is an open system far from equilibrium.In the last section the agreement between the theoretical prediction of the total protein quantity in a mouse cells population and the observed quantity will be shown, measured via fluorescence microscopy.

Multicomponent nanodevices based on molecular and nanocrystal moieties

Nanoscience is an emerging and fast-growing field of science with the aim of manipulating nanometric objects with dimension below 100 nm. Top down approach is currently used to build these type of architectures (e.g microchips). The miniaturization process cannot proceed indefinitely due to physical and technical limitations. Those limits are focusing the interest on the bottom-up approach and construction of nano-objects starting from “nano-bricks” like atoms, molecules or nanocrystals. Unlike atoms, molecules can be “fully programmable” and represent the best choice to build up nanostructures. In the past twenty years many examples of functional nano-devices able to perform simple actions have been reported. Nanocrystals which are often considered simply nanostructured materials, can be active part in the development of those nano-devices, in combination with functional molecules. The object of this dissertation is the photophysical and photochemical investigation of nano-objects bearing molecules and semiconductor nanocrystals (QDs) as components. The first part focuses on the characterization of a bistable rotaxane. This study, in collaboration with the group of Prof. J.F. Stoddart (Northwestern University, Evanston, Illinois, USA) who made the synthesis of the compounds, shows the ability of this artificial machine to operate as bistable molecular-level memory under kinetic control. The second part concerns the study of the surface properties of luminescent semiconductor nanocrystals (QDs) and in particular the effect of acid and base on the spectroscopical properties of those nanoparticles. In this section is also reported the work carried out in the laboratory of Prof H. Mattoussi (Florida State University, Tallahassee, Florida, USA), where I developed a novel method for the surface decoration of QDs with lipoic acid-based ligands involving the photoreduction of the di-thiolane moiety.

Synthetic strategies of new spin-labelled supramolecular systems

Molecular recognition and self-assembly represent fundamental issues for the construction of supramolecular systems, structures in which the components are held together through non-covalent interactions. The study of host-guest complexes and mechanical interlocked molecules, important examples in this field, is necessary in order to characterize self-assembly processes, achieve more control over the molecular organization and develop sophisticated structures by using properly designed building blocks. The introduction of paramagnetic species, or spin labelling, represents an attractive opportunity that allows their detection and characterization by the Electron Spin Resonance spectroscopy, a valuable technique that provides additional information to those obtained by traditional methods. In this Thesis, recent progresses in the design and the synthesis of new paramagnetic host-guest complexes and rotaxanes characterized by the presence of nitroxide radicals and their investigation by ESR spectroscopy are reported. In Chapter 1 a brief overview of the principal concepts of supramolecular chemistry, the spin labelling approach and the development of ESR methods applied to paramagnetic systems are described. Chapter 2 and 3 are focused on the introduction of radicals in macrocycles as Cucurbiturils and Pillar[n]arenes, due to the interesting binding properties and the potential employment in rotaxanes, in order to investigate their structures and recognition properties. Chapter 4 deals with one of the most studied mechanical interlocked molecules, the bistable [2]rotaxane reported by Stoddart and Heath based on the ciclobis (paraquat-p-phenylene) CBPQT4+, that represents a well known example of molecular switch driven by external stimuli. The spin labelling of analogous architectures allows the monitoring by ESR spectroscopy of the switch mechanism involving the ring compound by tuning the spin exchange interaction. Finally, Chapter 5 contains the experimental procedures used for the synthesis of some of the compounds described in Chapter 2-4.