Boolean logical frameworks are ubiquitous in Computer Science. For computers, information is binary; the circuits (microprocessors, digital signal processors known as DSP, ...) that process it are entirely based on boolean logic; the algorithms and programs that transmit it, emph{encrypt} it, correct it, are based on binary logic (in fact, often on extensions of binary finite fields). Many large-scale combinatorial problems are attacked by algorithms and software directed at solving constraint satisfaction problems, which are none other than basic extensions of the boolean paradigm. Most of the problems arising from boolean frameworks are computationally highly intensive. The implication is the following: When faced with a given (large) problem instance, a great many algorithmic alternatives are possible, but it is hard to know in advance which one should best be used. What is badly needed are methodologies dedicated at predicting with sufficient likelihood which strategy is most likely to succeed in each specific context. In an ideal world, one would then like to have a way of matching accessible structural indicators of boolean problems with the probabilistic behaviour of the complexity of solution strategies. This is precisely where our proposal fits. Our global objective is to quantify the expressive power of major boolean frameworks. By this we mean developing models with which to predict what should hold or not hold in a high number of cases; in other words we want to elucidate statistical properties of boolean structures. This approach departs significantly from the usual worst-case analyses. The eventual goal is to provide powerful guidance in the choice of the best representation or algorithm adapted to a given context, and, in a number of cases, develop new algorithmic strategies based on sound analytic results. In summary, we propose to develop a coherent and general-purpose mathematical toolbox with which to measure and compare quantitative (probabilistic) properties of boolean frameworks regarding computational aspects. The coherence of our proposal stems both from the fact that boolean functions form the common basis of all our investigations, and from the methods that we propose to use and develop, based on analytic combinatorics and probability theory. So far, the scientific communities engaged in research regarding boolean functions and random discrete structures have had very few interactions, and the techniques that we shall develop have been very rarely applied to boolean objects. To summarise, we believe that the novelty of our project is twofold: the introduction of new methods in the analysis of boolean frameworks; the extension of analytic combinatorics and probabilistic methods to a new kind of discrete structures.

1-Scientific background and objectives : The Human Genome Project revealed more than 500 proteases. The substrate specificity, the trigger events and sites of activation are still unknown for most of them. We know however that proteolysis is involved in many physiological situations like inflammation, coagulation, fibrinolysis or tissue turnover. Pathological proteolysis has been observed in many diseases like cystic fibrosis, emphysema, rheumatoid arthritis, bacterial, viral and parasitic infections, tumour and metastasis spreading or pancreatitis. In the intracellular compartment apoptosis also involves specific proteolysis. At present there is no non-invasive method to monitor proteolysis in deep-seated tissues suitable for humans or small animals. This project consequently aims at designing an imaging method for specific proteolysis activity based on Magnetic Resonance Imaging (MRI). Such a method would be a breakthrough for diagnosis and follow-up of treatments in many diseases. Moreover, this new tool would help understanding the activation and persistence of any given protease in correlation with many physiological events, development for instance, a much needed knowledge in fundamental protease research. 2-Description of the project, methodology : The project aims to develop a new method for monitoring the proteolysis in vivo using MRI technique enhanced by Dynamic Nuclear Polarization (DNP, Overhauser effect). The choice of the MRI technique relies on its capacities for exploring deep tissues, for its good resolution, and for its 3D accuracy. Enhancement due to the DNP will provide us with higher sensitivity and generate higher contrast between the tissues under proteolysis and the unaffected tissues. With DNP, calculations predict a 100-fold increase for the signal and preliminary in vitro experiment with our setup yielded a 54-fold increase of the signal. The project will span on four steps. First, peptides carrying nitroxides will be designed and prepared in such way that, after proteolysis, the released nitroxides will exhibit Electronic Paramagnetic Resonance (EPR) signature altered from the one of nitroxides attached to peptides. EPR studies will be performed with model nitroxides to determine the importance of EPR signature difference between free nitroxides and attached nitroxides in media mimicking biological environment. Second, the best candidates will be used to set the imaging system (DNP-MRI) in the way to obtain the highest sensitivity and to observe the highest contrasts on model systems mimicking biological conditions. Third, when the best conditions will be achieved, 3D imaging of the protease activity will be performed on in vitro and ex vivo models, and fourth, in vivo on small animals. 3-Expected results : The fields array of proteases involvement is very broad and so are the potential applications. The imaging technique (DNP-MRI) developed in this project will first be applied in three main fields: i) diagnosis, e.g., proteases/inhibitors balance for cystic fibrosis, or rheumatoid arthritis, or early detection of proteases in tumour surrounding, ii) monitoring the efficiency of anti-protease drugs, iii) fundamental research on proteolysis in normal and pathological situations. Consequently, the DNP-MRI technique has the potential to become a unique tool for studying, in a non-invasive manner, the pivotal and ill-known role of the proteolysis in physiological events or diseases whatever the organs or the body parts where it occurs.

In the last 20 years, the study of Reactive Oxygen and Nitrogen Species (RONS) has appeared as an increasing field of research. This interest is related to their roles as critical mediators in various physiological processes (second messengers and regulators of signal transduction) and pathologies (cancers, neurodegenerative diseases, ischemia / reperfusion damages, diabetes, atherosclerosis). Among these species, the oxygen-centered free radicals are of particular biomedical importance. An accurate technique for the detection of these species is critical for a better understanding of the physiological and pathophysiological processes in which they are implicated. Many methods have been developed to assay O2'- in biochemical, cellular, and in vivo systems such as: cytochrome c reduction, chemiluminescence, fluorescent dyes, hydroethidine oxidation, and nitroblue tetrazolium reduction. Unfortunately, using these techniques it is difficult to discriminate between biological oxidants and O2'-. Beside to above-mentioned techniques, the technique of Electron Paramagnetic Resonance (EPR) spectroscopy allows the detection of free radicals and an extensive characterization of their generation, kinetics, and reactions. Thus, EPR spectroscopy is one of the most powerful techniques for the study of free radicals. However, combinations of rapid decays and low steady-state concentration prevent the direct EPR detection of most biological relevant radicals (e.g., O2'-, NO', HO') under physiological conditions. To circumvent these difficulties, the implication of free radicals can be inferred using the spin trapping technique in which short-lived free radicals specifically react with a nitrone spin trap to form a persistent radical adduct that is conveniently detected by EPR spectroscopy. In 1999, Swartz et al. evaluated DEPMPO, a new spin trap, as spin trapping agent in biological systems and the authors concluded that DEPMPO is a potentially good candidate for trapping radical in biological systems and represents an improvement over DMPO. And more recently, Swartz et al. evaluated the effects of DMPO, CMPO, EMPO, BMPO, and DEPMPO on CHO cells and the stability of the radical adducts in the presence of cells. As a conclusion, the authors indicated that with appropriate controls and selection of spin traps, the spin trapping of reactive free radicals in biological systems is likely to be effective using the newly available spin traps. Since this work, few experiments have been performed successfully in biological systems. The longest half-lives for radical adducts in vivo have been estimated to be around 1 or 2 min with DEPMPO. For the study of free radicals in biological systems, the technique is still limited by the following main drawbacks: - the short lifetime of the superoxide spin adducts, - the readily reduction of nitroxide spin adducts to EPR silent compounds - the low rate constants observed for the trapping of superoxide radical ' the very low steady-state concentration of superoxide radical - the low concentration of the spin trap on the spot of the radical event As a consequence, application of spin trapping technique to in vivo systems is still challenging. There is still a growing need in tools that can provide information on free radical processes occurring in biological systems, and this is illustrated by the incorporation of organic chemists in leading groups involved in free radical biology and in grant proposals focused on this topic. We propose to study the spin trapping of reactive oxygen species in biological systems at the level of theoretical calculations, in buffer solution, in subcellular fractions, in cells and in small animals. These combined studies should allow us to gain insight into the way nitrone spin traps distribute in cells, in the intrinsic mechanism of degradation of the superoxide adduct, on the metabolism of the spin adducts, on the trapping rate of superoxide radical and on the valuable strategies useful to protect the spin adducts in biological systems. We hope to get the following major results: - to develop new spin traps with improved superoxide trapping rate, - to develop a theoretical model to correlate the hyperfine coupling constants to the structure and reactivity of the spin adducts, - to highlight the metabolic pathways of spin adducts, - to enhance the lifetime of the paramagnetic adducts in biological systems, - to determine the important molecular features governing the spin trapping efficiency, - to prepare efficient spin traps to investigate free radical processes in functional biological systems. A the end of the program, we expect that we will be able to offer new improved spin traps for biological systems to the scientific community. The tools and methods that will be developed during the project will be useful also in other fields such as: imaging techniques (PEDRI, OMRI, DNP), therapeutic studies, and immuno-spin trapping technique.