Rigorous results in Spin Glasses and Monomer-Dimer systems

In this work I reported recent results in the field of Statistical Mechanics of Equilibrium, and in particular in Spin Glass models and Monomer Dimer models . We start giving the mathematical background and the general formalism for Spin (Disordered) Models with some of their applications to physical and mathematical problems. Next we move on general aspects of the theory of spin glasses, in particular to the Sherrington-Kirkpatrick model which is of fundamental interest for the work. In Chapter 3, we introduce the Multi-species Sherrington-Kirkpatrick model (MSK), we prove the existence of the thermodynamical limit and the Guerra's Bound for the quenched pressure together with a detailed analysis of the annealed and the replica symmetric regime. The result is a multidimensional generalization of the Parisi's theory. Finally we brie y illustrate the strategy of the Panchenko's proof of the lower bound. In Chapter 4 we discuss the Aizenmann-Contucci and the Ghirlanda-Guerra identities for a wide class of Spin Glass models. As an example of application, we discuss the role of these identities in the proof of the lower bound. In Chapter 5 we introduce the basic mathematical formalism of Monomer Dimer models. We introduce a Gaussian representation of the partition function that will be fundamental in the rest of the work. In Chapter 6, we introduce an interacting Monomer-Dimer model. Its exact solution is derived and a detailed study of its analytical properties and related physical quantities is performed. In Chapter 7, we introduce a quenched randomness in the Monomer Dimer model and show that, under suitable conditions the pressure is a self averaging quantity. The main result is that, if we consider randomness only in the monomer activity, the model is exactly solvable.

New perspectives in statistical mechanics and high-dimensional inference

The main purpose of this thesis is to go beyond two usual assumptions that accompany theoretical analysis in spin-glasses and inference: the i.i.d. (independently and identically distributed) hypothesis on the noise elements and the finite rank regime. The first one appears since the early birth of spin-glasses. The second one instead concerns the inference viewpoint. Disordered systems and Bayesian inference have a well-established relation, evidenced by their continuous cross-fertilization. The thesis makes use of techniques coming both from the rigorous mathematical machinery of spin-glasses, such as the interpolation scheme, and from Statistical Physics, such as the replica method. The first chapter contains an introduction to the Sherrington-Kirkpatrick and spiked Wigner models. The first is a mean field spin-glass where the couplings are i.i.d. Gaussian random variables. The second instead amounts to establish the information theoretical limits in the reconstruction of a fixed low rank matrix, the “spike”, blurred by additive Gaussian noise. In chapters 2 and 3 the i.i.d. hypothesis on the noise is broken by assuming a noise with inhomogeneous variance profile. In spin-glasses this leads to multi-species models. The inferential counterpart is called spatial coupling. All the previous models are usually studied in the Bayes-optimal setting, where everything is known about the generating process of the data. In chapter 4 instead we study the spiked Wigner model where the prior on the signal to reconstruct is ignored. In chapter 5 we analyze the statistical limits of a spiked Wigner model where the noise is no longer Gaussian, but drawn from a random matrix ensemble, which makes its elements dependent. The thesis ends with chapter 6, where the challenging problem of high-rank probabilistic matrix factorization is tackled. Here we introduce a new procedure called "decimation" and we show that it is theoretically to perform matrix factorization through it.

Predictability in Social Science, The statistical mechanics approach.

The subject of this work concerns the study of the immigration phenomenon, with emphasis on the aspects related to the integration of an immigrant population in a hosting one. Aim of this work is to show the forecasting ability of a recent finding where the behavior of integration quantifiers was analyzed and investigated with a mathematical model of statistical physics origins (a generalization of the monomer dimer model). After providing a detailed literature review of the model, we show that not only such a model is able to identify the social mechanism that drives a particular integration process, but it also provides correct forecast. The research reported here proves that the proposed model of integration and its forecast framework are simple and effective tools to reduce uncertainties about how integration phenomena emerge and how they are likely to develop in response to increased migration levels in the future.

Geometric and Combinatorial Aspects of NonEquilibrium Statistical Mechanics

Non-Equilibrium Statistical Mechanics is a broad subject. Grossly speaking, it deals with systems which have not yet relaxed to an equilibrium state, or else with systems which are in a steady non-equilibrium state, or with more general situations. They are characterized by external forcing and internal fluxes, resulting in a net production of entropy which quantifies dissipation and the extent by which, by the Second Law of Thermodynamics, time-reversal invariance is broken. In this thesis we discuss some of the mathematical structures involved with generic discrete-state-space non-equilibrium systems, that we depict with networks in all analogous to electrical networks. We define suitable observables and derive their linear regime relationships, we discuss a duality between external and internal observables that reverses the role of the system and of the environment, we show that network observables serve as constraints for a derivation of the minimum entropy production principle. We dwell on deep combinatorial aspects regarding linear response determinants, which are related to spanning tree polynomials in graph theory, and we give a geometrical interpretation of observables in terms of Wilson loops of a connection and gauge degrees of freedom. We specialize the formalism to continuous-time Markov chains, we give a physical interpretation for observables in terms of locally detailed balanced rates, we prove many variants of the fluctuation theorem, and show that a well-known expression for the entropy production due to Schnakenberg descends from considerations of gauge invariance, where the gauge symmetry is related to the freedom in the choice of a prior probability distribution. As an additional topic of geometrical flavor related to continuous-time Markov chains, we discuss the Fisher-Rao geometry of nonequilibrium decay modes, showing that the Fisher matrix contains information about many aspects of non-equilibrium behavior, including non-equilibrium phase transitions and superposition of modes. We establish a sort of statistical equivalence principle and discuss the behavior of the Fisher matrix under time-reversal. To conclude, we propose that geometry and combinatorics might greatly increase our understanding of nonequilibrium phenomena.

Statistical mechanics formalism and methods for the analysis of real networks

This thesis provides a thoroughly theoretical background in network theory and shows novel applications to real problems and data. In the first chapter a general introduction to network ensembles is given, and the relations with “standard” equilibrium statistical mechanics are described. Moreover, an entropy measure is considered to analyze statistical properties of the integrated PPI-signalling-mRNA expression networks in different cases. In the second chapter multilayer networks are introduced to evaluate and quantify the correlations between real interdependent networks. Multiplex networks describing citation-collaboration interactions and patterns in colorectal cancer are presented. The last chapter is completely dedicated to control theory and its relation with network theory. We characterise how the structural controllability of a network is affected by the fraction of low in-degree and low out-degree nodes. Finally, we present a novel approach to the controllability of multiplex networks

New approaches to open problems in gene expression microarray data

In the past decade, the advent of efficient genome sequencing tools and high-throughput experimental biotechnology has lead to enormous progress in the life science. Among the most important innovations is the microarray tecnology. It allows to quantify the expression for thousands of genes simultaneously by measurin the hybridization from a tissue of interest to probes on a small glass or plastic slide. The characteristics of these data include a fair amount of random noise, a predictor dimension in the thousand, and a sample noise in the dozens. One of the most exciting areas to which microarray technology has been applied is the challenge of deciphering complex disease such as cancer. In these studies, samples are taken from two or more groups of individuals with heterogeneous phenotypes, pathologies, or clinical outcomes. these samples are hybridized to microarrays in an effort to find a small number of genes which are strongly correlated with the group of individuals. Eventhough today methods to analyse the data are welle developed and close to reach a standard organization (through the effort of preposed International project like Microarray Gene Expression Data -MGED- Society [1]) it is not unfrequant to stumble in a clinician's question that do not have a compelling statistical method that could permit to answer it.The contribution of this dissertation in deciphering disease regards the development of new approaches aiming at handle open problems posed by clinicians in handle specific experimental designs. In Chapter 1 starting from a biological necessary introduction, we revise the microarray tecnologies and all the important steps that involve an experiment from the production of the array, to the quality controls ending with preprocessing steps that will be used into the data analysis in the rest of the dissertation. While in Chapter 2 a critical review of standard analysis methods are provided stressing most of problems that In Chapter 3 is introduced a method to adress the issue of unbalanced design of miacroarray experiments. In microarray experiments, experimental design is a crucial starting-point for obtaining reasonable results. In a two-class problem, an equal or similar number of samples it should be collected between the two classes. However in some cases, e.g. rare pathologies, the approach to be taken is less evident. We propose to address this issue by applying a modified version of SAM [2]. MultiSAM consists in a reiterated application of a SAM analysis, comparing the less populated class (LPC) with 1,000 random samplings of the same size from the more populated class (MPC) A list of the differentially expressed genes is generated for each SAM application. After 1,000 reiterations, each single probe given a "score" ranging from 0 to 1,000 based on its recurrence in the 1,000 lists as differentially expressed. The performance of MultiSAM was compared to the performance of SAM and LIMMA [3] over two simulated data sets via beta and exponential distribution. The results of all three algorithms over low- noise data sets seems acceptable However, on a real unbalanced two-channel data set reagardin Chronic Lymphocitic Leukemia, LIMMA finds no significant probe, SAM finds 23 significantly changed probes but cannot separate the two classes, while MultiSAM finds 122 probes with score >300 and separates the data into two clusters by hierarchical clustering. We also report extra-assay validation in terms of differentially expressed genes Although standard algorithms perform well over low-noise simulated data sets, multi-SAM seems to be the only one able to reveal subtle differences in gene expression profiles on real unbalanced data. In Chapter 4 a method to adress similarities evaluation in a three-class prblem by means of Relevance Vector Machine [4] is described. In fact, looking at microarray data in a prognostic and diagnostic clinical framework, not only differences could have a crucial role. In some cases similarities can give useful and, sometimes even more, important information. The goal, given three classes, could be to establish, with a certain level of confidence, if the third one is similar to the first or the second one. In this work we show that Relevance Vector Machine (RVM) [2] could be a possible solutions to the limitation of standard supervised classification. In fact, RVM offers many advantages compared, for example, with his well-known precursor (Support Vector Machine - SVM [3]). Among these advantages, the estimate of posterior probability of class membership represents a key feature to address the similarity issue. This is a highly important, but often overlooked, option of any practical pattern recognition system. We focused on Tumor-Grade-three-class problem, so we have 67 samples of grade I (G1), 54 samples of grade 3 (G3) and 100 samples of grade 2 (G2). The goal is to find a model able to separate G1 from G3, then evaluate the third class G2 as test-set to obtain the probability for samples of G2 to be member of class G1 or class G3. The analysis showed that breast cancer samples of grade II have a molecular profile more similar to breast cancer samples of grade I. Looking at the literature this result have been guessed, but no measure of significance was gived before.

Mass spectrometry-based protein profiling strategies for biomarker discovery in liver and inflammatory bowel diseases

The study of protein expression profiles for biomarker discovery in serum and in mammalian cell populations needs the continuous improvement and combination of proteins/peptides separation techniques, mass spectrometry, statistical and bioinformatic approaches. In this thesis work two different mass spectrometry-based protein profiling strategies have been developed and applied to liver and inflammatory bowel diseases (IBDs) for the discovery of new biomarkers. The first of them, based on bulk solid-phase extraction combined with matrix-assisted laser desorption/ionization - Time of Flight mass spectrometry (MALDI-TOF MS) and chemometric analysis of serum samples, was applied to the study of serum protein expression profiles both in IBDs (Crohn’s disease and ulcerative colitis) and in liver diseases (cirrhosis, hepatocellular carcinoma, viral hepatitis). The approach allowed the enrichment of serum proteins/peptides due to the high interaction surface between analytes and solid phase and the high recovery due to the elution step performed directly on the MALDI-target plate. Furthermore the use of chemometric algorithm for the selection of the variables with higher discriminant power permitted to evaluate patterns of 20-30 proteins involved in the differentiation and classification of serum samples from healthy donors and diseased patients. These proteins profiles permit to discriminate among the pathologies with an optimum classification and prediction abilities. In particular in the study of inflammatory bowel diseases, after the analysis using C18 of 129 serum samples from healthy donors and Crohn’s disease, ulcerative colitis and inflammatory controls patients, a 90.7% of classification ability and a 72.9% prediction ability were obtained. In the study of liver diseases (hepatocellular carcinoma, viral hepatitis and cirrhosis) a 80.6% of prediction ability was achieved using IDA-Cu(II) as extraction procedure. The identification of the selected proteins by MALDITOF/ TOF MS analysis or by their selective enrichment followed by enzymatic digestion and MS/MS analysis may give useful information in order to identify new biomarkers involved in the diseases. The second mass spectrometry-based protein profiling strategy developed was based on a label-free liquid chromatography electrospray ionization quadrupole - time of flight differential analysis approach (LC ESI-QTOF MS), combined with targeted MS/MS analysis of only identified differences. The strategy was used for biomarker discovery in IBDs, and in particular of Crohn’s disease. The enriched serum peptidome and the subcellular fractions of intestinal epithelial cells (IECs) from healthy donors and Crohn’s disease patients were analysed. The combining of the low molecular weight serum proteins enrichment step and the LCMS approach allowed to evaluate a pattern of peptides derived from specific exoprotease activity in the coagulation and complement activation pathways. Among these peptides, particularly interesting was the discovery of clusters of peptides from fibrinopeptide A, Apolipoprotein E and A4, and complement C3 and C4. Further studies need to be performed to evaluate the specificity of these clusters and validate the results, in order to develop a rapid serum diagnostic test. The analysis by label-free LC ESI-QTOF MS differential analysis of the subcellular fractions of IECs from Crohn’s disease patients and healthy donors permitted to find many proteins that could be involved in the inflammation process. Among them heat shock protein 70, tryptase alpha-1 precursor and proteins whose upregulation can be explained by the increased activity of IECs in Crohn’s disease were identified. Follow-up studies for the validation of the results and the in-depth investigation of the inflammation pathways involved in the disease will be performed. Both the developed mass spectrometry-based protein profiling strategies have been proved to be useful tools for the discovery of disease biomarkers that need to be validated in further studies.

The uncertainty in standardised sound power measurements: complying with ISO 17025

In the context of “testing laboratory” one of the most important aspect to deal with is the measurement result. Whenever decisions are based on measurement results, it is important to have some indication of the quality of the results. In every area concerning with noise measurement many standards are available but without an expression of uncertainty, it is impossible to judge whether two results are in compliance or not. ISO/IEC 17025 is an international standard related with the competence of calibration and testing laboratories. It contains the requirements that testing and calibration laboratories have to meet if they wish to demonstrate that they operate to a quality system, are technically competent and are able to generate technically valid results. ISO/IEC 17025 deals specifically with the requirements for the competence of laboratories performing testing and calibration and for the reporting of the results, which may or may not contain opinions and interpretations of the results. The standard requires appropriate methods of analysis to be used for estimating uncertainty of measurement. In this point of view, for a testing laboratory performing sound power measurement according to specific ISO standards and European Directives, the measurement of uncertainties is the most important factor to deal with. Sound power level measurement, according to ISO 3744:1994 , performed with a limited number of microphones distributed over a surface enveloping a source is affected by a certain systematic error and a related standard deviation. Making a comparison of measurement carried out with different microphone arrays is difficult because results are affected by systematic errors and standard deviation that are peculiarities of the number of microphones disposed on the surface, their spatial position and the complexity of the sound field. A statistical approach could give an overview of the difference between sound power level evaluated with different microphone arrays and an evaluation of errors that afflict this kind of measurement. Despite the classical approach that tend to follow the ISO GUM this thesis present a different point of view of the problem related to the comparison of result obtained from different microphone arrays.

Impiego della spettrometria di massa nell'analisi di fumonisine in alimenti e matrici di origine animale

Mycotoxins are heterogeneous chemical compounds characterized by a low molecular weight and synthesized by the secondary metabolism of different molds. Fumonisins are water-soluble mycotoxins produced by Fusarium species spoiling corn and derived produc ts. These mycotoxins can be a health hazard when consuming contaminated cereals, but they can reach humans also indirectly through the consumption of food products derived from animals fed with contaminated feed. Fumonisins have been associated with several animal and human diseases: they are suspected risk factors for esophageal and liver cancers, neural tube defects and cardiovascular problems. Improved methods are needed to accurately assess fumonisins concentrations in food of vegetable and animal origin, in order to prevent acute and chronic human exposure. The aim of the present work was to evaluate the versatility and the performances of mass spectrometry, coupled with liquid chromatography, in fumonisins analysis from foods and matrices of animal origin. Different methods for the identification and quantification of fumonisins and related products have been developed and validated to determine fumonisin B1 in milk, fumonisin B1, fumonisin B2 and their complete hydrolyzed products (HFB1 and HFB2) in pig liver and fumonisins B1 and B2 in complete and complementary dry dog food. The experimental procedures have been carefully studied, considering matrices features, number and type of molecules to detect. Therefore, several extraction, clean up and separation techniques were tested in order to obtain the better conditions of sample processing. The fit for purpose sample preparation, matched with high mass spectrometry sensibility and specificity, have allowed to achieve good results in any tested animal matrices. Hence, the developed methods were validated and have shown a high accuracy, sensibility and precision, fulfilling performance requirements of Decision 2002/657/EC and of European Project Standard, Measuring and Testing (SMT). In any developed method, the analytes were identified and quantified even at very low concentrations : the limits of quantification resulted lower than other similar works, performed with different detectors. These methods were applied to some commercial samples and to some samples collected for research projects in the Department of Veterinary Public Health and Animal Pathology (DVPHAP) of University of Bologna. Although the disclosed data must be considered completely preliminary and without statistical significance, they emphasize the presence of mycotoxins in animal products. The outcomes obtained from the processed samples (bovine milk, pig liver and dry dog food) suggest the efficacy of these methods also on other food matrices, confirming the versatility and the performances of mass spectrometry, coupled with liquid chromatography, in fumonisins analysis. Moreover the results underline the need to set up a large scale monitoring in order to evaluate the presence of fumonisins in food of animal origin for human consumption.

Ricostruzione della colonna anteriore dopo resezione tumorale: Utilizzo di gabbia in titanio/carbonio o innesto osseo allogenico armato in titanio?

Il rachide è stato suddiviso in tre colonne da Denis: anteriore e centrale comprendono la metà anteriore del corpo vertebrale, la metà posteriore e l’inizio dei peduncoli, mentre la colonna posteriore comprende l’arco e i peduncoli stessi. In caso di resezione o lesione della colonna anteriore e media è indicata la ricostruzione. Diverse tecniche e materiali possono essere usati per ricostruire il corpo vertebrale. Innesti vascolarizzati, autograft, allograft sono stati usati, così come impianti sintetici di titanio o materiale plastico come il PEEK (Poly etere etere ketone). Tutti questi materiali hanno vantaggi e svantaggi in termini di proprietà intrinseche, resistenza meccanica, modulo di elasticità, possibilità di trasmissione malattie, capacità di fondersi con l’osso ospite o meno. Le soluzioni più usate sono le cage in titanio o carbonio, il PMMA ( Poli methil metacrilato), gli innesti ossei massivi. Si è effettuato uno studio di coorte retrospettivo paragonando due gruppi di pazienti oncologici spinali trattati da due chirurghi esperti in un centro di riferimento, con vertebrectomia e ricostruzione della colonna anteriore: un gruppo con cage in carbonio o titanio, l’altro gruppo con allograft massivo armato di innesto autoplastico o mesh in titanio. Si sono confrontati i risultati in termini di cifosi segmenterai evolutiva, fusione ossea e qualità di vita del paziente. Il gruppo delle cage in carbonio / titanio ha avuto risultati leggermente migliori dal punto di vista biomeccanico ma non statisticamente significativo, mentre dal punto di vista della qualità di vita i risultati sono stati migliori nel gruppo allograft. Non ci sono stati fallimenti meccanici della colonna anteriore in entrambi i gruppi, con un Fu tra 12 e 60 mesi. Si sono paragonati anche i costi delle due tecniche. In conclusione l’allogar è una tecnica sicura ed efficace, con proprietà meccaniche solide, soprattutto se armato con autograft o mesi in titanio.