Statistical mechanics for biological applications: focusing on the immune system

The emergence in the last decades of a huge amount of data in many fields of biology triggered also an increase of the interest by quantitative disciplines for life sciences. Mathematics, physics and informatics have been providing quantitative models and advanced statistical tools in order to help the understanding of many biological problems. Statistical mechanics is a field that particularly contributed to quantitative biology because of its intrinsic predisposition in dealing with systems of many strongly interacting agents, noise, information processing and statistical inference. In this Thesis a collection of works at the interphase between statistical mechanics and biology is presented. In particular they are related to biological problems that can be mainly reconducted to the biology of the immune system. Beyond the unification key given by statistical mechanics of discrete systems and quantitative modeling and analysis of the immune system, the works presented here are quite diversified. The origin of this heterogeneity resides in the intent of using and learning many different techniques during the lapse of time needed for the preparation of the work reviewed in this Thesis. In fact the work presented in Chapter 3 mainly deals with statistical mechanics, networks theory and networks numerical simulations and analysis; Chapter 4 presents a mathematical physics oriented work; Chapter 5 and 6 deal with data analysis and in particular wth clinical data and amino acid sequences data sets, requiring the use of both analytical and numerical techniques. The Thesis is conceptually organized in two main parts. The first part (Chapters 1 and 2) is dedicated to the review of known results both in statistical mechanics and biology, while in the second part (Chapters 3, 4 and 6) the original works are presented together with briefs insights into the research fields in which they can be embedded. In particular, in Chapter 1 some of the most relevant models and techniques in statistical mechanics of mean field spin systems are reviewed, starting with the Ising model and then passing to the Sherrington-Kirkpatrik model for spin glasses and to the Hopfield model for attractors neural networks. The replica method is presented together with the stochastic stability method as a mathematically rigorous alternative to replicas. Chapter 2 is dedicated to a very schematic overview of the biology of the immune system. In Chapter 3, Section 3.1 is dedicated to the presentation of a mathematical phenomenological model for the study of the idiotypic network while Section 3.2 serves as a review of the statistical mechanics based models proposed by Elena 1 2 Introduction Agliari and Adriano Barra as toy models meant to underline the possible role of complex networks within the immune system. In Chapter 4 the mathematical model of an analogue neural network on a diluted graph is studied. It is shown how the problem can be mapped in a bipartite diluted spin glass. The model is rigorously solved at the replica symmetric level with the use of the stochastic stability technique and fluctuations analysis is used to study the spin glass transition of the system. A topological analysis of the network is also performed and different topological regimes are proven to emerge though the tuning of the model parameters. In Chapter 5 a model for the analysis of clinical records of testing sets of patients is presented. The model is based on a Markov chain over the space of clinical states. The machinery is applied to data concerning the insurgence of Tuberculosis and Non-Tuberculous Infections as side effects in patients treated with Tumor Necrosis Factor inhibitors. The analysis procedure is capable of capturing clinical details of the behaviors of different drugs. Lastly, Chapter 6 is dedicated to a statistical inference analysis on deep sequencing data of an antibodies repertoire with the purpose of studying the problem of antibodies affinity maturation. A partial antibodies repertoire from a HIV-1 infected donor presenting broadly neutralizing serum is used to infer a probability distribution in the space of sequences that is compared with neutralization power measurements and with the deposited crystallographic structure of a deeply matured antibody. The work is still in progress, but preliminary results are encouraging and are presented here.