Statistical mechanics models for biological systems: cooperativity in biochemistry and affinity maturation of antibodies

Abstract

Statistical Mechanics provides useful tools and concepts to deal with collective behavior of many strongly interacting agents. Overlooking the detailed and the specific description of the interactions to focus on the very key features allows to ask different questions concerning the global systemic properties of biological systems. The information processing and statistical inference approach has became more urgent in the last decades due to the large amount of data coming from the exploit of different new experimental techniques. Concepts such as entropy, phase transition and criticality has entered the unavoidable terminology to describe the nature of biological systems at very different level of complexity: from the animal collective behaviour, the physiological apparatuses as nervous system and immune system to the biochemical processes in cells. The studies presented in this thesis are placed in this interdisciplinary border context.

The thesis is divided in three main parts.The first is devoted to the more formal aspect of statistical mechanics models of spin systems. We review briefly, in the first chapter, three milestone models of spin systems: the Curie-Weiss, the Sherrington-Kirkpatrick and the Hopfield model. These models constitute the paradigmatic examples of mean-field Statistical Mechanics and will constitute the ground for the studies in biochemical kinetics and immunology presented in the following parts. In the second chapter we report a detailed study of a generalization of the Hopfield model with diluted and correlated patterns. We investigate the topology of the emergent interactions network. We find an exact expression of the coupling distribution that allows to distinguish different regimes varying the dilution parameter. Moreover we study the thermodynamic properties of the model, obtaining explicitly the replica symmetric free-energy coupled with its self-consistence equations. Considering the small overlap expansion of these self consistencies equations we get the critical surface dividing the ergodic phase to the spin-glass one. The second part of the thesis focus on the investigation of the cooperative behavior in biochemical kinetics through mean field statistical mechanics. Cooperativity is one of the most important properties of molecular interactions in biological systems as it is often invoked to account for collective features in binding phenomena. It constitutes a fundamental tool that nature developed to modulate the chemical response of biological systems to varying stimuli. Statistical mechanics offers a valuable approach as, from its first principles, it aims to figure out collective phenomena, allowing a unified and broader theory for complex chemical kinetics. In this way different cooperative behaviors, described by the related binding curves, can be analysed in an unified framework. We compare the theoretical curves predicted by the model with experimental data found in literature, finding an overall good agreement and extrapolating the values of the effective interactions between the binding sites, which can be put in direct correspondence with the standard coefficient that measure cooperativity (Hill number). Moreover, an extension of the model allows to take into account heterogeneity that can affect both the couplings between the multiple active sites (allosteric regulation) and the chemical potentials in the binding of the ligands.

The last part is dedicated to a statistical inference analysis on deep sequencing data of an antibodies repertoire with the purpose of studying the process 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 . The idea is to use the model to study the structure of the affinity with an antigen as a function of the antibody sequence. We test this strategy using neutralization power measurements and the deposited crystallographic structure of a deeply matured antibody. The work is still in progress, but preliminary results are encouraging and are presented here.