Mean field models: rigorous results on spin glasses and biological applications to cooperative systems

Abstract

This thesis is split in two main parts, having in common the study of mean field models in Statistical Physics and Complex Systems. The first part is dedicated to an application of some of the methods of this science to the analysis of cooperative phenomena in biological systems, while the second part deals with mean field spin glass models, focusing on more formal aspects. Cooperativity is a widespread phenomenon in biochemical reactions: many biological functions involve the interactions of small molecules with specific sites on larger biopolymers, and the formation of some kind on non-covalent bond between the small molecule, called ligand, and the proper binding site; the binding of a ligand to one site can influence the affinity of other sites for the same kind of ligand, and in this case the binding is said to be cooperative. Cooperativity is said to be positive if the affinity of other sites increases after a ligand has bound, viceversa it is said to be negative, functioning a device to sharpen or dampen the responsiveness of a system to the changes in a stimulus. In our approach, the cooperative system is mapped into a spin model, with couplings generated by binary strings associated to each site, similarly to the Hopfield model. The simple non-cooperative systems are mapped into models with zero interactions, while positive and negative cooperativity are described in terms of models with, respectively, ferromagnetic and antiferromagnetic interactions. Thus, many different cooperative behaviors, described by the related binding curves, can be analyzed, in an unified vision, in terms of properties of the free energy for the corresponding spin model, by properly tuning the couplings. We fitted the theoretical curves obtained in this way with some experimental data found in literature, extrapolating the values of the effective interactions between the binding sites, which can be put in direct correspondence with some of the most used coefficients that measure cooperativity. The second part focuses on mean field spin glass models. The purpose of this part of the work is to show the applications of some techniques recently introduced to give a more rigorous and firm ground to the beautiful heuristic results known in this field for many years, both to prove the validity of such powerful methods and to develop alternative mathematical tools to approach the study of complex systems. The rigorous proof that the Parisi formula for the free energy is correct, in fact, was established only some years ago, split across two works by Guerra and Talagrand, and many important rigorous results, such as the existence of the thermodynamic limit for the free energy, or the correctness of the ultrametric hypothesis for low-temperature states, are quite recent. Most of the techniques used for these recent breakthroughs are based on interpolation and coupling the given system with an auxiliary, properly chosen, one. Some of these methods can be formulated through an interesting formal analogy with a mechanical system, governed by a proper Hamilton-Jacobi equation. The analysis of this associated mechanical problem, whose potential is related to the fluctuations of the order parameter, allowed us to reconstruct, in particular, the free energy for the Sherrington-Kirkpatrick model and the p-spin glass model, up to the first step of broken replica symmetry.

Questa tesi è divisa in due parti, che hanno in comune lo studio di modelli di campo medio in meccanica statistica e sistemi complessi. La prima parte è dedicata all'applicazione di alcuni metodi di queste scienze all'analisi di fenomeni cooperativi in sistemi biologici, mentre la seconda parte riguarda modelli di campo medio per vetri di spin, ed è focalizzata su aspetti più formali. La cooperatività è un fenomeno molto diffuso nelle reazioni biochimiche e, nel nostro approccio, i sistemi cooperativi sono mappati in un modello di spin interagenti con accoppiamenti generati da strighe binarie, in maniera simile al modello di Hopfield. In questo modo viene modellizzata una possibile eterogeneità nelle interazioni, e possono essere descritti effetti cooperativi ed anticooperativi a seconda del segno dell'accoppiamento medio. Nella seconda parte viene mostrata l'applicazione di alcune tecniche sviluppate di recente nell'ambito dello studio dei modelli di vetri di spin, inscritte in un approccio che ha consentito di dare una base rigorosa ai risultati euristici noti in questo contesto da molti anni (limite termodinamico, validità della soluzione di Parisi, ecc.). Queste tecniche, basate su un'analogia formale con sistemi meccanici, consentono di ricostruire l'energia libera per il modello di Sherrington-Kirkpatrick e per il p-spin glass utilizzando delle funzioni di partizione interpolanti con un sistema ausiliario, in cui agiscono una serie di campi esterni random.